Springer Proceedings in Mathematics & Statistics
Alexey N. Karapetyants
Igor V. Pavlov
Albert N. Shiryaev Editors
Operator
Theory and
Harmonic
Analysis
OTHA 2020, Part II –
Probability-Analytical Models, Methods
and Applications
Springer Proceedings in Mathematics &
Statistics
Volume 358
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Alexey N. Karapetyants • Igor V. Pavlov •
Albert N. Shiryaev
Editors
Operator Theory and
Harmonic Analysis
OTHA 2020, Part II – Probability-Analytical
Models, Methods and Applications
Editors
Alexey N. Karapetyants
Institute of Mathematics, Mechanic
and Computer Sciences and Regional
Mathematical Center
Southern Federal University
Rostov-on-Don, Russia
Igor V. Pavlov
Department of Higher Mathematics
Don State Technical University
Rostov-on-Don, Russia
Albert N. Shiryaev
Steklov Mathematical Institute
Russian Academy of Sciences
Moscow, Russia
ISSN 2194-1009
ISSN 2194-1017 (electronic)
Springer Proceedings in Mathematics & Statistics
ISBN 978-3-030-76828-7
ISBN 978-3-030-76829-4 (eBook)
https://doi.org/10.1007/978-3-030-76829-4
Mathematics Subject Classification: 26-02, 30-02, 32-02, 35-02, 42-02, 45-02, 46-02, 47-02, 60-02,
92-02
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Preface
This is the second volume of the two-volume series entitled
Operator Theory and Harmonic Analysis.
Vol. 1:
New General Trends and Advances of the Theory
and
Vol. 2:
Probability-Analytical Models, Methods, and Applications.
Volume 1 is devoted to harmonic analysis and its applications in general, while
Volume 2 is focused on probabilistic and mathematical (statistical) methods in
applied sciences, but still in the context of general harmonic analysis and its
numerous applications.
The volumes’ readership is the pool of researchers interested in various aspects
of harmonic analysis and operator theory: real and complex variable methods,
applications to PDE’s, mathematical modeling based on applied harmonic analysis
and probability-analytical methods, and exploration of new themes and trends.
The contributions to both volumes are based on the matter supposed to be
presented at the annual International Scientific Conference on Modern Methods
and Problems of Operator Theory and Harmonic Analysis and Their Applications
(OTHA-2020, http://otha.sfedu.ru/), canceled due to Covid19 restrictions.
The Editors are very grateful to all the authors for their valuable contributions and
for a strong willingness to support mathematical activities and communications in
the hope of the soonest resumption of regular conferences and safe mutual visits.
The Editors express an immense sorrow on the occasion of the recent loss of
remarkable scientists and brilliant persons, Hrachik Hayrapetyan (Armenia), who
is one of the authors of the first volume, and Vladimir Pilidi (Russia), who was an
active member of Program Committees of OTHA conferences, and Vladimir Nogin
(Russia), who was colleague and teacher of quite a few participants of OTHA.
v
vi
Preface
The first volume contains words in memoriam of our dear friends Hrachik
Hayrapetyan, Vladimir Pilidi, and Vladimir Nogin.
Rostov-on-Don, Russia
Moscow, Russia
A. Karapetyants
I. Pavlov
A. Shiryaev
Contents
A Conditional Functional Limit Theorem for a Decomposable
Branching Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
V. I. Afanasyev
A Probabilistic Interpretation of Conservation and Balance Laws. . . . . . . . .
Ya. I. Belopolskaya
1
19
Hierarchical Schrödinger Type Operators: The Case of Locally
Bounded Potentials.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Alexander Bendikov, Alexander Grigor’yan, and Stanislav Molchanov
43
A Model for the Outbreak of COVID-19: Vaccine Effectiveness
in a Case Study of Italy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Vasiliki Bitsouni, Nikolaos Gialelis and Ioannis G. Stratis
91
Rate of Convergence to the Poisson Law of the Numbers of
Cycles in the Generalized Random Graphs . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 109
Sergey G. Bobkov, Maria A. Danshina, and Vladimir V. Ulyanov
Combinatorial Identities with Binomial Coefficients . . . .. . . . . . . . . . . . . . . . . . . . 135
Ya. M. Erusalimsky
Random Tempered Distributions on Locally Compact Separable
Abelian Groups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 147
Manuel L. Esquível and Nadezhda P. Krasii
On Solutions of Stochastic Equations with Current and Osmotic
Velocities . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 167
Yuri E. Gliklikh
Stochastic Methods in Investigation of Modern Networks . . . . . . . . . . . . . . . . . . 185
Vladimir A. Gorlov and Alla V. Makarova
vii
viii
Contents
Double-Barrier Option Pricing under the Hyper-Exponential
Jump Diffusion Model .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 197
S. M. Grudsky and O. A. Mendez-Lara
Single Jump Filtrations: Preservation of the Local Martingale
Property with Respect to the Filtration Generated by the Local
Martingale . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 219
Alexander A. Gushchin and Assylliya K. Zhunussova
Local Time and Local Reflection of the Wiener Process .. . . . . . . . . . . . . . . . . . . . 233
I. A. Ibragimov, N. V. Smorodina, and M. M. Faddeev
Random Harmonic Processes with New Properties . . . . . .. . . . . . . . . . . . . . . . . . . . 243
Elena Karachanskaya
On Solvability of One Nonlinear Integral Equation Arising in
Modelling of Geographical Spread of Epidemics . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 253
A. Kh. Khachatryan
A Simple Wiener-Hopf Factorization Approach for Pricing
Double Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 273
Oleg Kudryavtsev
New Procedure for Applying the Cramér–von Mises Test for
Parametric Families of Distributions . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 293
Gennady Martynov
CVaR Hedging in Defaultable Jump-Diffusion Markets . . . . . . . . . . . . . . . . . . . . 309
Alexander Melnikov and Hongxi Wan
Out-of-Sample Utility Bounds for Empirically Optimal Portfolios
in a Single-Period Investment Problem .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 335
Dmitry B. Rokhlin
A Guaranteed Deterministic Approach to Superhedging:
Optimal Mixed Strategies of the Market and Their Supports . . . . . . . . . . . . . . 355
Sergey N. Smirnov
Some Properties of Regularly Varying Functions and Series in
the Orthant .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 373
A. L. Yakymiv
Influence of the Configuration of Particle Generation Sources on
the Behavior of Branching Walks: A Case Study . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 387
E. B. Yarovaya
Lanchester Model with the Random Coefficients . . . . . . . .. . . . . . . . . . . . . . . . . . . . 407
V. G. Zadorozhniy
A Conditional Functional Limit Theorem
for a Decomposable Branching Process
V. I. Afanasyev
Abstract A decomposable Galton–Watson branching process with two particle
types is studied. It is assumed that a particle of the first type produces equal numbers
of particles of the first and second types, while a particle of the second type produces
only particles of their own type. Under the condition that the total number of
particles of the second type is greater than N → ∞, a functional limit theorem
for the process describing the number of particles of the second type in different
generations is proved.
Keywords Decomposable Galton–Watson branching process · Feller diffusion ·
Local time of a Brownian excursion · Functional limit theorems
1 Introduction and Statement of Main Result
Consider a decomposable Galton–Watson branching process with two particle
types. It is assumed that a particle of the first type produces equal numbers of
particles of the first and second types, while a particle of the second type produces
only particles of their own type.
Let ϕ (s) and ψ (s), s ≥ 0, be the generating functions of nonnegative integervalued random variables ξ and η, respectively. Suppose that the maximal span of
the distribution of the random variable ξ is equal to 1. Suppose also that
Eξ = 1, Varξ : =σ12 ∈ (0, ∞) ; Eη = 1, Varη : =σ22 ∈ (0, ∞)
(1)
(note that, along with the symbols σ12 and σ22 , we use the notations b1 = σ12 /2 and
b2 = σ22 /2). Let us introduce the generating functions for the descendants of one
V. I. Afanasyev ()
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
e-mail: viafan@mi-ras.ru
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_1
1
2
V. I. Afanasyev
particle of the first and second types, respectively: for s1 , s2 ≥ 0,
f1 (s1 , s2 ) = ϕ (s1 s2 ) , f2 (s1 , s2 ) = ψ (s2 ) .
Let ξn and ηn denote the numbers of particles of the first and second types,
respectively, in the nth generation of the branching process under consideration.
It is assumed that ξ0 = 1 and η0 = 0. We set
1 =
∞
n=1
ξn , 2 =
∞
n=1
ηn .
Decomposable Galton–Watson branching processes with several particle types were
first studied under the nonextinction condition (for a detailed bibliography, see [1])
or under the condition of extinction at a fixed moment of time (see [1] and [2]).
In [3–7], decomposable Galton–Watson branching processes with two particle
types were considered under conditions imposed on the total number of particles
of the first or second types. For instance, various functional limit theorems for the
process {(ξn , ηn ) , n ∈ N0 } considered under the condition that 1 = N, where
N → ∞, proved in [3] with restrictions (1) and in [4] with more general restrictions
for which, in particular, that the values σ12 and σ22 can be infinite.
x
We set 2 (x) = n=0 ηn for x ≥ 0. In [5, 6], decomposable Galton–Watson
branching processes with two particle types were considered under the condition
that 2 (aN) > N 2 , where a is a constant belonging (0, ∞] and N → ∞ (N ∈ N).
In particular, in [6], under restrictions (1), a functional limit theorem was proved
for the process {ξn , n ∈ N0 } considered under the condition that 2 > N, where
N → ∞. In [7], under more general restrictions than (1), a one-dimensional version
of this theorem was proved, and in addition, a similar statement is obtained when
replacing the condition {2 > N} with the condition {2 = N}, where N → ∞. In
the present paper, we study the process {ηn , n ∈ N0 } considered under the condition
that 2 (aN) > N 2 , where N → ∞.
Note that the random process {ξn , n ∈ N0 } is a critical Galton–Watson branching
process in which the generating function of the descendants of one particle is ϕ (·).
For a fixed process {ξn , n ∈ N0 }, the process {ηn , n ∈ N0 } is a critical Galton–
Watson branching process with ξn immigrants in the nth generation for each n ∈ N.
Note that here, the generating function of the descendants of one particle (including
those of each immigrant) is ψ (·).
Let {Y (t) , t ≥ 0} be a Feller diffusion, i.e., a homogeneous Markov random
process with continuous trajectories starting from a positive state and having an
absorbing state at 0 whose transition probability density p(t, x, y), where t, x, y >
0, satisfies the equality
∞
e
−λy
0
for t, x > 0 and λ ≥ 0.
x
λx
p (t, x, y) dy = exp −
− exp −
1 + λt
t
A Conditional Functional Limit Theorem for a Decomposable Branching Process
3
We set for a ∈ (0, ∞]
Sa =
a
Y (b2 t) dt.
0
It is further assumed that the Feller diffusion starts from state 1 (i.e., Y (0) = 1 a.s.),
if the opposite is not specified. It is proved in [8] that the random variable Sa is finite
and positive and its distribution is absolutely continuous. Moreover, for λ ≥ 0,
E exp (−λSa ) =
√
√
exp −x λ/b2 tanh a λ/b2
√
exp −x λ/b2 ,
, a ∈ (0, ∞) ;
a = ∞.
In the special case, when a = ∞, this random variable S∞ has absolutely
continuous distribution with the probability density
1
2
f (z) = √
e−1/ 2σ2 z , z > 0.
3/2
2πσ2 z
We set for a ∈ (0, ∞]
c (a) =
∞
0
P z2 Sa/z > 1
dz.
z3/2
Note that for a ∈ (0, ∞],
0 < c (a) ≤
∞
0
∞ P z2 S∞ > 1
4
dz
=
2
P
S
>
y
dy = 2E 4 S∞ < ∞
∞
z3/2
0
(the latter inequality is easy to obtain using the explicit representation for f (z),
z > 0). We introduce for a ∈ (0, ∞] the probability density
pa (z) =
1 P z2 Sa/z > 1
, z > 0.
c (a)
z3/2
Let us assume that convergence of a sequence of random processes in distribution
in the space D (0, ∞) means convergence in distribution in the space D[v, +∞)
with the Skorokhod topology for each v > 0. Now, we formulate the main result.
Theorem 1 If restrictions (1) are valid and a ∈ (0, ∞], then, as N → ∞,
D
ηt N
, t > 0 2 (aN) > N 2 → { Y (b2 t) , t > 0| Sa > 1} ,
N
(2)
where Y (0) has absolutely continuous distribution with the probability density
D
pa (·) and the symbol → means convergence in distribution in the space D (0, ∞).
4
V. I. Afanasyev
2 Auxiliary Statements
We set for t ≥ 0
YN (t) =
ηt N
.
N
It is proved in [3, Theorem 3] that, if restrictions (1) are valid, then, as N → ∞,
D
{ YN (t) , t > 0| 1 = N} → {Y (b2 t) , t > 0} ,
(3)
D
where the symbol → means convergence in distribution in the space D (0, ∞).
Denote for a ∈ (0, ∞]
ζN (a) =
2 (aN)
.
N2
It is proved in [5, Theorem 1] that, if restrictions (1) are valid and 0 < b < a ≤ ∞,
then, as N → ∞,
D
D
{ ζN (a) | 1 = N} → Sa , { ζN (a) − ζN (b) | 1 = N} → Sa − Sb .
(4)
In addition to relations (3) and (4), we prove the following result.
Lemma 1 If restrictions (1) are valid and a ∈ (0, ∞], then, as N → ∞,
D
{ YN (t) , t > 0; ζN (a) | 1 = N} → {Y (b2 t) , t > 0; Sa } ,
(5)
D
where the symbol → means convergence in distribution in the space D (0, ∞) × R.
Proof We fix such v and u that 0 < v < u < ∞ and introduce random processes
Y = {Y (b2 t) , t ∈ [v, u]} ; YN = {YN (t) , t ∈ [v, u]} , N ∈ N.
Let f be a continuous bounded mapping of the space D [v, u] into R. It is required
to show that, as N → ∞,
D
{ f (YN ) , ζN (a) | 1 = N} → (f (Y ) , Sa ) .
Let numbers x ∈ R and y ∈ (0, ∞) be fixed. Set
P (N) = P ( f (YN ) ≤ x, ζN (a) ≥ y| 1 = N) .
(6)
A Conditional Functional Limit Theorem for a Decomposable Branching Process
5
Note that
(1)
2 (aN) = N
(2)
v + N
(3)
u ,
(3)
u =
v , u + N
(7)
where 0 < v < u < a and
(1)
N
v =
v N
(2)
ηn , N
u N
v ,u =
n=1
ηn , N
n=v N+1
aN
ηn .
n=u N+1
In view of (7),
P1 N, v , u ≤ P (N) ≤ P2 N, v , u , ε + P3 N, v , u , ε ,
(8)
where ε ∈ (0, y) and
P2 N, v , u , ε = P f (YN ) ≤ x,
Set v = min v, v
space D v , u :
v ,u
N2
≥ y 1 = N ,
(2)
v ,u
N2
≥ y − ε 1 = N ,
(2)
P1 N, v , u = P f (YN ) ≤ x,
P3 N, v , u , ε = P
N
(1)
N
N
(3)
v + N
N2
≥ ε 1 = N .
u
and u = max u, u . We consider two functionals on the
z → f ({z (t) , t ∈ [v, u]}) , z →
u
z (t) dt,
v
where z ∈ D v , u . These functionals are continuous for
such z ∈ D v , u ,
which are themselves continuous functions on the segment v , u . Since trajectories of a Feller diffusion are continuous a.s., the mapping
z → f ({z (t) , t ∈ [v, u]}) ,
u
z (t) dt ,
v
6
V. I. Afanasyev
where z ∈ D v , u , is continuous on almost all trajectories of the Feller diffusion.
Hence (see Theorem 5.1 in [9]), in view of (3), as N → ∞,
u
u
D
YN (t) dt 1 = N → f (Y ) ,
Y (b2 t) dt .
(9)
f (YN ) ,
v
v
Note that
(2)
N v ,u
N2
−
u
v
Y v +Y u
N
N
YN (t) dt ≤
N
(10)
and (see (3)), as n → ∞,
YN v + YN u
N
D
1 = N → 0.
(11)
It follows from (9)–(11) that, as N → ∞,
(2)
f (YN ) ,
N
v ,u
N2
u
D
Y (b2 t) dt .
1 = N → f (Y ) ,
v
(12)
Since convergence in distribution implies complete convergence of corresponding
distribution functions, we obtain from (12) that
lim P1 N, v , u = P f (Y ) ≤ x,
N→∞
Y (b2 t) dt ≥ y ,
(13)
v
lim P2 N, v , u , ε = P f (Y ) ≤ x,
N→∞
u
u
Y (b2 t) dt ≥ y − ε
(14)
v
and relations (13) and (14) are valid for all but a countable set of (x, y) ∈
R× (0, ∞).
Further,
P3 N, v , u , ε
(1)
N v
≤P
N2
(3)
N u
ε ≥ 1 = N + P
2
N2
ε ≥ 1 = N . (15)
2
A Conditional Functional Limit Theorem for a Decomposable Branching Process
7
From (15) in view of relation (4), we obtain that
lim sup P3 N, v , u , ε ≤ P
v
0
N→∞
ε
Y (b2 t) dt ≥
2
+P
a
ε
Y (b2 t) dt ≥
2
u
and, hence,
lim sup P3 N, v , u , ε = 0.
lim
(16)
v →0,u →a N→∞
It follows from (8), (13), and (14) that
P f (Y ) ≤ x,
u
Y (b2 t) dt ≥ y
v
≤ P f (Y ) ≤ x,
u
≤ lim inf P (N) ≤ lim sup P (N)
N→∞
N→∞
Y (b2 t) dt ≥ y − ε + lim sup P3 N, v , u , ε .
v
N→∞
Therefore, passing to the limit as v → 0, u → a and taking into account (16), we
find that
P (f (Y ) ≤ x, Sa ≥ y) ≤ lim inf P (N)
N→∞
≤ lim sup P (N) ≤ P (f (Y ) ≤ x, Sa ≥ y − ε) .
N→∞
Now, passing to the limit as ε → 0, we obtain
lim P (N) = P (f (Y ) ≤ x, Sa ≥ y) .
N→∞
Thus, the required relation (6) is established. The lemma is proved.
Set for z > 0 and a ∈ (0, ∞]
Sa(z) = z2 Sa/z ;
Y
(z)
(t) = zY
b2 t
z
, t ≥ 0.
It is known (see, for instance, [10]) that a Feller diffusion has the property of selfsimilarity, according to which
D
Y (z) (t) , t ≥ 0 Y (0) = 1 = { Y (t) , t ≥ 0| Y (0) = z} .
(17)
8
V. I. Afanasyev
In particular,
D
Sa(z) Y (0) = 1 = { Sa | Y (0) = z} .
(18)
Moreover, relations (17) and (18) can be combined into one relation.
Lemma 2 If restrictions (1) are valid and a ∈ (0, ∞], then for any z > 0, as
N → ∞,
D
{ YN (t) , t > 0; ζN (a) | 1 = zN} → { Y (b2 t) , t > 0; Sa | Y (0) = z} ,
(19)
D
where the symbol → means convergence in distribution in the space D (0, ∞) × R.
Proof Set KN = zN; then, for N ≥ z−1 ,
t
γ −1 ,
YN (t) = zYKN γN
z N
(20)
a
γN−2 ,
ζN (a) = z ζKN γN
z
(21)
2
where γN = zN/ zN. Note that
0 ≤ γN − 1 ≤ 1/KN .
(22)
By Lemma 1, as N → ∞,
zYKN
t
a D
(z)
(z)
, t > 0; z2ζKN
=
K
→
Y
.
,
t
>
0;
S
(t)
1
N
a
z
z (23)
We fix such v and u that 0 < v < u < ∞. Introduce a modulus of continuity for a
function x ∈ D [v, u]:
wx (δ; v, u) =
sup
t,s:|t −s|≤δ,
|x (t) − x (s)| , δ > 0.
t,s∈[v,u]
In view of (22),
t
t 1 u v 2u
sup YKN γN
; ,
− YKN
.
≤ wYKN
z
z KN z z z
t ∈[v,u]
(24)
Let number δ > 0 be fixed. Note that a functional that maps an element x of the
space D [v, u] to its modulus of continuity wx (δ; v, u) is continuous for such x,
A Conditional Functional Limit Theorem for a Decomposable Branching Process
9
which are themselves continuous functions on the segment [v, u]. Since trajectories
of a Feller diffusion are continuous a.s., then due to (23), as N → ∞,
v 2u v 2u
D
wY (z) δ; ,
,
,
(25)
δ;
=
K
→
w
(z)
1
N
Y
KN
z z z z
where
(z)
YN = zYN
t
, t ≥ 0 , Y (z) = Y (z) (t) , t ≥ 0 .
z
It follows from (25) that, as N → ∞,
wY (z)
KN
1 u v 2u D
1 = KN → 0,
; ,
KN z z z
(26)
since limδ→0 wx (δ; v/z, 2u/z) = 0 for a continuous function x on the segment
[v/z, 2u/z]. We obtain from (24) and (26) that, as N → ∞,
t
t
D
1 = KN →
− YKN
sup YKN γN
0.
(27)
z
z
t ∈[v,u]
Note that in view of (22),
a
a
a
a
− ζK N
≤ ζKN (1 + 1/KN )
− ζK N
.
0 ≤ ζK N γN
z
z
z
z
Therefore, if δ > 0 and N is large enough, then
a
a+δ
a
a
0 ≤ ζK N γN
− ζK N
≤ ζK N
− ζK N
.
z
z
z
z
(28)
By virtue of (4), as N → ∞,
a+δ
a
D (z)
z ζK N
− ζK N
1 = KN → Sa+δ − Sa(z) .
z
z
2
(29)
Finally, in view of the continuity of almost all trajectories of a Feller diffusion,
(z)
lim Sa+δ − Sa(z) = 0.
δ→0
It follows from (28)–(30) that, as N → ∞,
a D
ζ K γ N a − ζ K
=
K
→ 0.
1
N
N
N
z
z (30)
(31)
10
V. I. Afanasyev
From relation (23), given (27) and (31), we find that, as N → ∞,
t
a , t ∈ [v, u] ; z2 ζKN γN
=
K
1
N
z
z D
→ Y (z) (t) , t ∈ [v, u] ; Sa(z) .
zYKN
γN
(32)
From (32), remembering relations (20)–(22), we obtain that, as N → ∞,
D
{ YN (t) , t ∈ [v, u] ; ζN (a) | 1 = zN} → Y (z) (t) , t ∈ [v, u] ; Sa(z) .
Therefore,
D
{ YN (t) , t > 0; ζN (a) | 1 = zN} → Y (z) (t) , t > 0; Sa(z) ,
whence, in view of the relations (17) and (18), it follows the statement of the lemma.
We also present the corrected formulations of Theorems 2 and 3 from [5].
Lemma 3 If restrictions (1) are valid and a ∈ (0, ∞], then, as N → ∞,
c (a)
1
P 2 (aN) > N 2 ∼ √ ,
N
√
where c1 (a) = c (a) / 2 πb1 .
Lemma 4 If restrictions (1) are valid and a ∈ (0, ∞], then, for any x > 0,
x
lim P 1 ≤ xN| 2 (aN) > N 2 =
pa (z) dz,
N→∞
0
where pa (·) is the function defined in Sect. 1.
3 Proof of Theorem 1
Let’s rewrite relation (2) in the equivalent form: as N → ∞,
D
{ YN (t) , t > 0| ζN (a) > 1} → { Y (b2 t) , t > 0| Sa > 1} ,
(33)
where Y (0) has absolutely continuous distribution with the probability density
pa (·). We fix such v and u that 0 < v < u < ∞. Let f be a continuous bounded
and positive mapping of the space D [v, u] in R. For validity of (33), it is required
A Conditional Functional Limit Theorem for a Decomposable Branching Process
11
to show that
lim E ( f (YN ) | ζN (a) > 1) =
N→∞
∞
E(z) ( f (Y ) | Sa > 1) pa (z) dz,
(34)
0
where, as before, YN = {YN (t) , t ∈ [v, u]} for N ∈ N, and Y =
{Y (b2 t) , t ∈ [v, u]}; the notation E(z) , used instead of the notation E, indicates
that Y (0) = z a.s.
Note that for arbitrary positive numbers b and B (b < B),
E ( f (YN ) | ζN (a) > 1) = E1 (N, b, B) + E2 (N, b, B) ,
(35)
where
E1 (N, b, B) = E ( f (YN ) I (1 ∈ [bN, BN]) | ζN (a) > 1) ,
/ [bN, BN]) | ζN (a) > 1)
E2 (N, b, B) = E ( f (YN ) I (1 ∈
(here, I (A) is the indicator function of a random event A). It is clear that
|E2 (N, b, B)| ≤ LP ( 1 ∈
/ [bN, BN] | ζN (a) > 1) ,
where L = supx∈D[v,u] |f (x)|. Whence, in view of Lemma 4, we find that
lim
lim sup |E2 (N, b, B)| = 0.
(36)
b→0,B→∞ N→∞
We show that
lim
lim E1 (N, b, B) =
b→0,B→∞ N→∞
∞
E(z) ( f (Y ) | Sa > 1) pa (z) dz.
(37)
0
Note that
E1 (N, b, B) =
K∈[bN,BN]
=
E (f (YN ) ; ζN (a) > 1, 1 = K)
P (ζN (a) > 1)
E ( f (YN ) I (ζN (a) > 1) | 1 = K)
K∈[bN,BN]
P (1 = K)
P (ζN (a) > 1)
(the summation is performed using natural numbers K). It is known (see, for
instance, [11, Chapter 2]) that, as n → ∞,
P (1 = n) ∼
1
2 πb1 n3
.
(38)
12
V. I. Afanasyev
Therefore, as N → ∞,
2 πb1 lim E1 (N, b, B)
N→∞
= lim
N→∞
= lim
E ( f (YN ) I (ζN (a) > 1) | 1 = K)
K 3/2P (ζN (a) > 1)
K∈[bN,BN]
B
N→∞ b
E ( f (YN ) I (ζN (a) > 1) | 1 = zN) dz
√
,
z3/2
N P (ζN (a) > 1)
(39)
if at least one of these three limits exists.
We consider a mapping that matches a pair (x, y), where x ∈ D [v, u], y ∈ R,
the number f (x) I {y > 1}. This mapping is continuous for such (x, y) for which
y = 1. Note that P(z) (Sa = 1) = 0. Therefore (see [9], Theorem 5.1]), we conclude
using the Lemma 2 that
lim E ( f (YN ) I (ζN (a) > 1) | 1 = zN) = E(z) (f (Y ) ; Sa > 1) .
N→∞
By Lemma 3,
lim
N→∞
√
N P (ζN (a) > 1) = c1 (a) .
Hence, by the dominated convergence theorem,
B
lim
N→∞ b
=
B
b
E ( f (YN ) I (ζN (a) > 1) | 1 = zN) dz
√
z3/2
NP (ζN (a) > 1)
E(z) (f (Y ) ; Sa > 1) dz
.
c1 (a)
z3/2
(40)
We obtain from (39) and (40) that
lim E1 (N, b, B) =
N→∞
B
b
E(z) (f (Y ) ; Sa > 1)
√
dz.
2 πb1 c1 (a) z3/2
(41)
Note, using the Lemma 3, relation (18), and the definition of pa (z), that
E(z) (f (Y ) ; Sa > 1)
√
= E(z) ( f (Y ) | Sa > 1) pa (z) .
2 πb1 c1 (a) z3/2
(42)
It follows from (41) and (42) that
lim E1 (N, b, B) =
N→∞
B
b
E(z) ( f (Y ) | Sa > 1) pa (z) dz.
(43)
A Conditional Functional Limit Theorem for a Decomposable Branching Process
13
Passing in (43) to the limit as b → 0 and B → ∞, we get the required relation (37).
From (35)–(37) follows relation (34). The theorem is proved.
Remark 1 Note that Lemma 4 was not used to prove (43). For f ≡ 1, relation (43)
means that
lim P ( 1 ∈ [bN, BN] | ζN (a) > 1) =
N→∞
B
pa (z) dz.
b
The latter relation allows us to establish Lemma 4 in a shorter way than in [5].
4 Conclusion
The statement of Theorem 1 for a = ∞ can be given the following form: as N →
∞,
D
, t > 0 2 > N → { Y (b2 t) , t > 0| S∞ > 1} ,
√
N
η √ t N
(44)
where Y (0) has absolutely continuous distribution with the probability density
p∞ (·).
It is interesting to compare (44) with the corresponding limit theorem for a
particle in
critical branching Galton–Watson process {Zn , n ∈ N0 }, starting with one the zero generation and satisfying the condition that > n, where = ∞
n=0 Zn .
Recall that, for a fixed process {ξn , n ∈ N0 }, the random process {ηn , n ∈ N0 } is
also a critical branching Galton–Watson process; however, it is not created by one
particle in the zero generation but ξn immigrants in the nth generation for each
n ∈ N.
Let {W(t) , t ∈ [0, 1]} be a standard Brownian motion. Recall that a Brownian
excursion W0+ (t) , t ∈ [0, 1] is a limit in distribution in the space C [0, 1] as ε ↓ 0
of the Brownian motion considered under the condition that inft ∈[0,1] W (t) ≥ −ε
and W (1) ≤ ε. The local time l0+ (y) at a level y ≥ 0 of the Brownian excursion
+
W0 (t) , t ∈ [0, 1] is defined by
1
ε→0 ε
l0+ (y) = lim
1
0
I[y,y+ε] W0+ (t) dt,
where I[y,y+ε] (·) is the indicator function of the segment [y, y + ε]. Note
that there
exists a continuous modification of the random process l0+ (y) , y ≥ 0 .
14
V. I. Afanasyev
It is proved in [12, Theorem 4.10] that, if the maximal span of the distribution of
the random variable Z1 is equal to 1 and EZ1 = 1, VarZ1 : =σ 2 ∈ (0, ∞), then, as
N → ∞,
⎫
⎧ ⎬
⎨ Z t √N
σ + σ D
l0
t ,t ≥ 0 ,
(45)
√
, t ≥ 0 = N →
⎭
⎩
2
2
N
D
where the symbol → means convergence in distribution in the space D [0, ∞) with
the Skorokhod topology.
As a corollary of relation (45), we establish the following limit theorem.
Theorem 2 If the maximal span of the distribution of the random variable Z1 is
equal to 1 and EZ1 = 1 and VarZ1 : =σ 2 ∈ (0, ∞), then, as N → ∞,
⎧ ⎫
⎨ Z t √N
⎬
σ + σ D
, t ≥ 0 > N →
l0
tα , t ≥ 0 ,
√
⎩
⎭
2α
2
N
(46)
where α is a random variable
uniformly
distributed on the interval (0, 1) and
independent of the process l0+ (t) , t ≥ 0 .
Proof Note that for an arbitrary z > 0, as N → ∞,
⎫
⎧ ⎬
⎨ Z t √N
D
, t ≥ 0 = zN →
√
⎭
⎩
N
√
σ
σ z +
l0
t
,
t
≥
0
.
√
2
2 z
(47)
The proof of relation (47) is based on (45) and is carried out similarly to the proof
of Lemma 2 and is therefore not given here.
Set for z > 0 and t ≥ 0
V
(z)
Z √ √
t N
σ
σ z +
l0
, N ∈ N.
(t) =
√ t ; UN (t) = √
2
2 z
N
Fix such v and u that 0 < v < u < +∞. Let f be a continuous bounded and
positive mapping of the space D [v, u] in R. For validity (46), it is required to show
that
−2
,
(48)
lim E ( f (UN ) | > N) = Ef V α
N→∞
where UN = {UN (t) , t ≥ 0} and V (z) = V (z) (t) , t ≥ 0 .
For B > 1,
E ( f (UN ) | > N) = E1 (N, B) + E2 (N, B),
(49)
A Conditional Functional Limit Theorem for a Decomposable Branching Process
15
where
E1 (N, B) = E ( f (UN ) I ( ≤ BN) | > N) ,
E2 (N, B) = E ( f (UN ) I ( > BN) | > N) .
Note that for B > 1,
|E2 (N, B)| ≤ LP ( > BN| > N) = L
P ( > BN)
,
P ( > N)
(50)
where L = supx∈D[v,u] |f (x)|. Since (see (38)), as N → ∞,
1
P ( = N) ∼ √
,
2 πbN 3
(51)
1
P ( > N) ∼ √
πbN
(52)
where b = σ 2 /2, then,
and
1
P ( > BN)
= √ .
P ( > N)
B
(53)
lim lim sup |E2 (N, B)| = 0.
(54)
lim
N→∞
It follows from (50) and (53) that
B→∞ N→∞
Further,
E1 (N, B) =
BN
K=1
E ( f (UN ) | = N + K)
P ( = N + K)
.
P ( > N)
(55)
In view of relations (51) and (52), we obtain from (55) that
lim E1 (N, B) = lim
N→∞
BN
K=1
N→∞
= lim
B+1
N→∞ 1
E ( f (UN ) | = N + K)
E ( f (UN ) | = zN)
√
N
2 (N + K)3
dz
,
2z3/2
(56)
16
V. I. Afanasyev
if at least one of these three limits exists. By virtue of (47),
lim E ( f (UN ) | = zN) = Ef V (z) .
N→∞
(57)
We obtain from (56) and (57) by the dominated convergence theorem that
lim E1 (N, B) =
B+1
N→∞
1
dz
Ef V (z)
2z3/2
and, therefore,
lim
lim E1 (N, B) =
B→∞ N→∞
∞
1
dz
Ef V (z)
.
2z3/2
(58)
It follows from (49), (54), and (58) that
lim E ( f (UN ) | > N) =
N→∞
∞
1
dz
Ef V (z)
.
2z3/2
(59)
Note that
∞
1
1
dz
u−2
α −2
Ef V (z)
=
Ef
V
du
=
Ef
V
.
2z3/2
0
(60)
Due to relations (59) and (60), we obtain (48). The theorem is proved.
Comparing relations (44) and (46), we see a significant difference in the
asymptotic behavior of the random processes
, t > 0 2 > N ,
√
N
η √ t N
⎫
⎧ ⎬
⎨ Z t √N
, t ≥ 0 > N ,
√
⎭
⎩
N
although, in each of these processes, the reproduction of particles is carried out
according to probabilistic laws with the mathematical expectation 1 and a finite
variance. This is due to the fact that the conditions for the forming of these processes
differ. The first of them (with a fixed process {ξn }) is formed due to immigrants, and
the second due to a single initial particle.
References
1. Vatutin, V.A., D’yakonova, E.E.: Decomposable branching processes with a fixed extinction
moment. Proc. Steklov Inst. Math. 290, 103–124 (2015)
2. Vatutin, V.A., D’yakonova, E.E.: Extinction of decomposable branching processes. Discrete
Math. Appl. 26(3), 183–192 (2016)
A Conditional Functional Limit Theorem for a Decomposable Branching Process
17
3. Afanasyev, V.I.: Functional limit theorems for the decomposable branching process with two
types of particles. Discrete Math. Appl. 26(2), 71–88 (2016)
4. Vatutin, V.A.: A conditional functional limit theorem for decomposable branching processes
with two types of particles. Math. Notes. 101(5), 778–789 (2017)
5. Afanasyev, V.I.: On a decomposable branching process with two types of particles. Proc.
Steklov Inst. Math. 294, 1–12 (2016)
6. Afanasyev, V.I.: A functional limit theorem for decomposable branching processes with two
particle types. Math. Notes. 103(3), 337–347 (2018)
7. Vatutin, V.A., D’yakonova, E.E.: Decomposable branching processes with two types of
particles. Discrete Math. Appl. 28(2), 119–130 (2018)
8. Lindvall, T.: Limit theorems for some functionals of certain Galton-Watson branching processes. Adv. Appl. Probab. 6(2), 309–321 (1974)
9. Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)
10. Afanasyev, V.I.: Invariance principle for the critical Galton–Watson process attaining a high
level. Theory Probab. Appl. 55(4), 559–574 (2011)
11. Kolchin, V.F.: Random Mappings. Optimization Software, Publications Division, New York
(1986)
12. Drmota, M.: Random Trees. Springer, Wien (2009)
A Probabilistic Interpretation of
Conservation and Balance Laws
Ya. I. Belopolskaya
Abstract We construct stochastic systems associated with parabolic conservation
and balance laws written in the form of nonlinear parabolic systems and prove that
solution of stochastic systems allows to construct weak and mild solutions of the
Cauchy problem for original partial differential equations (PDE) partial differential
equations systems.
Keywords Stochastic differential equations · Hyperbolic and parabolic
conservation laws · Balance laws · Diffusion processes
1 Introduction
The laws of classical physics as a rule express conservation or balance of certain
quantities such as mass, momentum, angular momentum, and energy. A conservation law is an equation in divergence form
∂u +
∇yi f i (u) = 0,
∂t
d
(1.1)
i=1
which expresses the fact that the quantity of u(t, y) ∈ R d does not evolve in time
d
∂u
dt u(t, y(t)) = 0. We use notations ∇yi u = ∂yi .
A system of the form
1
∂um +
∇xi fmi (u) =
cmk (u)uk
∂t
d
d
i=1
k=1
(1.2)
is called a system of balance laws.
Ya. I. Belopolskaya ()
Saint Petersburg State University of Architecture and Civil Engineering, St Petersburg, Russia
Sirius University, Sochi, Russia
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_2
19
20
Ya. I. Belopolskaya
Including into consideration dissipation mechanisms, one comes to parabolic
systems of the form
1
1
∂um ij
+
∇yi fmi (u) =
∇y2i ,yj [Bmk (u)uk ] +
cmk (u)uk
∂t
d
d
i=1
k=1 i,j =1
d
d
(1.3)
k=1
(systems with cross-diffusion).
The aim of this article is to give a probabilistic interpretation to parabolic
conservation and balance laws that is to systems of the form (1.3). Recall that a
ij
probabilistic interpretation to systems of nonlinear parabolic equations with Gmk ≡
Gij (systems with self-diffusion) was developed in [3, 4]. Here, we construct and
analyze probabilistic representations of solutions of the Cauchy problem for the
system (1.3). In other words, we derive stochastic problems associated with the
Cauchy problem for these parabolic systems, investigate the stochastic problems,
and show connections between their solutions and solutions of the Cauchy problem
for (1.3).
Connections between scalar nonlinear parabolic equations and stochastic differential equations have been investigated by many authors since the pioneer papers
by McKean [9] and Freidlin [6]. Let us discuss a difference between these two
approaches analyzing constructions suggested in these papers and the possibility
to extend them to systems of nonlinear parabolic equations.
In [9], there was a constructed stochastic model associated with the Cauchy
problem for the Vlasov equation
∂u
1
= (A2 [u]u) − div(a[u]),
∂t
2
u(0, y) = u0 (y)
(1.4)
d ∂ 2
which arises in plasma physics. Here, =
i=1 ∂y 2 is the Laplace operator,
i
d
div a[u](y) = i=1 ∇yi ai [u](y), and coefficients A[u] ∈ R d ⊗ R d , a[u] ∈ R d
have the form
. . . ai (y, x1 , . . . , xd )u(x1 ) . . . u(xd )dx1 . . . dxd , i = 1, . . . , d.
ai [u](y) =
R
R
(1.5)
The problem (1.4) was treated in [9] as a forward Kolmogorov equation for a
distribution u(t, dy) = P {ξ(t) ∈ dy} of a stochastic process ξ(t) ∈ R d governed
by a stochastic equation
ξ(t) = ξ(0) +
t 0
+
t R
R
R
...
0
a(ξ(s), x1 , . . . , xd )u(s, dx1 ) . . . u(s, dxd ) ds+
...
A(ξ(s), x1 , . . . , xd )u(s, dx1 ) . . . u(s, dxd ) dw(s),
R
A Probabilistic Interpretation of Conservation and Balance Laws
21
where w(t) ∈ R d is the Wiener process defined on a probability space (, F, P ),
ξ(0) does not depend on w(t), and P (ξ(0) ∈ dy) = u0 (y)dy.
An alternative probabilistic approach to the Cauchy problem for semilinear
parabolic equations was suggested in [6]. To explain this approach, let us consider
the Cauchy problem
1
∂u
= T rA(x, u)∇ 2 uA(x, u) + a(x, u) · ∇u + c(x, u)u,
∂t
2
u(0, x) = u0 (x),
(1.6)
where
T rA(x, u)∇ 2 uA(x, u) =
d
d
Aik (x, u)∇x2i ,xj uAkj (x, u), a(x, u)·∇u =
i,j,k=1
ai (x, u)∇xi u.
i=1
If u(t, x) solves (1.6), then a function vT (t, x) = u(t, x) satisfies the Cauchy
problem
∂vT
1
+ T rA(x, vT )∇ 2 vT A∗ (x, vT ) + a(x, vT )∇vT + c(x, vT )vT = 0,
∂t
2
vT (T , x) = u0 (x).
(1.7)
One can treat (1.7) as the backward Kolmogorov equation and hence to reduce its
solution to solution of a stochastic system
dξ(θ ) = a(ξ(θ ), vT (θ, ξ(θ )))dθ + A(ξ(θ ), vT (θ, ξ(θ )))dw(θ ),
ξ(t) = x,
(1.8)
%
#
$T
vT (t, x) = E u0 (ξt,x (T ))e t c(ξt,x (τ ),vT (τ,ξt,x (τ )))dτ .
(1.9)
Provided one can prove the existence and uniqueness of a solution to the systems
(1.8), (1.9) and in addition prove that the function vT (t, x) is twice differentiable in
x, we can verify that this function satisfies the Cauchy problem (1.7) in the classical
sense. As a result, one obtains a probabilistic representation of a classical solution
for the Cauchy problem (1.7), and hence, for a solution u(t, x) of the Cauchy
problem (1.4), we get
%
#
$T
u(t, x) = vT (t, x) = E u0 (ξt,x (T ))e t c(ξt,x (τ ),vT (τ,ξt,x (τ )))dτ .
It is important to underline that the Cauchy problem (1.4) admits a natural
measure-valued solution that provided initial data u0 is a measure while (1.6) does
not admit such a solution. On the other hand, in applications, it is natural to interpret
a solution to (1.8) as a density of a probability measure. This goal requires an
approach to the Cauchy problem (1.6) alternative to [6].
22
Ya. I. Belopolskaya
To derive a stochastic model which is compatible with the interpretation of
a solution (1.6) as a density of a certain probability measure, one can apply an
approach developed in [7, 8]. Within this approach, one considers a regularized
Cauchy problem
d
d
1 2
∂v
=
∇xi xj ([Aik Akj ](x, ρ ∗v)v)+
∇xi (ai (x, ρ ∗v)v)+c(x, ρ ∗v)v,
∂t
2
i,j,k=1
i=1
(1.10)
v(0, x) = u0 (x),
$
where ρ ∗ v(x) = R d ρ (x − y)v(y)dy with ρ (x) being a mollifier and u0 (x)
is a density of a probability measure. To construct a stochastic interpretation of
(1.10), one may apply the McKean approach to the regularized Eq. (1.10) and
finally prove the existence of a limit of a solution to a stochastic system associated
with this equation as ρ → δ. At the end, one has to verify that a solution to the
limiting stochastic system allows to obtain the required probabilistic interpretation
of a solution to the Cauchy problem (1.8).
New phenomena arise when we turn to systems of (nonlinear) parabolic equations. While constructing stochastic models for these systems, we reveal that one
can separate a group of systems of the form (1.3); namely, systems with diffusion
ij
coefficients Gmk ≡ Gij such that the Cauchy problem (1.3) can be reduced to the
Cauchy problem for a system of backward Kolmogorov equations, and hence, its
solution admits a probabilistic interpretation [3, 4]. The correspondent construction
is an extension of the Freidlin approach. On the other hand, systems of the form (1.3)
ij
with general coefficients Gmk do not admit this reduction and require an alternative
approach to construct a probabilistic interpretation of the Cauchy problem solution.
The correspondent approach which extends the McKean approach to systems of
nonlinear forward parabolic equations was developed in papers [1, 2, 5].
ij
In this paper, we consider a class of systems (1.3) with Gmk (u) =
d
kj
ik
k=1 Mm (u)Mm (u) which allows to extend results obtained in [7] for scalar
equations and construct stochastic counterparts to parabolic systems of this class.
Besides, we omit an essential assumption assumed in [7] that cm (u) are bounded
functions and prove that there exist both a solution of a stochastic system and weak
and mild solutions of the Cauchy problem for an associated mollified parabolic
system defined on a certain time interval. We postpone the proof of a possibility to
go to a limit as the mollifier goes to delta function till ongoing paper because of a
volume limitation.
The remaining part of the article is organized as follows. In Sect. 2, we consider
a particular case of a system similar to (1.3) and construct two equivalent stochastic
systems associated with the system under consideration. One of them allows to
obtain a probabilistic representations of a weak and a mild solution of the Cauchy
problem for a system of this type, while the other allows to obtain a constructive
way to prove the existence and uniqueness of a solution to stochastic system itself.
A Probabilistic Interpretation of Conservation and Balance Laws
23
The model system which motivates conditions on coefficients in this paper is the
SKT system proposed by Shigesada, Kawashima, and Teramoto [10]
∂um
= (A2m (y, u)um ) + cm (y, u)um ,
∂t
(1.11)
um (0, y) = u0m (y), m = 1, 2,
A2m (y, u) = αm +
2
αmk uk (y), cm (y, u) = cm −
k=1
2
cmk uk (y),
k=1
to model population dynamics. Here, αm , αmk , cm , andcmk are positive constants.
In Sect. 3, we prove the existence and uniqueness theorem for solutions of the
stochastic system associated with the original partial differential equations (PDE)
system assuming that all coefficients of the PDE system are bounded. In the last
section, we omit this assumption and prove that solutions um (t, y) to systems under
investigation are bounded at least for some time interval [0, T1 ] with T1 depending
on coefficients of the PDE system and initial data. This allows to apply the results
of the previous section to systems extending (1.11).
2 Systems of Conservation and Balance Laws
with Cross-Diffusion
In this section, we consider a particular case of conservation and balance law
systems of the form (1.4) and construct stochastic systems associated with their
weak and mild solutions.
Let us start with some preliminary notions and notations.
Denote by C0∞ (R d ) (C0∞ ([0, T ] × R d )) the space of infinite differentiable
functions defined on R d (on [0, T ] × R d ) with compact supports, and let Cb (R d )
be the space of bounded continuous functions f : R d → R with the norm f ∞ =
supx |f (x)|. We use notations C k (R d ) for the space of k times differentiable real
valued functions defined on R d . For functional spaces of a vector space X valued
functions f : R d → X, we use notations of the form Cb (R d ; X), C k (R d ; X), and
so on.
1
$
Denote by Lp the space functions f with the norm f p = R d |f (x)|p dx p <
∞, and let W k,p (R d ) denote the Sobolev space of order k in (Lp (R d ), · p ),
1 ≤ p ≤ ∞.
Let P(R d ) denote the space of Borel probability measures and C([0, T ]; P(R d ))
be the space of continuous functions defined on [0, T ] and valued in P(R d ).
24
Ya. I. Belopolskaya
Our aim is to construct a probabilistic representation of a solution to the Cauchy
problem
1
∂um
= T r∇ 2 (Bm (y, u)um ) − div(bm (y, u)um ) + cm (y, u)um ,
∂t
2
(2.1)
um (0, y) = u0m (y), m = 1, . . . , d1 ,
∗,
provided Bm = Mm Mm
Mm (y, u) = αm (y) +
d1
bm (y, u) = bm (y) +
αmq (y)uq (y),
q=1
d1
bmq (y)uq (y),
q=1
(2.2)
cm (y, u) = cm (y) −
d1
cmq (y)uq (y),
(2.3)
q=1
where αm (y), αmq (y) ∈ R d ⊗ R d , cm (y), cmq (y) ∈ R, and bm (y) and bmq (y) ∈
R d are bounded Lipschitz continuous functions. If cm ≡ 0, one can interpret the
system (2.1) as a system of nonlinear forward Kolmogorov equations for densities
of probability measures μm (t, dy) w.r.t the Lebesgue measure
$ since it is easy to
verify
that
its
solution
u
(t,
y)
is
positive
if
u
(y)
>
0
and
m
0m
R d um (t, y)dy = 1 if
$
u
(y)dy
=
1.
d
0m
R
Let [Avm ]∗ um = 12 T r∇ 2 (Bm (y, u)um ) − div(bm (y, u)um ) and Avm h =
1
2
2 T rBm (y, v)∇ h + bm (y, v) · ∇h. With this notations, we have
Rd
h(y)[Avm ]∗ um (t, y)dy =
Rd
(Avm h(y))um (t, y)dy.
(2.4)
To construct a stochastic counterpart of (2.1), we consider a fixed probability
space (, F, P ), independent Wiener processes wm (t) ∈ R d , and independent
random variables ξ0m defined on this probability space. We write ξ0m ∼ μ0m when
ξ0m has a distribution μ0m that is P (ξ0m ∈ dy) = μ0m (dy) = u0m (y)dy and assume
below that ξ0m do not depend on wm (t). We denote by L(ξm ) a probability measure
γm in (, F) generated by the process ξm (t), t ∈ [0, T ].
Consider a stochastic system
ξm (t) = ξ0m +
t
0
bm (ξm (s), u(s, ξm (s)))ds +
t
Mm (ξm (s), u(s, ξm (s)))dwm (s),
0
(2.5)
A Probabilistic Interpretation of Conservation and Balance Laws
25
where u = (u1 , . . . , um , . . . , ud1 ), P {ξ0,m ∈ dy} = u0m (y)dy, and um (t, y) =
μm (t,dy)
are a density of a measure μm (t, dy) defined by a relation
dy
t
h(y)μm (t, dy) = E h(ξm (t)) exp
cm (ξm (s), u(s, ξm (s)))ds
Rd
(2.6)
0
(called the Feynman-Kac formula) valid for all h ∈ C0∞ (R d ). If the measure
μm (t, dy) has a density um (t, y), the left-hand side of (2.6) can be rewritten as
Rd
t
h(y)um (t, y)dy = E h(ξm (t)) exp
cm (ξm (s), u(s, ξm (s)))ds .
0
Systems (2.5), (2.6) together make a closed system, but we need a more
constructive system associated with (2.1). To derive a required system, we consider
first a mollified system
ξm (t) = ξ0m +
t
bm (ξm (s), u(s, ξm (s)))ds +
0
t
Mm (ξm (s), u(s, ξm (s)))dwm (s),
ξ0 ∼ μ0 ,
0
(2.7)
t
um (t, y) = E ρ(y − ξm (t ))exp
cm (ξm (s), u(s, ξm (s)))ds ,
(2.8)
0
where ρ : R d → R+ is a mollifier. Denote by γm ∈ P(Cd ) a distribution generated
on = Cd by the process ξm (t), L(ξm ) = γm .
When ρ = δ, where δ is the Dirac delta function, then (2.7) and (2.8) can be
reduced to (2.5) and (2.6) at least formally, and thus, applying the Ito formula,
we can verify that um (t, y) is a weak solution to (2.1). Below, we prove that if
μm (t, dy) = um (t, y)dy and um satisfy (2.8) with ρ = δ, then it is connected with
a weak solution of the mollified system
1
∂μm
= T r∇ 2 (Bm (y, ρ∗v)vm )−div(bm (y, ρ∗μ)μm )+cm (y, ρ∗μ)μm ,
∂t
2
(2.9)
μm (0, dy) = μ0m (dy), m = 1, . . . , d1 ,
$
by a relation um = ρ ∗ μm = R d ρ(· − x)μm (dx).
To verify this, we need some conditions.
We say that condition C 2.1 holds if coefficients bm : R d × R d1 → R d , Mm :
R d × R d1 → R d ⊗ R d , cm : R d × R d1 → R are Lipschitz continuous and bounded
(on each compact) with respect to space variables x ∈ R d and uniformly bounded
in (x, u) ∈ R d × R d1 by constants Kb , KM , and Kc .
26
Ya. I. Belopolskaya
Denote by Cd = C([0, T ]; R d ) the space of bounded functions with sup&1
norm · and by C = C([0, T ]; dm=1
Xm ) where Xm = R d . Let F (f )(y) =
$
−2πix·y
dx be the Fourier transform of a real valued function f defined on
R d f (x)e
Rd .
γ
Lemma 1 Assume C 2.1 holds and there exists a solution (ξm (t), μm (t, dy)) of a
system
ξm (t) = ξ0m +
t
bm (ξm (s), (ρ ∗ μγ )(s, ξm (s)))ds +
t
0
Mm (ξm (s), (ρ ∗ μγ )(s, ξm (s)))dwm (s),
0
(2.10)
ξ0m ∼ μ0m ,
γ
h(y)μm (t, dy) =
Rd
$t
Cd
γ
h(ξm (t, ω))e 0 cm (ξm (s,ω),ρ∗μ (s,ξm (s,ω)))ds γm (dω),
L(ξm ) = γm
(2.11)
γ
Then, there exists a solution (ξm (t), um (t, y)) of a system
ξm (t) = ξ0m +
t
b(ξm (s), uγ (s, ξm (s)))ds+
t
0
Mm (ξm (s), uγ (s, ξm (s)))dwm (s),
0
(2.12)
ξ0m ∼ μ0m ,
γ
um (t, y) =
Cd
$t
γ
ρ(y −ξm (t, ω))e 0 cm (ξm (s,ω),u (s,ξm (s,ω)))ds γm (dω),
L(ξm ) = γm .
(2.13)
γ
γ
and um = ρ ∗ μm . If in addition the Lebesgue measure of the measurable set {x ∈
R d : F(ρ)(x) = 0} is equal to zero, then the measure μm (t) satisfying (2.11) is
γ
uniquely determined by the function um (t, ·) satisfying (2.13) and vice versa.
Proof The proof of Lemma 1 is very close to the proof of Theorem 6.1 in [7]. Let
γ
γ
(ξm , um ) satisfy (2.12) and (2.13); then, the Fourier transform of um has the form
γ
F (um )(t, x) = F (ρ)(x)κ γ (t, x),
where
κ (t, x) =
γ
e
Cd
−2πix·ξm (t,ω)
t
exp
γ
cm (ξm (s), u (s, ξm (s)))ds γm (dω).
0
A Probabilistic Interpretation of Conservation and Balance Laws
27
γ
Since by assumption cm are bounded, then κm are bounded, nonnegative definite
functions. Choosing a sequence of complex numbers ak and points xk ∈ R d andk =
1, . . . , d,, we obtain that
d d
ak āj e −iy·(xk −xj ) =
d
k=1 j =1
⎞
d
⎛ d
−iy·xk ⎝
−iy·x
−iy·x
k
j⎠ = |
|2 ,
ak e
aj e
ak e
j =1
k=1
k=1
γ
that yields κm ≥ 0. This by the Bochner theorem results that there exists a finite
γ
nonnegative
Borel measure μm (t) on R d such that for all x ∈ R d , κm (t, x) =
$
γ
γ
−2πix·y
μm (t, dy). It remains to prove that μm (t, dy) satisfy (2.11).
Rd e
γ
Note that μm (t, dy) are the Schwartz (tempered) distributions
such that
$
γ
γ
γ
−1
F (κm ) = μm (t), and for any test function h, we have | R d h(y)μm (t, dy)| ≤
γ
γ
γ
h∞ μm (t, R d ) < ∞. Thus, from F (um )(t) = F (ρ)F (μm (t)), we deduce that
γ
γ
um (t) = ρ ∗ μm (t). On the other hand, for all test functions h,
Rd
=
F −1 (h)(x)
Cd
Rd
=
=
γ
h(y)μm (t, dy) =
γ
Rd
e−2πix·ξm (t,ω) exp
t
κm (x)F −1 (h)(x)dx =
γ
Rd
cm (ξm (s, ω), uγ (s, ξm (s, ω)))ds γm (dω) dx =
0
$
t
γ
F −1 (h(x))e−2πix·ξm (t,ω) dx e 0 cm (ξm (s),u (s,ξm (s)))ds γm (dω) =
Cd
Rd
$
t
γ
F −1 (h(x))e−2πix·ξm (t,ω) dx e 0 cm (ξm (s,ω),(ρ∗μ )(s,ξm (s,ω)))ds γm (dω) =
Cd
Rd
t
=
F −1 (κm )(x)h(x)dx =
Cd
h(ξm (t)) exp
cm (ξm (s, ω), (ρ ∗ μγ )(s, ξm (s, ω)))ds γm (dω),
0
thus, we proved that if (ξm (t), um ) solve (2.12) and (2.13) , then (ξm (t), μm ) solve
(2.10) and (2.11) .
To prove the converse statement, we assume that (ξm (t), μm ) solve (2.10) and
γ
γ
γ
(2.11). Setting um (t, y) = ρ ∗ κm (t), we verify that (2.13) holds. Since μm (t) is a
finite measure, we derive Eq. (2.13) choosing ρ(x − ·) for a test function.
γ
γ
Finally, note that from F (um )(t) = F (ρ)F (μm (t)), we have that if the Lebesgue
γ
d
measure of the set {x ∈ R : F(ρ)(x) = 0} is equal to zero, then F(μm (t)) =
γ
γ
γ
F(um (t ))
F(ρ) a.e. for t ∈ [0, T ]. Hence, μm is uniquely defined by um and vice versa. 28
Ya. I. Belopolskaya
Now, we describe connections between the stochastic system
ξm (t ) = ξ0m +
t
b(ξm (s), (ρ ∗ μ)(s, ξm (s)))ds +
0
t
Mm (ξm (s), ρ ∗ μ(s, ξm (s)))dwm(s),
0
(2.14)
γ
Rd
h(y)μm (t, dy) =
t
Cd
h(ξm (t, ω)) exp
cm (ξm (s, ω), (ρ ∗ μm )(s, ξm (s, ω)))ds γm (dω),
0
(2.15)
with
P {ξm (0) ∈ dy} = u0m (y)dy,
L(ξm ) = γm
and the PDE system (2.9).
We say that measures μm (t) ∈ M(R d ), m = 1, . . . , d1 are weak measure-valued
solutions of the system (2.9) with initial data μm (0, dy) = u0m (y)dy if for any
h ∈ C0∞ ([0, T ] × R d ) equalities
T
0
∂h 1
2
+ T rBm (y, v)∇ h + bm (y, v) · ∇h + cm (y, v)h dt+
μm (t, dy)
∂t
2
Rd
(2.16)
+
h(0, y)μm (0, dy) = 0
Rd
are valid.
Theorem 1 Assume C 2.1 holds and there exists a unique solution (ξm (t),
μm (t, dy)) of (2.14) and (2.15). Then, μ = (μ1 , . . . , μm , . . . , μd1 ) is the unique
weak measure-valued solution of the system (2.9).
Proof Consider processes ζ(t) = h(ξm (t))ηm (t), where h ∈ C0∞ (R d ) is a test
function and ξm (t), ηm (t) satisfy (2.14) and (2.15) and apply the Ito formula to
obtain
E[h(ξm (t))ηm (t)] = E[h(ξ0m )]+ E[h(ξm (s))cm (ξm (s), (ρ∗μm )(s, ξm (s)))ηm (s)]ds+
0t
+
t
E[∇h(ξm (s)) · bm (ξm (s), (ρ ∗ μm )(s, ξm (s)))ηm (s)]ds+
0
+
1
2
t
0
E[T rBm ∇ 2 h(ξm (s), (ρ ∗ μm )(s, ξm (s)))ηm (s)]ds.
A Probabilistic Interpretation of Conservation and Balance Laws
29
Substituting it into the right-hand side of (2.15), we get
Rd
h(y)μm (t, dy) =
+
t
Rd
h(x)μ0m (dx)+ E[h(ξm (s))cm (ξm (s), (ρ∗μm )(s, ξm (s)))ηm (s)]ds+
0t
E[∇h(ξm (s)) · bm (ξm (s), (ρ ∗ μm )(s, ξm (s)))ηm (s)]ds+
0
+
1
2
t
E[T rBm ∇ 2 h(ξm (s), (ρ ∗ μm )(s, ξm (s)))ηm (s)]ds.
0
3 Existence and Uniqueness of a Solution to (2.12) and (2.13)
Let for a fixed m the process ξm (t) be a canonical process on Cd = C([0, T ]; R d ),
ξm (t, ω) = ω(t), Bt (Cd ) = σ {ξm (s), 0 ≤ s ≤ t}, and γm = L(ξm ) be a distribution
of ξm . Denote by Pr (Cd ) the set of Borel probability measures on Cd with finite
moments up to the order r and by Cb (Cd ) the space of bounded continuous real
&1
valued functions on Cd . Let C = C([0, T ]; dm=1
Xm ) for Xm = R d .
d
Let the space P2 (R ) be equipped by the Wasserstein distance Wt of order 2
Wt2 (γm , γ̃m ) = infπ∈(γm ,γ̃m )
2
sup ξm (s, ω) − ξm (s, ω̃) π(dω, d ω̃))
Cd ×Cd 0≤s≤t
for all t ∈ [0, T ] where (γm , γ̃m ) denotes the set of Borel probability measures on
Cd ×Cd , π ∈ P(Cd ×Cd ) with fixed marginals π(F ×Cd ) = γm (F )andπ(Cd ×F ) =
γ̃m (F ), F ⊂ Cd .
We use notations B∞ (0, K) and B(0, K) for centered balls of radius K in spaces
L1 (R d ; R d1 ) and L∞&([0, τ ] × R d ; R d1 ), τ < T , respectively.
Denote by =
m , m = Cd and Fm = σ {ξm (s); 0 ≤ s ≤ t}. Let ξm (t)
be canonical processes on (m , Fm ) = C([0, T ]; R d ), that is, ξm : Cd → Cd and
ξm (t) = ωm (t), t ≥ 0, ωm ∈ Cd .
Consider the system
ξm (t) = ξ0m +
t
bm (ξm (s), uγ (s, ξm (s)))ds +
0
t
Mm (ξm (s), uγ (s, ξm (s)))dwm (s),
ξ 0 ∼ μ0 ,
0
(3.1)
γ
um (t, y) =
Cd
ρ(y − ξm (t, ω))ηm (t, ω)γm (dω).
(3.2)
30
Ya. I. Belopolskaya
In this section, we discuss the existence and uniqueness of a solution to this
system.
When cm ≡ 0, Eq. (3.2) can be treated as an equation linking the law of the
process ξm (t) and the function um . If the mollifier ρ = δ is the Dirac delta function,
then um (t) is the density of the marginal law of ξm (t). When cm = 0, then um (t) is
implicitly connected to the law γm of the whole path of ξm (·) and (3.2) associates a
probability law γm on Cd to a real valued function um (t).
Given a probability measure γm ∈ P(Cd ), we consider the equation
t
γ
γ
um (t, y) =
ρ(y − ξm (t, ω))exp
cm (ξm (s), u (s, ξm (s)))ds γm (dω),
Cd
0
(3.3)
where ξm is a canonical process.
Theorem 2 Assume that C 2.1 holds and sup ρ(x) ≤ K0 . Given probability measures γm ∈ P(Cd ), the system (3.2) admits a unique solution u =
(u1 , . . . , um , . . . , ud1 ).
Proof Denote by Z1 the linear space of real valued continuous processes
gm (t), m = 1, . . . , d1 , t ∈ [0, T ] defined on the canonical space Cd such that
gm ∞,1 = E γm [ sup |gm (s)|] =
sup |gm (s, ω)|γm (dω) < ∞.
0≤t ≤T1
Cd 0≤t ≤T1
Let Z1,d1 be the linear space of R d1 -valued continuous processes g defined on the
canonical space C such that
g∞,d1 = E γ [ sup g(s)] =
0≤t ≤T
sup g(s, ω)γ (dω) < ∞.
C 0≤t ≤T1
The spaces (C∞,1 , · ∞,1 ) and (C∞,d1 , · ∞,d1 ) are Banach spaces. For K ≥ 0,
denote by · K
∞,1 an equivalent norm in Z1
γm
−Kt
gm K
|gm (t)|].
∞,1 = E [ sup e
0≤t ≤T1
Denote by
Gm (t, ξm (ω), u(ξm (ω))) = exp{
t
cm (ξm (τ ), u(τ, ξm (τ )))dτ }.
0
γ
Given γm (t) ∈ P(Cd ), we denote by m : C1,d1 → C([0, T ] × R d × R d1 ; R) a map
defined by
γ
m (v)(t, y) =
Cd
ρ(y − ξm (t, ω))Gm (t, ξm (ω), v(ω))γm (dω)
(3.4)
A Probabilistic Interpretation of Conservation and Balance Laws
31
and by κ : C([0, T1 ] × R d × R d1 ; R) → C1,d1 a map given by
κ(v)(t, ω) = vm (t, ω(t)).
(3.5)
It should be mentioned that κ ◦ γ : C1,d1 → C1,d1 , that is, g = κ(γ (g)). Let us
prove the existence and uniqueness of a solution to the equation
u = (γ ◦ κ)(u)
(3.6)
which is equivalent to the system (3.2).
Assume first that the map κ ◦ γ has a fixed point g ∈ C1,d1 . Then, choose
γ
v = γ (g). Since g is a fixed point of the map κ ◦ γ , we have
g = κ(γ (g)),
(3.7)
hence, v γ is a solution to (3.6).
To prove uniqueness of a solution to (3.6), we assume on the contrary that there
exist two solutions v and g of this equation such that v = (γ ◦ κ)(v) and g =
(γ ◦ κ)(g). Set X = κ(v), Y = κ(g). Since v = γ (X), we have X = κ(v) =
κ(γ (X)). In a similar way, we obtain Y = κ(g) = κ(γ (Y )). Recall that X and Y
are fixed points of κ ◦ γ , and thus, the equality X = Y holds γ almost everywhere.
It remains to prove that the map κ ◦ γ has a unique fixed point X ∈ C1,d1 as it is
required.
Given a pair (X, Y ) ∈ C1,d1 × C1,d1 for any pair (t, y) ∈ [0, T ] × R d , we have
γ
γ
|m (X) − m (Y )|(t, y) =
=
Cd
ρ(y − ξm (t, ω))[Gm (t, ξm (ω), X(ω)) − Gm (t, ξm (ω), Y (ω))]γm (dω) ≤
≤ K0 etKc Lc E
t
t
X(s) − Y (s)ds ≤ K0 etKc Lc E
eKs e−Ks X(s) − Y (s)ds
0
0
≤ K0 et Kc Lc
t
+
eKs E
0
≤ K0 e
t Kc
,
sup e−τ K X(τ ) − Y (τ ) ds ≤
0≤τ ≤T
eKt − 1
Lc
K
,
+
sup
0≤τ ≤t
X − Y K
∞,1
.
32
Ya. I. Belopolskaya
γ
Next, we consider κ(m (X))(t)
γ
m (Y )(t, ξm (t)) and deduce
γ
γ
= m (X)(t, ξm (t)) and κ(m (Y ))(t) =
+
E
,
sup e
−Kt
0≤t ≤T1
γ
γ
|κ(m (X))(t) − κ(m (Y ))(t)|
=
,
+
=E
sup e
−Kt
0≤t ≤T1
γ
γ
|m (X)(t, ξm (t)) − m (Y )(t, ξm (t))
≤ K0 eKc T Lc
|≤
1
X − Y ∞,1 .
K
Taking sup in t ∈ [0, T ] and summing over m, we get
γ (X) − γ (Y ) ≤
d1
γ
γ
m (X) − m (Y ) ≤ d1 K0 eKc T Lc
m=1
1
X − Y ∞,1 .
K
Thus, we can choose K large enough, namely, K > d1 K0 et Kc Lc , and verify that
(κ ◦ γ ) is a contraction in (C1 , · K
∞,1 ). Hence, by fixed-point theorem, we obtain
that there exists a unique solution of Eq. (3.6).
We have proved that given a probability measure γm , there exists a unique
solution uγ of (3.6). Dependence of this solution on γm remains to be studied.
γ
Lemma 2 Let assumptions of Theorem 2 be valid. Then, for the solution um of
(3.6), an inequality
|uγ (t, y) − uγ̃ (t, ỹ)|2 ≤ L[y − ỹ2 +
d1
Wt2 (γm , γ̃m )]
(3.8)
m=1
holds for any couple γm , γ̃m ∈ P2 (Cd ) × P2 (Cd ) for all (t, y, ỹ) ∈ [0, T ] × Cd × Cd .
Proof Given (γm , γ̃m ) ∈ P2 (Cd ) × P2 (Cd ), we evaluate
γ
γ̃
γ
γ
γ
γ̃
um (t, y) − um (t, ỹ)2 ≤ 2[um (t, y) − um (t, ỹ)2 + um (t, ỹ) − um (t, ỹ)2 .
(3.9)
γ
Lipschitz continuity of um (t, y) with respect to y follows immediately from
Lipschitz continuity of the mollifier ρ and boundedness of the stochastic process
γ
Gm (t, ξm (ω), um (ω)). To estimate the second term on the right-hand side of (3.8),
A Probabilistic Interpretation of Conservation and Balance Laws
33
evaluate a difference
γ
γ̃
|um (t, ỹ) − um (t, ỹ)|2 =
−
Cd
u0m (ỹ − ξm (t, ω))Gm (t, ξm (ω), uγ (ξm (ω)))γ (dω)−
(3.10)
2
u0m (ỹ − ξm (t, ω̃))Gm (t, ξm (ω̃), u (ξm (ω̃)))γ̃ (d ω̃)| ≤
γ̃
Cd
≤
Cd ×Cd
|ρ(ỹ − ξm (t, ω))Gm (t, ξm (ω), uγ (ξm (ω)))−
−ρ(ỹ − ξm (t, ω̃))Gm (t, ξm (ω̃), uγ̃ (ξm (ω̃)))|2 π(dω, d ω̃)
for any π ∈ (γ , γ̃ ).
Keeping in mind Lipschitz continuity of ρ and properties of cm , we can verify
the estimate
|ρ(ỹ − ξm (t))Gm (t, y, z) − |ρ(ỹ − ξ̃m (t))Gm (t, ỹ, z̃)|2 ≤
≤ 2L2c e 2tKc ξm (t) − ξ̃m (t)2 + 4K02 L2c e 2tKc
t
[ξm (s) − ξ̃m (s)2 + z(s) − z̃(s)2 ]ds ≤
0
t
2
2
≤ C (1 + t) sup ξm (s) − ξ̃m (s) +
z(s) − z̃(s) ds .
s≤t
0
Combining the last inequalities and assuming that um (t, y) is Lipschitz continuous in y (which will be proved below), we get by replacing y and ỹ by ξm (t) and
ξ̃m (t)
γ
(3.11)
|um (t, ξm (t)) − uγ̃ (t, ξ̃m (t))|2 ≤
2
sup ξm (s) − ξ̃m (s)2 dπ(ω, ω̃)]+
≤ C[ξm (t) − ξ̃m (t) + (1 + t)
Cd ×Cd s≤t
t
+C
Cd ×Cd
|uγ̃ (s, ξm (s)) − uγ̃ (s, ξ̃m (s))|2 dsdπ(ω, ω̃)].
0
$
γ
γ̃
Set qm (s) = Cd ×Cd |um (s, ξm (s)) − um (s, ξ̃m (s))|2 π(dω, d ω̃). Integrating
(3.11) against π, we get
qm (t) ≤ C
t
d1
0 m=1
qm (s)ds + M(t)
Cd ×Cd
| sup ξm (s, ω) − ξm (s, ω̃)2 π(dω, d ω̃)
s≤t
34
Ya. I. Belopolskaya
for all t ∈ [0, T1 ]. Summing the last inequality in m and applying the Gronwall
1
the following inequality
lemma, we obtain for q(t) = dm=1
q(t) ≤ C
sup
d1
Cd ×Cd s≤t m=1
ξm (s, ω) − ξm (s, ω̃)2 π(dω, d ω̃).
Keeping in mind (3.10), we deduce
d1
+
γ
γ̃
|um (t, ỹ) − um (t, ỹ)|2 ≤ C
m=1
y − ỹ +
2
sup
d1
Cd ×Cd s≤t m=1
,
ξm (s, ω) − ξm (s, ω̃) π(dω, d ω̃) .
2
(3.12)
The above inequality holds for any π ∈ (γ , γ̃ ); thus, we can take infimum over
all π ∈ (γ , γ̃ ) that yields (3.8).
Now, we are ready to prove the following result.
Theorem 3 Assume that C 2.1 holds. Then, there exists a unique solution
(ξm (t), um (t, y)) of the system (3.1) and (3.2). The process ξm (t) ∈ R d is a
Markov process, and um (t, y) are bounded Lipschitz continuous in y ∈ R d real
valued functions.
Proof Let us fix the measures γm ∈ P2 (Cd ) considered above. We deduce from
γ
Theorem 2 and lemma 2 that there exists a unique solution um (t, y) of a system (3.4)
γ
and the functions um (t, y) are bounded and Lipschitz continuous both in y ∈ R d
and γm ∈ P2 (Cd ). This allows to assert that there exists a unique strong solution
γ
ξm (t) of the SDE (3.1) since coefficients of these equations are nonrandom and
satisfy conditions of classical theorem about existence and uniqueness of a strong
γ
solution to an SDE. Moreover, the processes ξm (t) are Markov processes. Applying
the Burkholder-Davis-Gundy theorem and Jensen inequality, we deduce that there
exists a positive constant C depending on constants in C 2.1 and T such that
γ
E[sups≤T1 ξm (t2 ] ≤ C(1 + E[ξ0m 2 ]). This results that the law L(ξm ) belongs
γ
to P2 (Cd ). Set m (γm ) = L(ξm ) and note that m : P2 (Cd ) → P2 (Cd ).
Our aim now is to prove that m is a contraction in P2 (Cd ) equipped with the
Wasserstein metric.
γ
γ
Let γm1 and γm2 belong to P2 (Cd ) and um1 andum2 be solutions of (3.2), associated
1
2
i
with γm and γm , respectively. Besides, let γm = L(ξmi ), where ξmi , i = 1, 2, are
solutions of
dξmi (t) = ξ0m +
t
0
bm (ξmi (s), uγi (s, ξmi (s)))ds +
t
0
i
Mm (ξmi (s), uγi (s, ξmi (s)))dwm
(s).
A Probabilistic Interpretation of Conservation and Balance Laws
35
From the Wasserstein metric definition, we deduce
|WT ((γm1 ), (γm2 ))|2 ≤ E[ sup ξm1 (t) − ξm2 (t)2 ].
(3.13)
0≤t ≤T
Keeping in mind estimates in C 2.1 and Lemma 2, we deduce that there exists a
positive constant C such that
+
E[ sup
0≤t≤τ
ξm1 (t ) − ξm2 (t )2] ≤ C
τ
E[ sup
0
0≤s≤t
ξm1 (s) − ξm2 (s)2 ]dt +
τ
0
,
Wt (γm1 , γm2 )dt
,
(3.14)
for any τ ∈ [0, T ].
By the Gronwall lemma, we obtain from (3.14)
E[ sup ξm1 (t) − ξm2 (t)2 ] ≤ eCT
0≤t ≤τ
τ
0
Wt (γm1 , γm2 )dt
and substituting it into the right-hand side of (3.12), we deduce
WT2 ((γm1 ), (γm2 )) ≤ T eCT WT2 (γm1 , γm2 ).
(3.15)
As a result, we get that is a contraction if T eCT < 1.
4 Existence and Uniqueness of a Stochastic System Solution
with Growing Coefficients
In this section, we omit the assumption that coefficients in (2.12) and (2.13) are
bounded and prove that there exists the unique solution ξm (t), um (t, y) of this
system such that um (t) are bounded over a certain interval [0, T1 ].
Let us preliminarily obtain some auxiliary results concerning the Cauchy problem solution of a linearized version of (2.1) and its stochastic counterparts.
Assume that v(t, y) ∈ R d1 is a given vector function with bounded Lipschitz
continuous in y ∈ R d positive components vm (t, y), m = 1, . . . d1 , and consider
the Cauchy problem
1
∂um
= T r∇ 2 (Bm (y, v)um ) + div(bm (y, v)um ) + cm (y, v)um ,
∂t
2
um (0, y) = u0 (y), m = 1, . . . , d1 .
(4.1)
36
Ya. I. Belopolskaya
We say that condition C 4.1 holds if there exist positive constants K0 , K, L0 ,
L, Lu,v , Lc , n0 , andn1 such that:
1. bm (y, v)2 + |Mm (y, v)|2 ≤ K[1 + v2 ], y ∈ R d , v ∈ R d1 .
2. bm (x, u) − bm (y, v)2 + |Mm (x, u) − Mm (y, v)|2 ≤ L[x − y2 + Lu,v u −
v2 ] x ∈ R d , where |Mm | = di,j =1 |M ij |2 is the Frobenius matrix norm.
3. supy u0m (y) ≤ K0 , u0m (y) − u0m (x) ≤ L0 x − y, x, y ∈ R d .
4. |cm (y, v)| ≤ n0 + n1 vand |cm (x, u) − cm (y, v)| ≤ Lc [x − y + Lu,v u −
v].
To construct a stochastic representation of a solution to (4.1), we consider SDEs
dξm (s) = bm (ξm (s), v(s, ξm (s)))ds + Mm (ξm (s), v(s, ξm (s)))dwm(s),
ξm (0) = x ∈ R d ,
(4.2)
where wm (t) ∈ R d are independent Wiener processes defined on a fixed probability
space (, F, P ). From general theory of SDEs, we have the following result [4].
Lemma 3 Assume that condition C 4.1 holds and vm (s, x) are positive functions,
such that
sup |vm (s, x)| ≤ Kv (s), |vm (s, x) − vm (s, y)| ≤ Lv (s)x − y,
x
where Kv (s) and Lv (s) are positive functions bounded over an interval [0, T1 ].
Then, there exist unique solutions ξm (t) ∈ R d of (4.2), t ∈ [0, T1 ] possessing the
Markov property. In addition, processes
ηm (t) = exp
t
(4.3)
cm (ξm (s), v(s, ξm (s)))ds
0
$t
satisfy estimates Eηm (t)2 ≤ exp{ 0 [n0 + n1 ]Kv (s)ds}.
For the Markov process ξm (t) satisfying (4.2), we denote by Pmv (0, x, t, dy) =
P {ξm (t) ∈ dy|ξm (0) = x} its transition probability.
We say that measures μm (t) ∈ P(R d ) are mild measure-valued solutions of (4.1)
with initial data μm (0, dy) = u0m (dy) if for any hm ∈ C0∞ (R d ), t ∈ [0, T ]
Rd
h(y)μm (t, dy) =
+
Rd
Rd
Pmv (0, x, t, dy)μm (0, dx)+
t
0
h(y)
Rd
Pmv (θ, zm , t, dy)cm (z, v(θ, , z))μm (θ, dz)dθ
.
(4.4)
A Probabilistic Interpretation of Conservation and Balance Laws
37
Lemma 4 Let vm be bounded Lipschitz continuous positive functions, m =
1, . . . , d1 , and condition C 4.1 holds. Then, the process ξm (t) satisfying (4.2) and
the process ηm (t) of the form (4.3) together give rise to measures μm (t) such that
Rd
h(y)μm (t, dy) = E [h(ξm (t))ηm (t)]
(4.5)
holds for any h ∈ C0∞ (R d ). In addition, measures μm (t, dy) are unique measurevalued mild solutions of the Cauchy problem (4.1).
Proof Since lemma conditions ensure the existence and uniqueness of processes
ξm (t) and ηm (t) satisfying (4.2) and (4.3), we can observe that for each fixed m,
the expression lm (h) = E[h(ξm (t))ηm (t)] is a bounded linear functional over the
space C0∞ (R d ). Approximating bounded functions on R d by C ∞ functions, we can
define a probability measure μm (t) considering lm (h) as a bounded linear functional
over Cb (R d ) which by the Riesz theorem defines a unique measure μm (t) on R d
satisfying (4.5).
To prove that the measure μm (t) defined by (4.5) is a mild measure-valued
solution of (4.1), we note that properties of the conditional expectations allow to
rewrite the right-hand side of (4.5) in the form
E [h(ξm (t))ηm (t)] = E[h(ξm (t))]+
t
+E
(4.6)
cm (ξm (s), v(s, ξm (s)))ηm (s)E[h(ξm (t)|ξm (s))]ds =
0
=
+E
t Rd
Rd
h(y)Pmv (0, x, t, dy)u0m(dx)+
Rd
0
Rd
h(y)Pmv (s, ξm (s), t, dy)cm (z, v(s, ξm (s)))ηm (s)ds
.
$
Since R d h(y)Pmv (s, z, t, dy)cm (z, v(s, , z)) is a bounded function, we choose it for
a test function and obtain an equality
t E
0
Rd
=
Rd
h(y)Pmv (s, ξm (s), t, dy)cm (z, v(s, ξm (s)))ηm (s)ds
t
0
Rd
=
Pmv (s, zm , t, dy)cm (z, v(s, , z))μm (s, dz)ds.
Substituting this equality into (4.6), we deduce that the right-hand side of (4.5)
coincides with the right-hand side of (4.4).
It remains to prove that the measure-valued mild solution to (4.1) is unique.
Assume on the contrary that there exist two measure-valued mild solutions
38
Ya. I. Belopolskaya
μm (t)andνm (t) ∈ P(R d ) of (4.1) and set κm (t) = μm (t) − νm (t). Since cm (x, u)
are bounded, we deduce that κm (t)T V < ∞. In addition to ∀h ∈ Cb (R d ), we
have
t
h(y)κm (t, dy) =
cm (z, v(θ, z))κm (θ, dz)
h(y)Pmv (θ, z, t, dy)dθ.
Rd
0
Rd
Rd
and hence, evaluating supremum over all h such that h∞ = supx |h(x)| ≤ 1, we
obtain
t
κm (θ, ·)T V dθ.
κm (t, ·)T V ≤ Kc
0
Finally, by the Gronwall lemma, we deduce that κm (t, ·) = 0 and hence μm (t) =
γm (t).
To construct a solution of the systems (2.12) and (2.13), we consider its
successive approximations
ξm(0) = x,
u(1)
m (t, y) = u0m (y),
dξm(1) (t ) = −bm (ξm(1) (t ), u0 (ξm(1) (t )))dt + Mm (ξm(1) (t ), u0 (ξm(1) (t )))dwm(t ),
ξm(1) (0) = ξ0m ,
t
(1)
(1)
(1)
u(2)
(t,
y)
=
E
u
(y
−
ξ
(t))
exp
c
(ξ
(t),
u
(ξ
(τ
)))dτ
,
0m
m m
0 m
m
m
0
and
dξm(k) (t) = −bm (ξm(k) (t), u(k) (t, ξm(k) (t)))dt + Mm (ξm(k) (t), u(k) (t, ξm(k) (t)))dwm (t),
ξm(k) (0) = ξ0m ,
(4.7)
t
(k+1)
(k)
(k)
(k)
u(k+1)
(t,
y)
=
E
u
(y
−
ξ
(t))
exp
c
(ξ
(τ
),
u
(τ,
ξ
(τ
)))dτ
0m
m
m
m
m
m
0
(4.8)
and prove that (ξmk (t), uk+1
m (t, y)) converge as k → ∞ to a solution of the
systems (2.12) and (2.13) if C 4.1 holds. To prove the convergence of successive
approximations (4.7) and (4.8) to processes ξm (t) and bounded Lipschitz continuous
functions um (t, y) as k → ∞ on some time interval [0, T1 ], we study first a
linearized stochastic system.
Denote by Kv (t) = supy |v(t, y)| = v(t)∞ and by K0m = u0m ∞ .
Given a Lipschitz continuous bounded function v(t) ∈ R d1 , we consider a system
including (4.2) and an equation
t
um (t, y) = E ρ(y − ξm (t)) exp
cm (ξm (τ ), v(τ, ξm (τ )))dτ .
0
(4.9)
A Probabilistic Interpretation of Conservation and Balance Laws
39
Lemma 5 Assume that C 4.1 holds. Then, functions um (t) defined by (4.7) possess
the following properties: there exists an interval [0, T1 ], such that
T1 <
1
n0
log 1 +
n0
n1 K0
and a positive bounded function α(t) defined on [0, T1 ] such that
Ku (t) =
d1
um (t)∞ ≤ α(t)
d1
if
m=1
vm (t)∞ = Kv (t) ≤ α(t)
m=1
for t ∈ [0, T1 ]. In addition, functions um (t, y) are Lipschitz continuous if u0m are
Lipschitz continuous.
Proof Due to general results of the SDE theory condition, C 4.1 ensures the
existence and uniqueness of the process ξm (t) satisfying (4.2). From (4.9), we
deduce that
+
Kum (t) ≤ K0m E exp
t
[cm +
0
Setting K0 = supm K0m and
(4.10) in m, we get
d1
,
cmk vk (τ, ξm (τ ))]dτ
.
(4.10)
k=1
Ku (τ ) =
Ku (t) ≤ d1 K0 exp
t
d1
k=1 Kuk (τ ) and summing inequality
[n0 + n1
0
d1
Kvk (τ )]dτ .
k=1
Consider an integral equation
α(t) = d1 K0 exp
t
[n0 + n1 α(τ )]dτ
0
and note that its solution satisfies the Cauchy problem for an ordinary differential
equation
dα(t)
= [n0 + n1 α(t)]α(t),
dt
α(0) = d1 K0 .
(4.11)
A solution to this Cauchy problem has the form
α(t) =
d1 K0 n0
.
n0 + K0 n1 − K0 n1 en0 t
(4.12)
40
Ya. I. Belopolskaya
Thus, an estimate Ku (t) ≤ α(t) holds if Kv (t) ≤ α(t), where the function
α(t) given
by (4.11)
is a bounded function over an interval [0, T1 ] with T1 <
n0
1
.
log
1
+
n0
K0 n1
Given a fixed γm which is a law of the process ξm satisfying (4.2), we obtain
|um (t, y) − um (t, y1 )| =
(4.13)
t
= E γm |u0m (y − ξm (t)) − u0m (y1 − ξm (t))| exp
cm (ξm (τ ), u(τ, ξm (τ )))dτ
≤
0
≤ L0 eKc T y − y1 .
Now, we can prove a result similar to the corresponding result for stochastic
systems associated with the system of nonlinear backward Kolmogorov equations
[4].
Theorem 4 Assume C 4.1 holds. Then, there exist the interval [0, T1 ] defined in
Lemma 5 and a unique solution (ξm (t), um (t, y)) of (2.12) and (2.13) defined on
the interval [0, T1 ] such that ξm (t) ∈ R d is a Markov process and um (t, y) is a
bounded Lipschitz continuous ( in y ∈ R d ) function.
Proof Obviously, relation (2.13) may be treated as a functional equation of the
form u(t, y) = (t, y, u), where the mapping acts in Cb ([0, T1 ] × R d ; R d1 ).
Successive approximations u(k) (t, y) = (t, y, u(k−1) ) are defined by (4.7)–(4.8).
Due to Lemma 5, we know that uk (t, y)m are uniformly bounded over the interval
[0, T1 ] and equicontinuous since Lipschitz constant for uk (t, y) in spatial argument
does not depend on k. Denote by nk (t, y) = u(k+1) (t, y) − u(k) (t, y). From
Lemma 3, we know that
n (t, y) ≤ K
k
t
0
sup nk−1 (s, y)ds
y
and hence κ k (t) = supy nk (t, y) satisfies the estimate
κ (t) ≤
k
t
t
...
0
0
u(2) (τ1 ) − u0 dτ1 . . . dτn ≤
Cn
n!
since all u(k) (τ ) are bounded. Here, C is a positive constant depending on constants
in C 2.1 and T1 . Thus, we have from Arzela-Ascoli theorem that there exists a
function u(t, y) = limk→∞ u(k) (t, y) and u(t, y) is Lipschitz continuous in y and
bounded on [0, T1 ].
A Probabilistic Interpretation of Conservation and Balance Laws
41
To prove that the limit function u(t, y) is unique, we assume on the contrary that
there exist two solutions of (3.9), namely, u(t, y) and g(t, y). Then, we have
u(t) − g(t)∞ ≤ const
t
u(s) − g(s)ds
0
which implies that u(t) − g(t)∞ = 0 by the Gronwall lemma. Hence, the solution
u(t, y) is unique.
As soon as we have proved that under condition 4.1, functions um (t, y) satisfying
(2.13) are bounded for t ∈ [0, T1 ], we deduce that coefficients cm (y, u) of the form
(2.3) are bounded and condition 2.1 holds and the results of the previous sections
are applicable.
Thus, we can state the final statement.
Theorem 5 Let condition C 4.1 hold and ρ be a mollifier. Then, there exists
a unique solution (ξm (t), um (t, y)) of the systems (2.12)–(2.13) and um (t, y) =
γ
ρ ∗ μm , m = 1, . . . , d1 , where measures μm (t, dy) satisfying (2.11) define a unique
mild measure-valued solution of the Cauchy problem for the system (2.9).
Acknowledgments Funded [in part] by Sirius University.
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Hierarchical Schrödinger Type
Operators: The Case of Locally Bounded
Potentials
Alexander Bendikov, Alexander Grigor’yan, and Stanislav Molchanov
Abstract The goal of this paper is the spectral analysis of the Schrödinger type
operator H = L + V , the perturbation of the Taibleson-Vladimirov multiplier
L = Dα by a potential V . Assuming that V belongs to a certain class of potentials,
we show that the discrete part of the spectrum of H may contain negative energies; it
also appears in the spectral gaps of L. We will split the spectrum of H in two parts:
high-energy part containing eigenvalues, which correspond to the eigenfunctions
located on the support of the potential V , and low-energy part, which lies in
the spectrum of certain bounded Schrödinger type operator acting on the Dyson
hierarchical lattice.
We pay special attention to the class of sparse potentials. In this case, we obtain
precise spectral asymptotics for H , provided the sequence of distances between
locations tends to infinity fast enough.
We also obtain certain results concerning localization theory for H subject to
(non-ergodic) random potential V . Examples illustrate our approach.
A. Bendikov was supported by the Polish National Science center, grant 2015/17/B/ST1/00062
and by SFB 1283 of the German Research Council. A. Grigor’yan was supported by SFB 1283 of
the German Research Council. S. Molchanov was supported by the Russian Science Foundation
(Projects: 20-11-20119 and 17-11-01098).
A. Bendikov ()
Institute of Mathematics, Wroclaw University, Wroclaw, Poland
e-mail: bendikov@math.uni.wroc.pl
A. Grigor’yan
Department of Mathematics, University of Bielefeld, Bielefeld, Germany
Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russian Federation
e-mail: grigor@math.uni-bielefeld.de
S. Molchanov
Department of Mathematics, University of North Carolina, Charlotte, NC, USA
National Research University Higher School of Economics, Moscow, Russian Federation
e-mail: smolchan@uncc.edu
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_3
43
44
A. Bendikov et al.
Keywords Ultrametric space · Symmetric Markov semigroup · Symmetric
Markov generator · Taibleson-Vladimirov operator · Schrödinger-type operators ·
Spectral asymptotics · Localization theory
2010 Mathematics Subject Classification Primary: 35P05, 35K08. Secondary:
28A80, 60J35
1 Introduction
The spectral theory of nested fractals similar to the Sierpinski gasket, i.e., the
spectral theory of the corresponding Laplacians, is well understood. It has several
important features: Cantor-like structure of the essential spectrum and, as result, the
large number of spectral gaps, presence of infinite number of eigenvalues each of
which has infinite multiplicity and compactly supported eigenstates, non-regularly
varying at infinity heat kernels which contain an oscillating in log t scale terms, etc.;
see [13, 17] and [8].
The spectral properties mentioned above occur in the very precise form for the
Taibleson-Vladimirov Laplacian Dα , the operator of fractional derivative of order
α. This operator can be introduced in several different forms (say, as L2 -multiplier
in the p-adic analysis setting, see [40]), but we select the geometric approach [4–
6, 14, 28, 29] and [7].
1.1 The Dyson Hierarchical Model
Let us fix an integer p ≥ 2 and consider the family {r : r ∈ Z} of partitions of the
set X = [0, +∞[ such that each r consists of all intervals I = [kpr , (k + 1)pr [,
k = 0, 1, . . .. We call r the rank of the partition r (respectively, the rank of the
interval I ∈ r ). Each interval of rank r is the union of p disjoint intervals of rank
(r − 1). Each point x ∈ X belongs to a certain interval Ir (x) of rank r, and the
intersection of all intervals Ir (x) is {x}.
Definition 1.1 Let B be the family of all intervals [kpr , (k+1)pr [. The hierarchical
distance d(x, y) is defined as infimum of lengths |I | of the intervals I ∈ B, which
contain both x and y.
It is easy to see that the function (x, y) → d(x, y) is nondegenerate, symmetric,
and for arbitrary x, y, and z:
d(x, y) ≤ max{d(x, z), d(z, y)},
i.e., d(x, y) is an ultrametric on X. It has the following properties:
• The ultrametric d(x, y) majorizes the Euclidean metric |x − y|, but these two
metrics are not equivalent. Indeed, by the very definition, d(x, y) ≥ |x − y| for
all x, y ∈ X, whereas d(1 − ε, 1) = p for all 0 < ε < 1 (Fig. 1).
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
45
Fig. 1 Comparison of two metrics: d (x, y) ≥ |x − y|
• (X, d) is a complete locally compact non-compact and separable metric space. In
this metric space, the set of all open balls coincides with the set of all intervals
I ∈ B. The next property says that (X, d) is a totally disconnected metric space.1
• Each open ball B in (X, d) is a closed compact set, and each point x of B can be
regarded as its center; any two balls either do not intersect or one is a subset of
another etc.
• The Borel σ -algebra generated by the ultrametric d(x, y) coincides with the
classical Borel σ -algebra generated by the Euclidean metric.
Definition 1.2 Let us fix a parameter κ ∈]0, 1[. Let m be the Lebesgue measure. In
the hierarchical Laplacian L, we introduce following [31] as a linear combination
of “elementary Laplacians”:
⎛
(Lf )(x) =
⎜
(1 − κ)κ r−1 ⎝f (x) −
r∈Z
1
|Ir (x)|
⎞
⎟
f dm⎠ .
(1.1)
Ir (x)
The series in (1.1) diverges in general, but it is finite and belongs to L2 (X, m) for
any f ∈ L2 (X, m) which takes constant values on intervals of any fixed rank r.
1 In particular, (X, d) is homeomorphic to the punctured Cantor set {0, 1}ℵ0 \{o}, see a survey on
totally disconnected metric spaces in [6, Section 1].
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A. Bendikov et al.
The operator L admits a complete system of compactly supported eigenfunctions. Indeed, let I be an interval of rank r and I1 , I2 , . . . , Ip be its subintervals of
rank r − 1. Set λ(r) = κ r−1 and consider p functions
fIi =
1I
1Ii
−
, i = 1, 2, . . . , p.
|Ii | |I |
Each function fIi belongs to the domain of the operator L and satisfies the equation
LfIi = λ(r)fIi .
The eigenspace H(I ) = span{fIi } has dimension p − 1 because
eigenspaces H(I ) and H(I ) are orthogonal provided I = I , and
L2 (X, m) =
/
r∈Z
⎛
⎝
/
i fIi = 0. The
⎞
H(I )⎠ .
I ∈r
In particular, L is an essentially self-adjoint operator having a pure point spectrum.
Clearly, each eigenvalue λ(r) = κ r−1 has infinite multiplicity, whence Spec(L)
coincides with its essential part Specess (L).
We shall see in Example 3.10 that writing κ = p−α the operator L can be
identified with the Taibleson-Vladimirov operator Dα , the operator of fractional
derivative of order α defined as L2 -multiplier in the p-adic analysis setting [22, 42].
According to [5], the operator L can be represented as a hypersingular integral
operator acting in L2 (0, ∞):
∞
Lf (x) =
(f (x) − f (y)) J (x, y)dy
0
where
J (x, y) =
1
κ −1 − 1
·
.
1 − κ/p d(x, y)1+α
The Markov semigroup (e−t L )t >0 is symmetric and admits a continuous heat kernel
p(t, x, y).2 The function p(t, x, y) can be estimated as follows:
p(t, x, y) t
.3
[t 1/α + d(x, y)]1+α
(1.2)
2 The function (x, y) → p(t, x, y) is continuous (and even locally Lipschitz continuous) w.r.t. the
ultrametric d(x, y), but it is discontinuous w.r.t. the Euclidean metric |x − y|.
3 We write f g if the ratio f/g is bounded from above and from below by positive constants for
a specified range of variables. We write f ∼ g if the ratio f/g tends to identity.
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
47
The function p(t) := p(t, x, x) does not depend on x. By [31, Proposition 2.3], it
can be represented in the form
p(t) = t −1/α A(logp t),
(1.3)
where A(τ ) is a continuous nonconstant α-periodic function; see also an extended
version of this result in [8]. In particular, in contrary to the classical case (symmetric
stable densities), the function t → p(t) does not vary regularly.
There are already several publications on the hierarchical Laplacian acting on a
general ultrametric measure space (X, d, m); see [1, 3–7, 28, 29]. By the general
theory developed in [4, 5] and [6], any hierarchical Laplacian L acts in L2 (X, m) as
essentially self-adjoint operator having a pure point spectrum. This operator can be
represented in the form
(f (x) − f (y))J (x, y)dm(y).
Lf (x) =
(1.4)
X
The Markov semigroup (e−t L )t >0 admits with respect to m a continuous transition
density p(t, x, y). It turns out that in terms of certain (intrinsically related to L)
ultrametric d∗ ,
1/d∗ (x,y)
J (x, y) =
N(x, τ )dτ,
(1.5)
N(x, τ ) exp(−tτ )dτ,
(1.6)
0
1/d∗ (x,y)
p(t, x, y) = t
0
where N(x, τ ) is the so-called spectral function related to L (which will be defined
later).
1.2 Outline
Let us describe the main body of the paper. In Sect. 2, we introduce the notion of
homogeneous hierarchical Laplacian L and list its basic properties: the spectrum
of the operator L is pure point; all eigenvalues of L have infinite multiplicity and
compactly supported eigenfunctions; the heat kernel p(t, x, y) exists, and it is a
continuous function having certain asymptotic properties; etc. As a special example,
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A. Bendikov et al.
we consider the case X = Qp , the ring of p-adic numbers endowed with its standard
ultrametric d(x, y) = |x − y|p , and the normed Haar measure m. The hierarchical
Laplacian L in our example coincides with the Taibleson-Vladimirov operator Dα ,
the operator of fractional derivative of order α; see [40, 42] and [22]. The most
complete source for the basic definitions and facts related to the p-adic analysis is
[21] and [39].
The Schrödinger type operator H = L + V with hierarchical Laplacian L
was studied in [11, 15, 26, 27, 29, 31, 32] (the hierarchical lattice of Dyson) and
in [22, 41, 42] (the field of p-adic numbers). In the next sections, we consider
the Schrödinger type operator acting on a homogeneous
ultrametric space X. We
assume that the potential V is of the form V =
σi 1Bi , where Bi are balls
which belong to a fixed horocycle H (i.e., all Bi have the same diameter). The
main aim here is to study the set Spec(H ). Under certain assumptions on V (e.g.,
V (x) → 0 at infinity , etc.), we conclude that the set Spec(H ) is pure point
(with possibly infinite number of limit points). We split the set Spec(H ) in two
disjoint parts: the first part consists of the point λ = 0 and the eigenvalues of
the operator L which correspond to the horocycle H (with compactly supported
eigenfunctions) and the second part is the closure of a countably infinite set of eigenvalues of the operator H (with non-compactly supported eigenfunctions).
In the case of sparse potential V , i.e., when d(Bi , Bj ) → ∞ fast enough, we
specify the structure of the set . In this connection, we would like to mention
here pioneering works of S. Molchanov [29], D. Krutikov [24, 25], and A. N.
Kochubei [22].
In the last section, we consider the potential V of the form V =
σi (ω)1Bi ,
where σi (ω), ω ∈ (, , P ) are i.i.d. random variables, and embark on the
localization theory. More precisely, we show that if the sequence of (nonrandom)
distances d(Bi , Bj ) between locations tends to infinity fast enough, then the
spectrum of H is pure point for P -a.a. ω ∈ .
In the case when X is discrete, L is the Dyson Laplacian, Bi are Singletons, and
V is ergodic; the localization theorem appeared first in the paper of Molchanov [29]
(σi (ω) are Cauchy random variables) and later (under more general assumptions on
σi (ω)) in the papers of Kritchevski [27] and [26]. The proof of this theorem is based
on the self-similarity of H . This approach is not applicable to the case of (random)
sparse potentials.
The proof of the localization theorem for (random) sparse potentials presented
in this paper is based on the abstract form of Simon-Wolff criterion [38] for pure
point spectrum, the technique of fractional moments, the decoupling lemma of
Molchanov, and Borel-Cantelli type arguments; see [1, 28].
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
49
2 Preliminaries
2.1 Homogeneous Ultrametric Space
Let (X, d) be an ultrametric space. Recall that a metric d is called an ultrametric if
it satisfies the ultrametric inequality
d(x, y) ≤ max{d(x, z), d(z, y)},
which is stronger than the usual triangle inequality. Henceforth, we assume that the
ultrametric space (X, d) is separable, non-compact, and proper, that is, each d-ball
is a compact set.
For any x ∈ X and r ≥ 0, let Br (x) = {y ∈ X : d (x, y) ≤ r} be a closed ball.
The basic consequence of the ultrametric property is that Br (x) is an open set for
any r > 0. Moreover, each point y ∈ Br (x) can be regarded as its center; any two
balls of the same radius are either disjoint or identical; etc. See a survey part in paper
[6, Section 1] and references therein.
To any ultrametric space (X, d), one can associate in a standard fashion a tree T.
The vertices of the tree are metric balls; the boundary ∂T can be identified with the
one-point compactification X ∪ { }. We refer to [6, Section 1] for a treatment of
the association between ultrametric space (X, d) and the tree T of its metric balls.
Definition 2.1 Let m be a Radon measure on X. The triple (X, d, m) we call a
homogeneous ultrametric measure space if the group of isometries of (X, d) acts
transitively on X and if the measure m is invariant w.r.t. the action of this group.
The following remarkable result is due to M. Del Muto and A. Figà-Talamanca;
see paper [12, Section 2].
Theorem 2.2 Any homogeneous ultrametric measure space (X, d, m) can be identified with a certain locally compact Abelian group G equipped with a translation
invariant ultrametric d and Haar measure m.
For example, the set X = [0, +∞[ equipped with the ultrametric structure
generated by p-adic intervals can be identified with Qp , the ring of p-adic numbers.
The identification in Theorem 2.2 is not unique. One possible way to define the
identification is to choose the sequence {an } of forward degrees associated with the
tree of balls T. This sequence is two-sided if X is non-compact and perfect (has no
isolated points), it is one-sided if X is compact and perfect, or if X is discrete. In the
first case, we identify X with a , the ring of a-adic numbers; in the 2nd case with
a ⊂ a , the ring of a-adic integers; and in the 3rd case with the discrete group
[a : a ] ≃ Za1 ⊕ Za2 ⊕ . . ., the weak sum of cyclic groups Zan (i.e., the set of
all eventually finite sequences z = (z1 , z2 , . . . , zl , 0, 0, . . .), ∃l = l(z)). We refer
the reader to [18] for the comprehensive treatment of the special groups a , a and
Za1 ⊕ Za2 ⊕ . . ..
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A. Bendikov et al.
2.2 Homogeneous Hierarchical Laplacian
Let (X, d, m) be a non-compact homogeneous ultrametric measure space. Let B be
the set of all open balls, B (x) ⊂ B the set of all balls centered at x, and C : B →
(0, ∞) a function satisfying the following conditions:
1. m(A) = m(B)
⇒ C(A) = C(B).
2. λ(B) :=
C(T ) < ∞.
T ∈B:B⊆T
3. supB∈B(x) λ(B) = ∞ for any non-isolated x.
The class of functions C(B) satisfying these conditions is rich enough. For
example, let us fix α > 0, and for any two nearest neighboring balls B ⊂ B ,
set
C(B) = m(B)−α − m(B )−α
and then
λ(B) = m(B)−α .
Definition 2.3 Let C : B → (0, ∞) be as above; we define the homogeneous
hierarchical Laplacian L pointwise as
⎛
⎞
1
Lf (x) :=
C(B) ⎝f (x) −
f dm⎠ .
(2.1)
m(B)
B∈B(x)
B
Let D be the set of all compactly supported locally constant functions.4 The
series in (2.1) diverges in general ,but for f ∈ D, it belongs to C∞ (X) ∩ L2 (X, m).
Moreover, the operator L admits a complete in L2 (X, m) system of eigenfunctions
fB =
1B
1B
−
,
m(B) m(B )
(2.2)
where the couple B ⊂ B runs over all nearest neighboring balls. The eigenvalue
corresponding to fB is the number λ(B ) defined at condition 2.
In particular, we conclude that L : D → L2 (X, m) is an essentially self-adjoint
operator. Each eigenvalue λ(B) has infinite multiplicity, so Spec(L) is pure point
and coincides with its essential part.
4 The set D is a dense subset in each of the Banach spaces C
∞ (X) and L
p (X, m), 1 ≤ p < ∞.
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
51
The intrinsic ultrametric d∗ (x, y) is defined as follows:
d∗ (x, y) :=
0
when x = y
,
1/λ(x y) when x = y
(2.3)
where x y is the minimal ball containing both x and y. In particular, for any open
ball B, we have
λ(B) =
1
.
diam∗ (B)
(2.4)
The spectral function τ → N(τ ), see Eq. (1.5), is defined as a left-continuous stepfunction having jumps at the points λ(B), and
N(λ(B)) = 1/m(B).
The volume function V (r) is defined by setting V (r) = m(B), where the ball B has
d∗ -radius r. It is easy to see that
N(τ ) = 1/V (1/τ ).
(2.5)
The Markov semigroup Pt = e−t L admits a continuous density p(t, x, y) with
respect to m; we call it the heat kernel. The function p(t, x, y) can be represented
in the form given by Eq. (1.6). Respectively, the Markov generator L admits the
representation given by Eqs. (1.4) and (1.5).
The resolvent operator (L + λI)−1 , λ > 0, admits a continuous strictly positive
kernel R(λ, x, y) with respect to the measure m. The resolvent operator is well
defined for λ = 0, i.e., the Markov semigroup (Pt )t >0 is transient, if and only if for
some (equivalently, for all) x ∈ X the function τ → 1/V (τ ) is integrable at ∞. Its
kernel R(x, y) := R(0, x, y), called also the Green function, is of the form
+∞
dτ
,r = d∗ (x, y).
V (τ )
R(x, y) =
(2.6)
r
Under certain Tauberian conditions, Eq. (2.6) takes the form
R(x, y) r
,r = d∗ (x, y).
V (r)
For all these facts, we refer the reader to [4, 5], and [6].
(2.7)
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A. Bendikov et al.
2.3 Subordination
Let : R+ → R+ be an increasing homeomorphism. For any two nearest
neighboring balls B ⊂ B , we define
C(B) = (1/m(B)) − 1/m(B ) .
(2.8)
The following properties hold true:
(i) λ(B) = (1/m(B)). In particular, the hierarchical Laplacians L and LI d
are related by the equation L = (LI d ).5
(ii) d∗ (x, y) = 1/ (1/m(x y)).
(iii) V (r) ≤ 1/−1 (1/r). Moreover, V (r) 1/−1 (1/r) whenever both and
−1 are doubling and m(B ) ≤ Cm(B) for some constant C > 0 and all
neighboring balls B ⊂ B . In turn, this yields the heat kernel estimates
1 −1
p(t, x, y) t · min
t
1
1
1
,
,
t
m(x y)
m(x y)
(2.9)
2.4 L2 -Multipliers
As a special case of the general construction, consider X = Qp , the ring of padic numbers equipped with its standard ultrametric d(x, y) = |x − y|p . Remind
that the ultrametric space (Qp , d) and the ultrametric space ([0, ∞), d) with nonEuclidean d (the Dyson’s model) are isometric.
Let m be the normed Haar measure on the Abelian group Qp and F : f → f0the
Fourier transform acting in L2 (Qp , m). It is known, see [22, 39, 42], that F : D → D
is a bijection.
Let : R+ → R+ be an increasing homeomorphism. The self-adjoint operator
(D) we define as L2 -multiplier, that is,
(ξ ) = (|ξ |p )f0(ξ ), ξ ∈ Qp .
(D)f
(2.10)
By [5, Theorem 3.1], (D) is a homogeneous hierarchical Laplacian. The eigenvalues λ(B) of the operator (D) are of the form
λ(B) = p
.
m(B)
5 In the case (τ ) is a Bernstein function the association L
well-known Bochner’s subordination theory [16].
(2.11)
= (LI d ) has been studied in the
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
53
Let p(t, x, y) be the heat kernel associated with the operator (D). Assume that
both and −1 are doubling; Then, Eq. (2.9) applies. Since for any x, y ∈ Qp ,
m(x y) = |x − y|p , we obtain
1 −1
p(t, x, y) t · min
t
1
1
1
.
,
|x − y|p
|x − y|p
t
(2.12)
The Taibleson-Vladimirov operator Dα is L2 -multiplier; it can be written as a
hypersingular integral operator
Dα f (x) =
1
p (−α)
f (y) − f (x)
Qp
|y − x|1+α
p
dm(y),
(2.13)
where p (z) = (1 − pz−1 )(1 − p−z )−1 is the p-adic gamma function; see [42,
VIII.2].
The heat kernel pα (t, x, y) of the operator Dα admits two-sided bounds
pα (t, x, y) t
.
(t 1/α + |x − y|p )1+α
(2.14)
In particular, the Markov semigroup (e−t D )t >0 is transient if and only if α < 1. In
the transient case, the Green function Rα (x, y) is of the form
α
Rα (x, y) =
1
1
.
p (α) |x − y|1−α
p
(2.15)
For all facts listed above, we refer the reader to [4, 5] and [6].
2.5 The Symbol of the Hierarchical Laplacian
Identifying X with a locally compact Abelian group, we can regard −L as an
isotropic Lévy generator. By (1.4), the operator L on D takes the form
(f (x) − f (y))J (x − y)dm(y),
Lf (x) =
(2.16)
X
or equivalently, in terms of the Fourier transform,
1 (θ ) = 0
0
Lf
L(θ ) · f0(θ ), θ ∈ X,
(2.17)
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A. Bendikov et al.
1p can be identified with Qp ) and
0 is the dual Abelian group (e.g., Q
where X
0 )=
L(θ
[1 − Re $h, θ %]J (h)dm(h).
(2.18)
X
The function 0
L(θ ) ≥ 0, the symbol of the Lévy generator −L, is a continuous
negative definite function [10]. In particular, the function 0
L(θ ) is subadditive. By
the subordination property [5, Theorem 3.1], the function 0
L(θ )2 is the symbol of
2
0
symmetric Lévy generator −L , so the function L(θ ) = 0
L(θ )2 is subadditive as
well, i.e., it satisfies the triangle inequality
0
L(θ1 + θ2 ) ≤ 0
L(θ1 ) + 0
L(θ2 ).
(2.19)
Since −L is an isotropic Lévy generator [5, Sec. 5.2], a stronger property holds true.
Theorem 2.4 The function 0
L(θ ) satisfies the ultrametric inequality
0
0 2 )}.
L(θ1 + θ2 ) ≤ max{0
L(θ1 ), L(θ
(2.20)
Proof In order to simplify notation, we assume that X = Qp , the ring of p-adic
numbers. Let B ⊂ B be two nearest neighboring balls centered at the neutral
element. Notice that both B and B are compact subgroups of the group Qp , say
B = p−k Zp and B = p−k−1 Zp . Applying the Fourier transform to the both sides
of equation LfB = λ(B )fB , we get
1
0
L(θ )f1
B (θ ) = λ(B )f
B (θ ).
(2.21)
The measure ωB = (1B m)/m(B) is the normalized Haar measure of the compact
subgroup B, similarly for ωB . Since for any locally compact Abelian group, the
Fourier transform of the normalized Haar measure of any compact subgroup A is
the indicator of its annihilator group A⊥ , and in our particular case B ⊥ = pk Zp and
(B )⊥ = pk+1 Zp , we obtain
f1
B (θ ) = 1B ⊥ (θ ) − 1(B )⊥ (θ ) = 1∂B ⊥ (θ ),
(2.22)
where ∂B ⊥ is the sphere B ⊥ \ (B )⊥ .
0 ) takes constant value
Equations (2.22) and (2.21) imply that the function L(θ
⊥
0
λ(B ) on the sphere ∂B , i.e., L(θ ) = ψ(θ p ) for some function ψ(τ ) such that
ψ(0) = 0 and ψ(+∞) = +∞. Since C ⊂ D implies λ(C) > λ(D), the function
ψ(τ ) can be chosen to be continuous and increasing, so 0
L(θ ) = ψ(θ p ) satisfies
the ultrametric Inequality (2.20) as claimed.
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
55
3 The Schrödinger Type Operator
3.1 General Properties
In what follows, we assume that (X, d, m) is a non-compact homogeneous ultrametric measure space. Let V be a real measurable function. Consider the Schrödinger
type operator
H u = Lu + V u,u ∈ D.
(3.1)
Our goal is to show that under certain mild conditions on V , the operator H defined
by Eq. (3.1) is essentially self-adjoint.
Theorem 3.1 Assume that V is locally bounded. Then, the following is true:
(i) The operator H = L + V is essentially self-adjoint.6
(ii) Assume that V (x) → +∞ as x → . Then the operator H has a compact
resolvent. Consequently, the spectrum of H is discrete.
(iii) Assume that V (x) → 0 as x → . Then, the essential spectrum of H
coincides with the spectrum of L. Thus, the spectrum of H is pure point,
and the negative part of the spectrum consists of isolated eigenvalues of finite
multiplicity.
Proof
(i) When X = Qp , the field of p-adic numbers, and L is the Taibleson-Vladimirov
operator, the essential self-adjointness of H = L + V has been proved in [22,
Sec. 3.2 and 3.4]; see also [2]. We provide a proof; in general case, (X, d, m) is
a homogeneous ultrametric measure space and L a homogeneous hierarchical
Laplacian acting on it.
As the potential V is locally bounded, H : D → L2 (X, m) is a well-defined
symmetric operator. Let us choose an open ball O, which contains the neutral
element, and write Eq. (2.16) in the form
⎞
⎛
Lf (x) = ⎝ + ⎠ [f (x) − f (x + y)]J (y)dm(y)
O
Oc
= LO f (x) + LO c f (x).
We have Hf = LO f + LO c f + Vf , where the operator V is the operator
of multiplication by the function V (x). The operator LO c f = J (O c )(f −
a ∗ f ), where a(y) = J (y)1O c (y)/J (O c ), is a bounded symmetric operator
6 Recall that, for the classical Schrödinger operator H = − + V in Rn , this statement is not true,
unless V satisfies a certain lower bound; see [9, Chapter II, Theorem 1.1, and Example 1.1].
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A. Bendikov et al.
in L2 (X, m) (as f → a ∗ f is the operator of convolution with probability
measure a(y)dm(y)) and thus does not influence self-adjointness. As LO is
minus Lévy generator, it is essentially self-adjoint (one more way to make this
conclusion is that the matrix of the operator LO is diagonal in the basis {fB }
of eigenfunctions of the operator L; see [23]).
For any ball B which belongs to the same horocycle H as O, we denote HB
the subspace of L2 (X, m), which consists of all functions f having support
in B. Since O is a subgroup of the Abelian group X and each ball B ∈ H is
a coset (i.e., belongs to the quotient group [X : O]), we conclude that HB is
an invariant subspace of the symmetric operator HO = LO + V . Moreover, by
symmetry HB reduces HO .
The ultrametric space X can be covered by a sequence of nonintersecting
balls Bn (recall that due to the ultrametric property, two balls of the same
diameter either coincide or do not intersect). This leads to the orthogonal
decomposition
L2 (X, m) =
/
H Bn
n
where each HBn reduces HO . The restriction of the essentially self-adjoint
operator LO to its invariant subspace HBn is an essentially self-adjoint operator,
while the restriction of the operator V is bounded. Thus, HO is essentially
self-adjoint as orthogonal sum of essentially self-adjoint operators HO,n , the
restriction of HO to HBn .
(ii) The proof is similar to the one for the Schrödinger operators given in [42,
Theorem X.3]; the main tools are boundedness from below of the operator H
and the Riesz-Rellich compactness criteria for subsets of L2 (X, m).
(iii) Let us show that the operator V is L−compact. Then, by [19, Theorem
IV.5.35], the essential spectrums of the operators H and L coincide. Recall
that L−compactness means that if a sequence {un } is such that both {un } and
{Lun } are bounded, then there exists a subsequence {un } ⊂ {un } such that the
sequence {V un } converges.
1. Denote vn = Lun + un . By assumption, the sequence {vn } is bounded and un =
R1 vn = r1 ∗ vn . It follows that the quantity
1/2
|un (x + h) − un (x)|2 dm(x)
≤ vn L2
|r1 (z + h) − r1 (z)| dm(z)
tends to zero uniformly in n as h tends to the neutral element. Thus, the sequence
{un } consists of equicontinuous on the whole in L2 (X, m) functions. The same
is true for the sequence {V un }. Indeed, for any ball B which contains the neutral
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
57
element, we write
1/2
|V (x + h)un (x + h) − V (x)un (x)| dm(x)
2
≤ I + I I + I I I,
where
1/2
|un (x + h) − un (x)|2 dm(x)
I = V L∞
I I = un L2
,
1/2
|V (x + h) − V (x)|2 dm(x)
,
B
I I I = un L2 sup |V (x + h) − V (x)| .
x∈B c
Clearly, I, I I , and I I I tend to zero uniformly in n as h tends to the neutral
element and B ' X.
2. The sequence {V un } consists of functions with equicontinuous L2 (X, m) integrals at infinity. Indeed, for any ball B which contains the neutral element, we
have
|V un (x)|2 dm(x) ≤ un L2 sup |V (x)| → 0
Bc
x∈B c
uniformly in n as B ' X.
Thus, the sequence {V un } is bounded in L2 (X, m), consisting of equicontinuous
on whole in L2 (X, m) functions with equicontinuous L2 (X, m) integrals at infinity.
By the Riesz-Kolmogorov criterion of compactness in L2 (X, m), the set {V un } is
compact, whence it contains a convergent subsequence {V un }, as claimed.
The Case (X, d, m) Is countably infinite In the case when the homogeneous
ultrametric measure space (X, d, m) is countably infinite, the statement (ii) of
Theorem 3.1 can be complemented as follows.
Theorem 3.2 Assume that (X, d, m) is countably infinite. Then, the following
statements are equivalent:
(i) The operator H has a discrete spectrum.
(ii) |V (x)| tend to infinity as x → .
Proof (ii) ⇒ (i) : Since X is discrete and L is a bounded symmetric operator, let
us set d := L. Suppose that |V (x)| tend to infinity as x → . Then, for every
given interval I = [a, b] and its neighborhood I = [a − d − 1, b + d + 1], there
exist a finite set A of points x such that V (x) ∈ I . Let us choose v ∈
/ I and define
58
A. Bendikov et al.
the operator H = L + V where
V (x) :=
V (x) if x ∈
/A
.
v if x ∈ A
The resolvent of the operator V : u(x) → V (x)u(x) is analytic inside of I , and, as
a result, the resolvent of H is analytic inside of I . Indeed, it is straightforward to
show that
2
2 2
2
2
2 2
2
2L(V − λI)−1 2 = 2(V − λI)−1 L2 ≤
d
< 1,
d +1
for any λ ∈ I . It follows that the operator
H − λI = (V − λI) E + L(V − λI)−1
is invertible. This in turn implies that the operator H has no spectrum inside the
interval I . But the difference H −H is an operator of finite rank. Hence, the operator
H has (in the same interval I ) not more than finite number of eigenvalues; see
Lemma 3.12. Thus, we have already proved that the spectrum of H is discrete.
(i) ⇒ (ii) : Suppose that the operator H has a discrete spectrum. Then, clearly
the spectrum of H 2 is also discrete. Let E1 ≤ E2 ≤ · · · be the eigenvalues of H 2 .
Then, by Courant’s min − max principle
En = min max{(ψ, H 2 ψ) : ψ ∈ span(ψ1 , . . . , ψn ), ψ = 1}.
ψ1 ,...,ψn
(3.2)
Assume that |V (x)| does not tend to +∞ as x → . Then, there exists a sequence
{xn } ⊂ X such that |V (xn )| ≤ C for some C > 0 and all n ≥ 1. It follows that
(ψ, H 2 ψ) ≤ 2(d 2 + C 2 ), ∀ψ ∈ span(δx1 , δx2 , δx3 , . . .), ψ = 1.
(3.3)
Equations (3.2) and (3.3) imply that the interval [0, 2(d 2 + C 2 )] contains at list one
limit point of the sequence {En }, i.e., the essential spectrum of H 2 (equivalently of
H ) is not empty. This fact contradicts the discreetness of the spectrum of H 2 (or
H ). This proves the second part of the theorem.
The Case (X, d, m) Is Perfect In the case when (X, d, m) is perfect (has no
isolated points), the situation with the equivalence of statements (i) and (ii) of
Theorem 3.2 is not so obvious. Consider a class K of potentials V of the form
V =
B∈H
σ (B)1B ,
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
59
where H is any fixed horocycle in the tree of balls. For a potential V ∈ K, let us
select the following Hilbert subspaces of L2 (X, m) :
• L+ = span{1B : B ∈ H}.
• LB = span{fT : T B}. /
LB
• L− = L2 (X, m) ( L+ =
B∈H
The proofs of the following three lemmas are left to the reader.
Lemma 3.3 The linear spaces L+ , LB , and L− are invariant subspaces for both
operators H and L. Let H+ , HB and H− (resp. L+ , LB , and L− ) be the restriction of
the operator H (resp. L) to L+ , LB , and L− , respectively. The following properties
hold true:
(i) H = H+ ⊕ H− .
(ii) HB = L
B + σ (B).
/
(LB + σ (B)).
(iii) H− =
B∈H
Remind that LfB = λ(B )fB for any open ball B. As B converges to a singleton,
λ(B ) → +∞, whence LB has discrete spectrum. By the homogeneity property,
Spec(LA ) is the same for all A ∈ H. Let us set
• SH := Spec(LA ).
• RV : =Range(V ).
Lemma 3.4 In the introduced notation
Spec(H− ) = SH + RV .
In particular, the operator H− has a pure point (not necessary discrete) spectrum.
Let us choose in each ball B ∈ H an element aB and consider a discrete
ultrametric space (X , m , d ) with X = {aB : B ∈ H} induced by (X, m, d).
Lemma 3.5 The operator L+ can be identified with certain hierarchical Laplacian
L acting on (X , m , d ); respectively, the operator H+ can be identified
with cer
tain Schrödinger type operator H = L + V with potential V = a∈X V (a)δa .
Theorem 3.3 For a potential V ∈ K, the statements (i) and (ii) of Theorem 3.2 are
related by the implication (i) ⇒ (ii). The inverse implication (ii) ⇒ (i) holds
true if and only if the set SH + RV has no accumulating points.
Proof If we assume that Spec(H ) is discrete, then the operator H+ (whence the
operator H ) has a discrete spectrum. Applying Theorem 3.2, we conclude that
|V (x)| → +∞, i.e., (i) ⇒ (ii) as claimed.
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A. Bendikov et al.
If the sequence {σ (B) : B ∈ H} contains a subsequence σ (Bk ) → −∞, then
it may well happen that the set Spec(H− ) = SH +RV will contain a number of
accumulating points, i.e., Spec(H ) in this case is not discrete. In particular, (ii) ⇒
(i) if and only if the set SH +RV has no accumulating points.
3.2 Rank One Perturbations
In this section, we assume that the homogeneous ultrametric measure space
(X, d, m) is countably infinite. In this case, X can be identified with a countable
Abelian group G equipped with an increasing sequence {Gn }n∈N of finite subgroups
such that ∩Gn = {0} and ∪Gn = G. Each ball in G is a set of the form g + Gn for
some g and n.
As an example, one can consider the group G = Z(p1 )⊕ Z(p2 ) ⊕ . . ., the weak
sum of cyclic groups, equipped with the sequence of its subgroups
Gn = Z(p1 ) ⊕ Z(p2 ) ⊕ . . . ⊕ Z(pn ) ⊕ {0} ⊕ . . . .
Let L be a homogeneous hierarchical Laplacian. We study spectral properties of the
Schrödinger type operator H = L + V with potential V (x) = −σ δa (x), σ > 0.
Clearly, H can be written in the form
Hf (x) = Lf (x) − σ (f, δa )δa (x),
that is, H can be regarded as a rank one perturbation of the operator L. In this
connection, let us recall an abstract form of the Simon-Wolff theorem [38, Theorems
2 and 2’] about pure point spectrum of rank one perturbations.
The Simon-Wolff Criterion Let A be a self-adjoint operator with simple spectrum
on a Hilbert space H, and let ϕ be a cyclic vector for A, that is, {(A − λ)−1 ϕ |
Im λ >0} is a total set for H. By the spectral theorem, H is unitary equivalent to
L2 (R, μ0 ) in such a way that A is multiplication by x with cyclic vector ϕ ≡ 1.
Here, μ0 is the spectral measure of ϕ for A. Let H = A + σ (ϕ, ·)ϕ be a rank one
perturbation of the operator A. Set
F (x) :=
2
22
2
2
(x − y)−2 dμ0 (y) = lim 2(A − (x + i)I)−1 ϕ 2 .
→0
Theorem 3.7 Fix an open interval ]a, b[. The following are equivalent:
(i) For a.e. σ , H has only pure point spectrum in ]a, b[.
(ii) For a.e. x ∈]a, b[, F (x) < ∞.
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
61
In general, if H0 is the closed subspace generated by vectors {(A − λI)−1 ϕ
|Im λ >0}, then its orthogonal complement (H0 )⊥ is an invariant space for H and
H = A on (H0 )⊥ . Thus, the extension from the cyclic to general case is clear.
The function ϕ = δa is not a cyclic vector for L because the operator L has many
compactly supported eigenfunctions φ having support outside of a. Indeed, for any
such φ, for all λ ∈ C with Im λ >0 and for some eigenvalue λk , we will have
((L − λI)−1 δa , φ) = (δa , (L − λI)−1 φ) = (δa , (λk − λ)−1 φ) = 0.
We use the Krein-type identity below to show that the spectrum of the operator
H = L − σ δa is pure point for all σ . Let ψ(x) = R(λ, x, y) be the solution of the
equation
Lψ(x) − λψ(x) = δy (x).
Let ψV (x) = RV (λ, x, y) be the solution of the equation
H ψV (x) − λψV (x) = δy (x).
Notice that L and H are symmetric operators, whence both (x, y) → R(λ, x, y)
and (x, y) → RV (λ, x, y) are symmetric functions.
Theorem 3.8 In the notation introduced above
RV (λ, x, y) = R(λ, x, y) +
σ R(λ, x, a)R(λ, a, y)
,
1 − σ R(λ, a, a)
(3.4)
RV (λ, a, y) =
R(λ, a, y)
1 − σ R(λ, a, a)
(3.5)
RV (λ, a, a) =
R(λ, a, a)
.
1 − σ R(λ, a, a)
(3.6)
and
Proof We have
LψV (x) − λψV (x) = δy (x) + σ δa (x)ψV (x)
= δy (x) + σ δa (x)ψV (a).
It follows that
ψV (x) = R(λ, x, y) + σ ψV (a)R(λ, x, a).
(3.7)
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A. Bendikov et al.
Setting x = a in the above equation, we obtain
ψV (a) = R(λ, a, y) + σ ψV (a)R(λ, a, a)
or
ψV (a)(1 − σ R(λ, a, a)) = R(λ, a, y).
Since ψV (a) = RV (λ, a, y), we obtain Eq. (3.6). In turn, Eqs. (3.6) and (3.7) imply
(3.4) and (3.5).
Theorem 3.9 The operator H = L −σ δa has a pure point Spectrum, which
consists of at most one negative eigenvalue and countably many positive eigenvalues
with accumulating point 0.
The operator H has precisely one negative eigenvalue λσ− if and only if σ >
0 and one of the following two conditions holds: (i) the semigroup (e−t L)t >0 is
recurrent, and (ii) the semigroup (e−t L )t >0 is transient and R(0, a, a) > 1/σ . If it
is the case, then Spec(H ) consists of numbers
λσ− < 0 < . . . < λk+1 < λσk < λk < . . . < λ2 < λσ1 < λ1 .
Otherwise, Spec(H ) consists of numbers
0 < . . . < λk+1 < λσk < λk < . . . < λ2 < λσ1 < λ1 .
If σ < 0, then Spec(H ) consists of numbers
0 < . . . < λk+1 < λσk < λk < . . . < λ2 < λσ1 < λ1 < λσ+ .
The eigenvalues λk are at the same time eigenvalues of the operator L. All λk have
infinite multiplicity and compactly supported eigenfunctions, the eigenfunctions of
the operator L, whose supports do not contain a.
The eigenvalue λσk (resp. λσ− , λσ+ ) is the unique solution of the equation
R(λ, a, a) = 1/σ
in the interval ]λk+1 , λk [ (resp. ] − ∞, 0[, ]λ1 , +∞[). Each λσk (resp. λσ− , λσ+ ) has
multiplicity one and non-compactly supported eigenfunction ψk (x) = R(λσk , x, a)
(resp. ψ− (x) = R(λσ− , x, a), ψ+ (x) = R(λσ+ , x, a)) (Fig. 2).
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
63
Fig. 2 The roots λσ∗ of the equation R (λ, a, a) = 1/σ. The dashed graph corresponds to a
recurrent case, the solid graph—to the transient case
Proof Let ϒ(X) be the tree of balls associated with the ultrametric space (X, d).
Consider in ϒ(X) the infinite geodesic path from a to : {a} = B0 B1 . . . Bk . . . . The series below converges uniformly and in L2 ,
δa =
1 B1
1 B0
−
m(B0 ) m(B1 )
+
1 B2
1 B1
−
m(B1 ) m(B2 )
+... =
∞
fBk .
(3.8)
k=0
Notice that all fBk are eigenfunctions of the operator L, i.e., LfBk = λ(Bk+1 )fBk =
λk+1 fBk . By definition, R(λ, x, y) = (L − λ)−1 δy (x), whence we obtain
1
1
fB (a) +
fB (a) + . . .
λ1 − λ 0
λ2 − λ 1
1
1
1
−
=
λ1 − λ m(B0 ) m(B1 )
1
1
1
+
−
+ ...,
λ2 − λ m(B1 ) m(B2 )
R(λ, a, a) =
or in the final form
∞
1
Ak
1
, Ak =
−
R(λ, a, a) =
.
λk − λ
m(Bk−1 ) m(Bk )
(3.9)
k=1
Since λ → R(λ, a, a) is an increasing function, the equation
1 − σ R(λ, a, a) = 0, σ = 0,
(3.10)
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A. Bendikov et al.
has precisely one solution λσk lying in each open interval ]λk+1 , λk [ ,
λk+1 < λσk < λk , k = 1, 2, . . . .
Claim 1 All numbers λσk are eigenvalues of the operator H . Indeed, the function
ψ(x) = R(λ, x, a) with λ = λσk satisfies the equation
H ψ(x) − λψ(x) = Lψ(x) − λψ(x) − σ δa (x)ψ(x)
= Lψ(x) − λψ(x) − σ δa (x)ψ(a)
= Lψ(x) − λψ(x) − δa (x) = 0.
Claim 2 All numbers λk are eigenvalues of the operator H . Indeed, for any ball B
which does not contain a but belongs to the horocycle Hk−1 , we have
HfB = LfB = λk fB .
When σ > 0, there may exist one more eigenvalue λσ− < 0, a solution of
the Eq. (3.10). Indeed, λ → R(λ, a, a) is an increasing function, continuous on
the interval ] − ∞, 0]. Since R(λ, a, a) → 0 as λ → −∞ and R(λ, a, a) →
R(0, a, a) ≤ +∞ as λ → −0, Eq. (3.10) has unique solution λ = λσ− < 0 in the
cases (i) and (ii). The proof of the theorem is finished.
Example 3.10 The Dyson’s Laplacian. Consider the set X={0, 1, 2, . . .} equipped
with the counting measure m and with the set of partitions {r : r = 0, 1, . . .}, each
of which consists of all rank r intervals Ir = {x ∈ X : kpr ≤ x < (k + 1)pr }. The
set of partitions {r } generates the ultrametric structure on X and the hierarchical
Laplacian
⎛
+∞
⎜
Dα f (x) =
(1 − κ)κ r−1 ⎝f (x) −
r=1
1
m(Ir (x))
⎞
⎟
f dm⎠ , κ = p−α ,
Ir (x)
where the sum is taken over all rank r intervals Ir (x) containing x.
The operator Dα admits a complete system of compactly supported eigenfunctions. Indeed, let I be an interval of rank r and I1 , I2 , . . . , Ip be its subintervals of
rank r − 1. Let us consider p functions
fIi =
1I
1Ii
−
, i = 1, 2, . . . , p.
m(Ii ) m(I )
Each function fIi belongs to the domain of the operator Dα and
Dα fIi = κ r−1 fIi .
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
65
When I runs over the set all p-adic intervals, the set of eigenfunctions fIi forms a
complete system in L2 (X, m). In particular, Dα is essentially self-adjoint operator
having pure point spectrum
Spec(Dα ) = {0} ∪ {κ r−1 : r ∈ N}.
Clearly, each eigenvalue λr = κ r−1 has infinite multiplicity. Let us compute the
value R(λ) := R(λ, 0, 0) of the resolvent kernel for Dα . By Eq. (3.9), we have
R(λ) =
Ak
1
= (p − 1)
.
k
λk − λ
p (λk − λ)
k≥1
k≥1
In particular, R(0) = +∞ if and only if α ≥ 1; Otherwise,
R(0) =
p − 1 1
p−1
=
.
k(1−α)
p
p − pα
p
k≥0
Consider the operator H = Dα − σ δ0 , σ > 0. Let us compute the number
Neg(H ) of negative eigenvalues of the operator H counted with their multiplicity.
By Theorem 3.9, the operator H has at most one negative eigenvalue. It has
exactly one negative eigenvalue if and only if either α ≥ 1 or 0 < α < 1 and
σ > (p − pα )(p − 1)−1 . If we denote the set of pairs (α, σ ) which satisfy the above
conditions by and by 0 = R2+ \ its complement, we obtain
Neg(H ) =
1 if(α, σ ) ∈ 0 if(α, σ ) ∈ 0
which is shown on the picture below (Fig. 3).
Fig. 3 Sets 0 and 66
A. Bendikov et al.
3.3 Finite Rank Perturbations
As in the previous section, the ultrametric measure space (X, d, m) is countably
infinite and homogeneous. For convenience, we assume that m(B) = diam(B) for
any non-singleton ball B.
Let L be a homogeneous hierarchical Laplacian. We study spectral
properties of
the Schrödinger type operator H = L + V with potential V (x) = − N
i=1 σi δai (x),
σi > 0. Clearly, H can be written in the form
Hf (x) = Lf (x) −
N
σi (f, δai )δai (x),
i=1
that is, H can be regarded as rank N perturbation of the operator L. Throughout this
section, we use the following notation:
• R(λ, x, y) is the solution of the equation Lψ(x) − λψ(x) = δy (x). We set
→
−
→ −
→
N
R(λ, x, −
a ) := (R(λ, x, ai ))N
i=1 , and R(λ, a , a ) := (R(λ, aj , ai ))i,j =1 .
• RV (λ, x, y) is the solution of the equation H ψ(x) − λψ(x) = δy (x). We set
→
→ −
−
→
N
RV (λ, x, −
a ) := (RV (λ, x, ai ))N
i=1 , and RV (λ, a , a ) := (RV (λ, aj , ai ))i,j =1 .
• := diag(σi : i = 1, . . . , N).
Theorem 3.11 The following properties hold true:
1. The set Spec(H ) is pure point; its essential part Specess (H ) coincides with
the set Spec(L) = {0} ∪ {λk }, and its discrete part Specd (H ) in each open
interval lying in the complement of Spec(L) consists of at most N distinct points,
solutions of the equation
→
→
det( −1 − R(λ, −
a ,−
a )) = 0.
(3.11)
2. For each k ∈ N, there exists δ > 1 such that mini =j d(ai , aj ) > δ implies that the
operator H has precisely N distinct eigenvalues in each open interval (λs+1 , λs ):
1 ≤ s ≤ k. Moreover, there exists precisely N distinct negative eigenvalues of
the operator H , provided one of the following two conditions is satisfied:
(2.1) The semigroup (e−t L )t >0 is recurrent.
(2.2) The semigroup (e−t L )t >0 is transient and all 1/σi < R(0, a, a).7
The proof of the first part of Theorem 3.11 is based on the Weyl’s theorem on the
essential spectrum of compactly perturbed symmetric operators, see [19, Theorem
IV.5.35], and on the following lemma.
7 Thanks to the homogenuity assumption R(λ, a, a) does not depend on a.
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
67
Lemma 3.12 Let A and B be two symmetric bounded operators and H = A + B.
Assume that B is an operator of rank N operator. Let (a, b) be an interval lying in
the complement of the set Spec(A). Then, the set Spec(H ) ∩ (a, b) consists of at
most N distinct points.
Proof By the Weyl’s essential spectrum theorem, Specess (H ) coincides with the
set Specess (L) = {0} ∪ {λk }. Hence, the set Spec(H ) ∩ (a, b) may contain only
finite number of eigenvalues each of which has finite multiplicity. Consider the case
N = 1, that is, the operator B is of the form
Bf = σ1 (f, f1 )f1 .
Let λ ∈ (a, b) and let f be a nontrivial solution of the equation Hf −λf = 0. Then,
f can be written in the form
f = −σ1 (f, f1 )Rλ f1
(3.12)
where Rλ = (A − λ)−1 is the resolvent operator. It follows that (f, f1 ) = 0 and
(f, f1 ) = −σ1 (f, f1 )(Rλ f1 , f1 )
or
σ1 (Rλ f1 , f1 ) + 1 = 0.
(3.13)
The function φ(λ) = (Rλ f1 , f1 ) is strictly increasing on the interval (a, b). Indeed,
applying the resolvent identity, we get
dφ(λ)
= (Rλ2 f1 , f1 ) = Rλ f1 2 > 0.
dλ
It follows that Eq. (3.13) has at most one solution lying in the interval (a, b).
Assuming that Eq. (3.13)
2 has a2solution, denote it λ∗ . Then, (3.12) implies that
the vector f∗ := Rλ∗ f1 / 2Rλ∗ f1 2 satisfies the equation
Hf∗ − λ∗ f∗ = 0.
Thus, the operator H has at most one eigenvalue in the interval (a, b).
Without loss of generality, we may provide the induction from N = 1 to N = 2.
Thus, assuming that the perturbation operator B is of the form
Bf = σ1 (f, f1 )f1 + σ2 (f, f2 )f2
we set
A f := Af + σ1 (f, f1 )f1
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A. Bendikov et al.
and
Hf := A f + σ2 (f, f2 )f2 .
Observe that the operator A may have in the interval (a, b) at most one eigenvalue
λ∗ . The2corresponding
eigenspace is one-dimensional; call it $f∗ %, where f∗ :=
2
Rλ∗ f1 / 2Rλ∗ f1 2. Let us consider two cases.
First Case Assume that f2 ⊥ f∗ . Then, Hf∗ = A f∗ = λ∗ f∗ , i.e., λ∗ is an
eigenvalue of the operator H . It follows that the orthogonal complement $f∗ %⊥ is
a joint invariant subspace of the operators H and A and that these operators being
restricted to $f∗ %⊥ , call them H⊥ and A⊥ , satisfy
H⊥ f = A⊥ f + σ2 (f, f2 )f2 .
The operator A⊥ has no eigenvalues in the interval (a, b). Hence, by what we
have already shown in the first part of the proof, the operator H⊥ has at most one
eigenvalue in the interval (a, b). It follows that the operator H has at most two
eigenvalues in the interval (a, b).
Second Case Assume that f2 and f∗ are not orthogonal. Let Rλ := (H − λI)−1 and
Rλ := (A − λI)−1 be the resolvent operators. The following identity holds true:
(Rλ f, g) = (Rλ f, g) −
σ2 (Rλ f, f2 )(Rλ f2 , g)
1 + σ2 (Rλ f2 , f2 )
(3.14)
for any f, g, and λ = λ∗ lying in (a, b). Using the spectral resolution formula for
the operator A , the fact that its spectral function Eλ in (a, b) has the only jump at
λ = λ∗ and that the value of the jump Eλ∗ is the operator of orthogonal projection
on the subspace $f∗ %, we get
(f∗ , f )2
+ O1 (1)
λ − λ∗
(3.15)
(f∗ , f )(f∗ , f2 )
+ O2 (1)
λ − λ∗
(3.16)
(Rλ f, f ) =
and
(Rλ f, f2 ) =
where Oi (1) are analytic functions. Substituting asymptotic equations (3.15) and
(3.16) in Eq. (3.14), we get analyticity of the function λ → (Rλ f, f ) at λ = λ∗ . In
particular, this shows that λ = λ∗ is not an eigenvalue of H .
On the other hand, λ∗ splits the interval (a, b) in two parts: (a, λ∗ ) and (λ∗ , b),
each of which does not contain eigenvalues of the operator A . Then, as we have
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
69
already shown, each of these intervals contains at most one eigenvalue of the
operator H . Since λ∗ is not an eigenvalue of the operator H , the number of distinct
eigenvalues of H in the interval (a, b) is at most two. The proof of the lemma is
finished.
Proof of Theorem 3.11 (Second part): Let λ ∈ Specd (H ) and let ψ(x) be the
corresponding eigenfunction, i.e.,
H ψ(x) − λψ(x) = 0.
We have
Lψ(x) − λψ(x) =
N
σi ψ(ai )δai (x)
i=1
or applying to this equation the resolvent operator (L − λ)−1 we get
N
ψ(x) =
σi ψ(ai )R(λ, x, ai ).
(3.17)
i=1
Taking consequently x = a1 , a2 , . . . , aN in Eq. (3.17), we obtain a homogeneous
system of N linear equations with N variables
ψ(aj ) =
N
σi ψ(ai )R(λ, aj , ai )
(3.18)
i=1
or in the vector form
→
→
" = R(λ, −
a ,−
a )",
(3.19)
where " = (ψ(ai ) : i = 1, . . . , N). The system (3.19) has a nontrivial solution if
and only if
→
→
a ,−
a ) = 0.
det( −1 − R(λ, −
(3.20)
Observe that the variable z := R(λ, ai , ai ) does not depend on ai , and its range
is the whole interval ] − ∞, ∞[ when λ takes values in each of the open interval
]λk+1 , λk [. Equation (3.20) can be written as characteristic equation
det(A − zI) = 0
(3.21)
70
A. Bendikov et al.
where A = (aij )N
i,j =1 is symmetric N × N matrix with entries
aij =
for i = j
1/σi
.
−R(λ, ai , aj ) for i = j
(3.22)
Let us compute R(λ, ai , aj ). For any two neighboring balls B ⊂ B , let us denote
A(B) =
1
1
−
.
m(B) m(B )
Remember that we normalize m so that m(B) = diam(B) for any non-singleton ball
B, whence for such B,
A(B) =
1
1
−
.
diam(B) diam(B )
(3.23)
Let ai aj be the minimal ball, which contains both ai and aj . Following the same
line of reasons as in the proof of Eq. (3.9), we obtain
R(λ, ai , ai ) =
B:ai ∈B
A(B)
.
λ(B) − λ
(3.24)
Similarly, for all i = j , we get
R(λ, ai , aj ) = −
d(ai , aj )−1
+
λ(ai aj ) − λ
B:ai aj ⊂B
A(B)
.
λ(B) − λ
(3.25)
Let λ > λ(ai aj ). Equations (3.23) and (3.25) and the fact S ⊂ T ⇒ λ(S) > λ(T )
imply that
R(λ, ai , aj ) =
>
=
d(ai , aj )−1
−
λ − λ(ai aj )
B:ai aj ⊂B
A(B)
λ − λ(B)
d(ai , aj )−1
1
−
λ − λ(ai aj ) λ − λ(ai aj )
1
λ − λ(ai aj )
A(B)
B:ai aj ⊂B
1
1
−
d(ai , aj ) diam(ai aj )
> 0.
Hence, for λ > λ(ai aj ), we obtain
0 < R(λ, ai , aj ) <
d(ai , aj )−1
.
λ − λ(ai aj )
(3.26)
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
71
Notice that λ(B) → 0 as diam(B) → ∞. Let us fix k and let us consider λ > λk+1 .
Let us choose δ > 1 such that if mini =j d(ai , aj ) ≥ δ, then λ(ai aj ) < λk /2.
Then, for all i = j , we get λ − λ(ai aj ) > λk /2 and thus
R(λ, ai , aj ) < 2 := ε(δ) .
δλk
N
(3.27)
Let us increase if necessary δ so that the intervals
{s : |1/σi − s| ≤ ε(δ)}, i = 1, 2, . . . , N,
do not intersect. By Gershgorin circle theorem, the matrix A admits N different
eigenvalues ai , each of which lies in the corresponding open interval
{s : |1/σi − s| < ε(δ)}, i = 1, 2, . . . , N.
The eigenvalues ai , i = 1, 2, . . . , N, are analytic functions of λ in each open
interval (λs+1 , λs ), 1 ≤ s ≤ k; see [35, Theorem XII.1]. Whence, in each interval
(λs+1 , λs ), the number of different solutions of the equations ai = R(λ, ai , ai ) is at
least N. By Lemma 3.12, the number of different solutions is at most N. Thus, the
number of different solutions is precisely N as claimed.
Theorem 3.13 The set Specd (H ) coincides with the set of solutions of Eq. (3.11).
Each eigenfunction ψλ (x) corresponding to λ ∈ Specd (H ) can be represented as
linear combination of functions R(λ, x, ai ), that is,
ψλ (x) =
N
ζi R(λ, x, ai ).
i=1
Thus, support of ψλ is the whole space X, whereas the eigenfunctions fB corresponding to the eigenvalues λ(B) ∈ Specess (H ) are compactly supported.
Proof The proof is straightforward: we apply Eqs. (3.17) and (3.18) to get the result;
see the first part of the proof of Theorem 3.11 (second statement).
Theorem 3.14 For λ ∈
/ Spec(H ), the following identities hold true:
→
→
→
→
RV (λ, x, y) = R(λ, x, y) + R(λ, x, −
a )( −1 − R(λ, −
a ,−
a ))−1 R(λ, −
a , y), 8
(3.28)
→
→
→
→
a , y) = ( −1 − R(λ, −
a ,−
a ))−1 R(λ, −
a , y)
RV (λ, −
8 For a matrix A and vectors ξ and η, we write ξ Aη :=
i,j aij ξi ηj .
(3.29)
72
A. Bendikov et al.
and
→
→
→
→
→
→
RV (λ, −
a ,−
a ) = ( −1 − R(λ, −
a ,−
a ))−1 R(λ, −
a ,−
a ).
(3.30)
In particular, the operator T (λ) := (H − λI)−1 − (L − λI)−1 is of finite rank N. Its
operator norm can be estimated as follows:
2
22
22
2
22
2
→
→
T (λ) ≤ 2( −1 − R(λ, −
a ,−
a ))−1 2 2(L − λI)−1 2 .
(3.31)
Proof Recall that Spec(H ) coincides with the union of two sets: Spec(L) and the
→
→
a ,−
a )) = 0. The proof of the theorem
set of those λ ∈ R for which det( −1 −R(λ, −
is similar to its one-dimensional version Theorem 3.8. Clearly, we can write the
following equation:
LRV (λ, x, y) − λRV (λ, x, y) = δy (x) +
N
σj δaj (x)RV (λ, x, y)
i=1
= δy (x) +
N
σj RV (λ, aj , y)δaj (x),
j =1
or equivalently
RV (λ, x, y) = R(λ, x, y) +
N
σj RV (λ, aj , y)R(λ, x, aj ).
(3.32)
j =1
Substituting consequently x = a1 , a2 , . . . , aN , we obtain system of N linear
equations with N variables
RV (λ, ai , y) = R(λ, ai , y) +
N
σj R(λ, ai , aj )RV (λ, aj , y)
j =1
or in the vector form
→
→
→
→
a , y) = R(λ, −
a , y).
(I − R(λ, −
a ,−
a ))RV (λ, −
(3.33)
→
→
Assuming that λ ∈
/ Spec(H ), in particular det(I − R(λ, −
a ,−
a )) = 0, we get
→
→
→
→
a , y) = (I − R(λ, −
a ,−
a ))−1 R(λ, −
a , y)
RV (λ, −
(3.34)
Evidently Eqs. (3.32) and (3.34) imply Eqs. (3.28), (3.29) and (3.30).
Equation T (λ) = (H − λI)−1 (L − H )(L − λI)−1 applies that T (λ) is of
rank N. Finally, Eq. (3.31) follows from Eq. (3.28). Indeed, for f ∈ L2 (X, m),
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
73
→
→
we introduce (finite-dimensional)
vectors R(λ, f, −
a ) := x f (x)R(λ, x, −
a ) and
−
→
−
→
R(λ, a , f ) := y f (x)R(λ, a , y) and then
(T (λ)f, f ) =
→
→
→
→
f (x)R(λ, x, −
a )( −1 − R(λ, −
a ,−
a ))−1 R(λ, −
a , y)f (y)
x,y
→
→
→
→
= R(λ, f, −
a )( −1 − R(λ, −
a ,−
a ))−1 R(λ, −
a , f ).
→
→
By symmetry, R(λ, −
a , f ) = R(λ, f, −
a ), whence
2
22
22
2
2
→
→
→
|(T (λ)f, f )| ≤ 2( −1 − R(λ, −
a , f )2
a ,−
a ))−1 2 2R(λ, −
22
2
22
22
2
2
→
→
≤ 2( −1 − R(λ, −
a ,−
a ))−1 2 2(L − λI)−1 f 2
2
22
22
2
22
2
→
→
≤ 2( −1 − R(λ, −
a ,−
a ))−1 2 2(L − λI)−1 2 f 2
as desired. The proof of the theorem is finished.
3.4 Sparse Potentials
We assume that the ultrametric measure space (X, d, m) is countably
infinite and
homogeneous. Our analysis of finite rank potentials V = − N
i=1 σi δai indicates
that in the case of increasing distances between locations {ai } of the bumps Vi =
−σi δai , their contributions to the spectrum of H = L + V are close to the union of
the contributions of the individual bumps Vi (each bump contributes one eigenvalue
in each gap (λm+1 , λm ) of the spectrum of the operator L).
The development
of this idea leads to consideration of the class of sparse
potentials V = − ∞
i=1 σi δai , where distances between locations {ai : i = 1, 2, . . .}
form a fast increasing sequence. In the classical theory, this idea goes back to D. B.
Pearson [33]; see also S. Molchanov [30] and A. Kiselev, J. Last, and B. Simon [20].
Throughout this section, we will assume that the sequence mini,j :≥n,i =j d(ai , aj )
tends to infinity with certain rate, which will be specified later.9 We will also assume
that α < σi < β for all i and for some α, β > 0. For λ ∈
/ Spec(L), we define the
following infinite vectors and matrices:
→
• R(λ, x, −
a ) := (R(λ, x, ai ) : i = 1, 2, . . .).
−
→
→
• R(λ, a , −
a ) := (R(λ, a , a ) : i, j = 1, 2, . . .).
i
j
• := diag(σi : i = 1, 2, . . .), −1 := diag(1/σi : i = 1, 2, . . .).
9 We choose the ultrametric d(x, y) such that it coincides with the measure m(B) of the minimal
ball B, which contains both x and y; see e.g. (3.23).
74
A. Bendikov et al.
Theorem 3.15 The following properties hold true:
→
(i) R(λ, x, −
a ) ∈ l2.
−
→
→
(ii) R(λ, a , −
a ), and −1 act in l 2 as bounded symmetric operators.
→
→
(iii) If the operator B(λ) = −1 − R(λ, −
a ,−
a ) has a bounded inverse, then
→
→
a )B(λ)−1 R(λ, −
a , y).
RV (λ, x, y) = R(λ, x, y) + R(λ, x, −
(3.35)
Proof
Let ξ = (ξi ) ∈ l 2 has finite number nonzero coordinates. Define function
f =
ξi δai . Evidently, f ∈ L2 = L2 (X, m) and f = ξ . Let Rλ = (L −
λI)−1 , λ ∈
/ Spec(L), be the resolvent. Then,
→
R(λ, x, −
a )ξ =
R(λ, x, y)f (y)dm(y) = Rλ f (x)
whence
→
R(λ, x, −
a )ξ ≤ Rλ f = Rλ ξ which clearly proves (i).To prove (ii), we write
→
→
ξ R(λ, −
a ,−
a )ξ =
f (x)R(λ, x, y)f (y)dm(y)dm(x)
= (f, Rλ f ) ≤ Rλ f 2 = Rλ ξ 2
→
→
which clearly proves boundedness of the symmetric operator R(λ, −
a ,−
a ) : l2 →
2
l . Since {σi } ∈ (α, β) for all i and some α, β > 0, boundedness of the operators and −1 follows.
(iii) Assume that λ is such that the self-adjoint operator B(λ) has a bounded
inverse; then, Eq. (3.35) follows from its finite dimensional version (3.28) by
passage to limit.
Theorem 3.16 Spec(L) ⊂ Specess (H ).
Proof Let V be the sum of all but finite number of bumps Vi and H = L + V .
By Weyl’s essential spectrum theorem, Specess (H ) = Specess (H ). It follows that
without loss of generality, we may assume that the sequence of distances n =
mini,j :≥n,i =j d(ai , aj ) strictly increases to ∞. Having this in mind, we can choose
for any given τ from the range of the distance function an infinite sequence {Bn } of
disjoint balls of diameter τ such that Bn ∩ {ai } = ∅ for all n. Thanks to our choice,
we obtain
HfT = LfT = λ(T )fT
for any ball T ⊂ Bn and for all n. In particular, each λ = λ(T ), such that T ⊆ Bn
for some n, is an eigenvalue of the operator H having infinite multiplicity, whence
it belongs to Specess (H ).
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
75
Theorem 3.17 Let σ∗ be a limit point of the sequence {σi }. Fix m ∈ N and let
λ∗m ∈ (λm+1 , λm ) be the unique solution of the equation
1
= R(λ, a, a).10
σ∗
(3.36)
Then, λ∗m belongs to the set Specess (H ).
Before we embark on the proof of Theorem 3.17, let us state the Weyl’s
characterization of the essential spectrum Specess (A) of a self-adjoint operator A;
see [43] and [36, Ch. IX, Sect. 2(133)].
Lemma 3.18 A real number λ belongs to the set Specess (A) if and only if there
exists a normed sequence {xi } ⊂ dom(A) such that xi → 0 weakly and Axi −λxi →
0 strongly.
Proof of Theorem 3.17 To show that λ∗m ∈ Specess (H ), we construct a λ∗m sequence {fim } via Lemma 3.18. Let λim ∈ (λm+1 , λm ) be the unique solution
of the equation 1/σi = R(λ, ai , ai ). Let ψim (x) = R(λim , x, ai )/ R(λim , ·, ai )2
be the normed solution of the equation Hi ψ = λim ψ, where Hi := L − σi δai is a
one-bump perturbation of L. Clearly, λim → λ∗m .
Passing if necessary to a subsequence of {σi }, we can assume that d(ai , 0) → ∞
monotonically. Let us put fim := ψim · 1Bi , where Bi is the maximal ball centered
at ai , which does not contain ai−1 and ai+1 . Thanks to our choice fim → 0 weakly
and
2
fim 2 =
|ψim |2 dm → 1.
Bi
Thus, what is left is to show that Hfim − λ∗m fim → 0 strongly. We have
Hfim − λ∗m fim 2 ≤ Hfim − λim fim 2 + fim 2 |λim − λ∗m |
≤ Hfim − λim fim 2 + |λim − λ∗m |
= Hfim − λim fim 2 + o(1),
Hfim − λim fim 2 ≤ H ψim − λim ψim 2 + (H − λim I)(fim − ψim )2
≤ H ψim − λim ψim 2 + (H − λim I)||||(fim − ψim )2
= H ψim − λim ψim 2 + o(1),
2
2
2
2
2
2
2
H ψim − λim ψim 2 ≤ Hi ψim − λim ψim 2 + 2
σ
δ
ψ
j aj im 2
2
2 j =i
2
2
10 Recall that the function λ → R(λ, a, a) does not depend on a.
76
A. Bendikov et al.
and
2
2
3
2
2
3
2
2
2
2
2
2
σj δaj ψim 2
σj |ψim (aj )| ≤ sup{σj }
|ψim |2 dm.
2
2 =
X\B
2 j =i
2
i
j =i
2
The right-hand side of this inequality tends to zero as i → ∞, and we finally
conclude that {fim } is the desired λ∗m -sequence in the sense of Lemma 3.18. The
proof is finished.
Let us introduce the following notation:
• ∗ is the set of limit points of the sequence {σi }.
• 1/∗ := {1/σ∗ : σ∗ ∈ ∗ }.
• R−1 (1/∗ ) := {λ : R(λ, a, a) ∈ 1/∗ }.
Theorem 3.19 Assume that the following condition holds:
lim sup
N→∞ i≥N
j ≥N:j =i
1
= 0,
d(ai , aj )
(3.37)
and then
Specess (H ) = Spec(L) ∪ R−1 (1/∗ ).
(3.38)
Proof That Spec(L) and R−1 (1/∗ ) are subsets of Specess (H ) follows from
Theorem 3.16 and Theorem 3.17. We are left to prove that
Specess (H ) ⊂ Spec(L) ∪ R−1 (1/∗ ).
Let us fix m ∈ N and choose a closed interval I from the spectral gap (λm+1 , λm ).
We claim that
I ∩ Specess (H ) = ∅.
Indeed, since R(λ) := R(λ, a, a) is strictly increasing and continuous in the interval
(λm+1 , λm ), closed sets R(I) and 1/∗ do not intersect. Hence, there exists only a
finite number of σi such that 1/σi ∈ R(I). Let us choose N big enough so that the
sets {1/σi : i > N} and R(I) do not intersect. Let us write H = H + V where V
is a finite number of bumps −σi δai , i ≤ N. By Weyl’s essential spectrum theorem,
Specess (H ) = Specess (H ).
Notice however that the sets Specd (H ) and Specd (H ), discrete parts of Spec(H )
and Spec(H ), may well be quite different. Observe that for the operators H and
H , the sets of limit points, the function R, the set of gaps, etc. are the same. Thus,
in all our further considerations, we may assume that {1/σi } ∩ R(I) = ∅.
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
77
→
→
Making this assumption, consider now the operator B(λ) = −1 −R(λ, −
a ,−
a ),
λ ∈ I. According to Identity (3.35), if B(λ) has a bounded inverse, then λ ∈
/
Spec(H ). Let us write
#
%
→
→
B(λ) = −1 − R(λ, −
a ,−
a ) := −1 − R(λ)I − 4
R(λ).
Since we assume that the closed bounded sets {1/σi } and R(I) do not intersect, the
−1
−1
operator A(λ)
2 :=
2 − R(λ)I has a bounded inverse A(λ) for all λ ∈ I. Clearly,
−1
2
2
the norm A(λ)
can be estimated by the reciprocal of the distance between sets
{1/σi } and R(I); denote it by C1 . Thus, writing for λ ∈ I the identity
B(λ) = A(λ)(I − A(λ)−14
R(λ)),
(3.39)
2
2
2
2
2
2
R(λ))2 ≤ C1 24
R(λ))2 .
2A(λ)−14
(3.40)
we get
Writing again H as H + V where V consists of a finite number, say N, of bumps
and applying Inequality (3.26) for the operator H :
R(λ, ai , aj ) <
1
1
, i = j, i, j ≥ N,
d(ai , aj ) λ − λ(ai aj )
we will get, thanks to the Gershgorin circle theorem and to our assumption (3.37),
the following inequality:
2
2
24
R(λ))2 ≤ C2 sup
i≥N j :j =i,j ≥N
1
1
<
d(ai , aj )
2C1
(3.41)
for some constant C2 > 0, which depends only on I, and for N chosen big
enough. Clearly, Inequalities (3.40) and (3.41) imply the fact that the operator
I − A(λ)−14
R(λ) has bounded inverse for all λ ∈ I:
−1 k
A(λ)−14
I − A(λ)−14
R(λ)
R(λ) .
=
k≥0
This fact, in turn, implies that the operator B(λ) given by Eq. (3.39) has bounded
inverse for all λ ∈ I; therefore, I ∩ Spec(H ) = ∅. In particular, since
Specess (H ) = Specess (H ) by Weyl’s essential spectrum theorem, we finally get
I ∩ Specess (H ) = ∅
as desired. The proof is finished.
78
A. Bendikov et al.
Remark 3.20 Theorem 3.19 does not contain any information about sets Specac (H )
and Specsc (H ), the absolutely continuous and singular continuous parts of
Spec(H ). In the next section, we will show that under more restrictive assumption
Specac (H ) and Specsc (H ) are indeed empty sets, that is, Spec(H ) is pure point.
Moreover, the eigenfunctions of H decay exponentially in certain metric at infinity.
This is the so-called spectral localization property.
3.5 Spectral Localization
As in the previous section, the ultrametric measure space (X, d, m) is countably
infinite and homogeneous. We consider the operator H = L + V where L, the
deterministic part of H , is a hierarchical Laplacian and
V =−
σ (a, ω)δa , ω ∈ (, F, P ),
a∈I
is a random potential defined by a family of locations I = {ai } and a family σ (ai , ω)
of i.i.d. random variables. Henceforth, we assume that the probability distribution
of σ (ai , ω) is absolutely continuous with respect to the Lebesgues measure and has
a bounded density supported by a finite interval [α, β].
In the case when X is the Dyson lattice and L =Dα , the Dyson Laplacian (see
Example 3.10), the perturbed operator
H = Dα −
σ (a, ω)δa
a∈X
has a pure point spectrum for P −a.s. ω. This statement (the localization theorem)
appeared first in the paper of Molchanov [29] (σ (a, ω) is the Cauchy random
variable) and later in a more general form in the papers of Kritchevski [27] and [26].
The proof of this statement is based on the self-similarity property of the operator
H.
The localization Theorem 3.23 concerns the case where the family of locations
I does not coincide with the whole space X, whence the operator H is not selfsimilar. The technique developed in [27, 29] and [26] does not apply here to prove
Theorem 3.23.
Our approach is based on a different technique: the abstract form of the
Aizenman-Molchanov criterion for pure point spectrum, the Krein-type identity
from the previous section, the technique of fractional moments, the decoupling
lemma of Molchanov, and Borel-Cantelli type arguments; see papers [1, 28].
The Aizenman-Molchanov Criterion Let H = H0 + V be a self-adjoint operator
in l 2 () ( is a countable set of sites) with H0 a bounded operator and V =
− a∈ σ (a, ω)δa . Assume that the collection of random variables {σ (a, ω) : a ∈
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
79
} has the property that for each site a the conditional probability distribution of
σ (a, ω) (conditioned on the values of the potential at all other sites) is absolutely
continuous with respect to the Lebesgues measure (in particular, this assumption
holds if {σ (a, ω) : a ∈ } are mutually independent random variables having
absolutely continuous
w.r.t. the Lebesgues measure l probability distributions).
$
Let H = λdEλ be the spectral resolution of the symmetric operator H . Let
G(λ, x, y) be the integral kernel of the operator (H − λI)−1 . Then, for any fixed
x, τ and = 0,
2
22 2
2
|G(τ + i, x, y)|2 = 2(H − (τ + i)I)−1 δx 2 =
y∈
d(Eλ δx , δx )
(λ − τ )2 + 2
(3.42)
As the left-hand side of Eq. (3.42) (as a function of ) decreases on the interval
]0, +∞[, the limit (finite or infinite) in Eq. (3.42) exists and equals
lim
↓0
|G(τ + i, x, y)| =
2
y∈
d(Eλ δx , δx )
.
(λ − τ )2
Theorem 3.21 If for any x ∈ and Lebesgues a.a. τ ∈ [a, b]:
lim
↓0
|G(τ + i, x, y)|2 < ∞,
(3.43)
y∈
for a.e. realizations of {σ (x, ·)}, then almost surely the operator H has only pure
point spectrum in the interval [a, b]. Furthermore, if under Condition (3.43) the
integral kernel
G(τ + i0, x, y) := lim G(τ + i, x, y)
↓0
(which exists a.e. τ ) decays exponentially at infinity (in some metric ρ(x, y) on ),
then do the eigenfunctions ϕτ (y), for τ ∈ [a, b].11
Proof The first part of the statement follows from Simon-Wolff Theorem 3.7. For
completeness of exposition, we comment on the proof. To prove the second Part,
one needs an ad hoc argument and we refer to the cited above paper [1, Theorems
3.1 and 3.3 in Sec. 3]).
α
α
Note that in the case of the Dyson-Vladimirov Laplacian D and H = D −
σi (ai , ω)δa , one can use the metric ρ(x, y) = ln(1 + d(x, y)), where d(x, y) is
the ultrametric generated by p-adic intervals as in Example 3.10. In this case, the
exponential decay of eigenfunctions in ρ−metric follows directly from two facts:
(1) each eigenfunction ϕτ (y) of H can be represented as a linear combination of
11 An even more versatile version can be found in [1, Theorems 3.1 and 3.3 in Sec. 3].
80
A. Bendikov et al.
functions R(τ, ai , y), where R(λ, x, y) is the resolvent kernel of Dα (see Theorem
3.13) and (2) R(λ, x, y) has an exponential decay because the heat kernel p(t, x, y)
does (see Eq. (1.2)).
By the spectral theory, one can represent l 2 () as the direct sum of three H invariant subspaces:
l 2 () = Hac ⊕ Hsc ⊕ Hpp ,
where Hac (resp. Hsc , Hpp ) is the set of all functions f ∈ l 2 () such that the spectral
measure
σ f (A) = 1A (λ)d(Eλ f, f )
is absolutely continuous (resp. singular continuous, pure point) with respect to the
Lebesgues measure. By Theorem 3.7, Condition (3.43) implies that for any x ∈ the probability measure
σ x (A) = 1A (λ)d(Eλ δx , δx )
is pure point, that is, σ x (A) = σ x (A ∩ Sx ) for any open set A and some at most
countable set Sx . Set S := ∪x∈ Sx ; then, for any f ∈ l 2 () and measurable set A,
f
σ (A) = 1A (λ)d(Eλ f, f ) = 1A (H )f 2
=
|f (x)|2 |(1A (H )f, δx )|2
x∈
and, if A lies in the complement of S,
|(1A (H )f, δx )|2 ≤ |(1A (H )f, f )| |(1A (H )δx , δx )|
= 1A (H )f 2 σ x (A) = 0.
Thus, for any f ∈ l 2 () the spectral measure σ f is pure point, that is, f ∈ Hpp .
That means that the operator H has a pure point spectrum.
Remark 3.22 The function z → G(z, x, y), analytic in the domain C+ , is
represented by the Borel-Stieltjes transform of a signed measure of finite variation
G(z, x, y) =
d(Eλ δx , δy )
.
λ−z
It follows that the limit G(τ + i0, x, y) exists and takes finite values for Lebesgues
a.e. τ ; see e.g. [37, Theorem 1.4]. Moreover, the limit G(τ + i0, x, y) exists even in
a more restrictive sense, as the non-tangential limit; see [34, Ch. III, Sec. 2.2, 3.1,
and 3.2]. We will apply this fact in the proof of Theorem 3.23.
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
81
The Localization Theorem Coming back to our setting, let H = L+V
where L is
a hierarchical Laplacian and V a random potential of the form V = − i σi (ω)δai .
Here, σi (ω) := σ (ai , ω) are i.i.d. random variables corresponding to the set of
locations I = {ai }.
Let d(x, y) be the ultrametric which is chosen such that it coincides with the
measure m(B) of the minimal ball B containing both x and y.
Let R(λ, x, y) be the integral kernel of the operator (L − λI)−1 , i.e., the solution
of the equation Lu − λu = δy . The function λ → R(λ, x, x) does not depend on x;
we denote its value R(λ). This is strictly increasing continuous in each spectral gap
function; we denote by R−1 (ν) its inverse function.
Theorem 3.23 The operator H has a pure point spectrum for P −a.s. ω provided
for some (whence for all) y ∈ X the sequence d(ai , y) eventually increases and for
some small r (say, 0 < r < 1/3):
lim sup
M→∞ i≥M
j ≥M:j =i
1
= 0.12
d(ai , aj )r
(3.44)
Proof The set of limit points of the sequence {σi (ω)} coincides (for P −a.s. ω)
with the whole interval [α, β]. Hence, by Theorem 3.19, the closed set Specess (H )
consists (for P -a.a. ω) of two parts: (1) the set Spec(L) and (2) the collection of
countably many disjoint closed intervals Ik = R−1 ([1/β, 1/α])∩ ]λk+1 , λk , [ and
the interval I− = R−1 ([1/β, 1/α])∩ ] − ∞, 0[, i.e.,
Specess (H ) = Spec(L) ∪ I− ∪ I1 ∪ I2 . . . .
Let RV (λ, x, y) be the integral kernel of the operator (H −λI)−1 , i.e., solution of the
equation H u − λu = δy . Due to the Aizenman-Molchanov criterion, the operator
H has only pure point spectrum (for P −a.s. ω) provided for each y ∈ X, for each
interval Ik , and for Lebesgues a.e. τ ∈ Ik :
lim
→+0
|RV (τ + i, x, y)|2 < ∞,
(3.45)
x
for a.e. realization of {σ (y, ω)}.We split the proof of Eq. (3.45) in seven steps.
Step I When V has a finite rank, Theorem 3.11(i) and Theorem 3.14 imply that for
each fixed ω, the function RV (τ +i0, x, y) = (H −τ I)−1 δy (x) belongs to L2 (X, m)
for each y ∈ X and for all but finitely many τ ∈ Ik (which are eigenvalues of H ).
In general,
when the rank of V is infinite, we split V in two parts: V = −σ1 δa1
and V = − i>1 σi δai . Writing the set of locations as {a} = {a1 } ∪ {ai : i > 1},
12 Clearly, this condition implies Condition (3.37).
82
A. Bendikov et al.
we get similarly to Eq. (3.35): for λ in the domain C+ ,
RV (λ, x, y) = RV (λ, x, y) + RV (λ, x, a1 )B(λ)−1 RV (λ, a1 , y),
where B(λ) = 1/σ1 − RV (λ, a1 , a1 ) is a nonconstant analytic in the domain C+
function. It follows that
RV (λ, ·, y)2 ≤ RV (λ, ·, y)2
+ |B(λ)|−1 RV (λ, ·, a1 )2 |RV (λ, a1 , y)|.
Hence, the function RV (λ, x, y) satisfies Condition (3.45), i.e., RV (τ + i0, ·, y)2
is finite for all y and a.e. τ , provided the function RV (λ, x, y) satisfies Condition
(3.45), i.e., RV (τ + i0, ·, a)2 is finite for all a and a.e. τ , and also one more
restriction on τ, it does not belong to the exceptional set
ϒ := {s : B(s + i0) = 0}.
The function B(λ), analytic in the domain C+ , admits non-tangential boundary
values B(s + i0) for a.e. s. By the Lusin-Privalov uniqueness theorem on boundaryvalues of analytic functions [34, Ch. IV, Sec. 2.5] (see also [37, Theorem 1.5]), the
Lebesgues measure of the exceptional set ϒ equals to zero. Thus, we come to the
conclusion that Condition (3.45) for the potential V can be reduced to the case of
truncated potential V .
Repeating this argument finitely many times, we come to the final conclusion:
in order to prove that (3.45) holds for V , we can consider, if necessary, any finitely
truncated potential V (the potential corresponding to the finitely truncated system
of locations {ai : i > k}) and to prove that (3.45) holds for V instead of V .
Step II Writing for λ ∈ C+ equation H u−λu = δy in the form Lu−λu = δy −V u,
we obtain
RV (λ, x, y) = R(λ, x, y) +
∞
σj R(λ, x, aj )RV (λ, aj , y).
(3.46)
j =1
Equation (3.46) shows that to estimate the function y → RV (λ, ·, y)2 , it is
enough to estimate the quantity |RV (λ, aj , y)| for j = 1, 2, . . . etc. Indeed, since
R(λ, ·, y)2 does not depend on y, we get
⎛
RV (λ, ·, y)2 ≤ R(λ, ·, y)2 ⎝1 + β
∞
j =1
⎞
|RV (λ, aj , y)|⎠ .
(3.47)
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
83
Choosing x = ai , i = 1, 2, . . . , in Eq. (3.46) and setting R(λ, ai , ai ) = R(λ), we
obtain
RV (λ, ai , y) =
σj R(λ, aj , ai )RV (λ, aj , y)
R(λ, ai , y)
+
.
1 − σi R(λ)
1 − σi R(λ)
(3.48)
j :j =i
Step III Applying in Eq. (3.48) the inequality
s
∞
∞
s
Zj , Zj ∈ C, 0 < s ≤ 1,
Z
≤
j
j =1 j =1
we will get
σj R(λ, aj , ai ) s R(λ, ai , y) s
RV (λ, aj , y)s
|RV (λ, ai , y)| ≤ +
1 − σi R(λ)
1 − σi R(λ)
s
j :j =i
R(λ, aj , ai ) s R(λ, ai , y) s
s
RV (λ, aj , y)s .
≤
+β
1 − σi R(λ)
1 − σi R(λ)
j :j =i
Taking the expectation over {σi }, we obtain the following Inequality:
s
1
R(λ, aj , y)s
E |RV (λ, ai , y)| ≤ E 1 − σi R(λ)
RV (λ, aj , y) s s
R(λ, aj , ai )s .
+β
E 1 − σi R(λ)
s
(3.49)
j :j =i
Step IV Due to Eq. (3.4), the random variable RV (λ, aj , y) can be represented in
the form
RV (λ, aj , y) = RV (λ, aj , y) +
σi RV (λ, aj , ai )RV (λ, ai , y)
aσi + b
:=
1 − σi RV (λ, ai , ai )
cσi + d
where the random variables V = − k:k =i σk (ω)δak , a, b, c , and d do not depend
on σi (but they of course depend on the truncated sequence {σk : k = i}). From this
observation and the following two general inequalities from Molchanov’s lectures
[28, Chapter II, Lemma 2.2]): There exist constants c0 , c1 > 0 such that for all
complex numbers a, b, c, d, σ
1
0
dσ
c0
, forall0 < s < 1,
≤
|σ − σ |s
1−s
84
A. Bendikov et al.
and
1
1
aσ + b s
aσ + b s
dσ
dσ, forall0 < s < 1/2,
|σ − σ |s ≤ c1
0 cσ + d
0 cσ + d
yield the following lemma, which is the fundamental point of our reasons.
Lemma 3.24 (Decoupling Lemma) There exist constants C0 , C0 > 0 which
depend on s, α, β, and k such that the inequalities
E s
1
≤ C0
1 − σi R(λ) and
RV (λ, aj , y) s
≤ C E RV (λ, aj , y)s
E
0
1 − σi R(λ)
hold for all 0 < s < 1/2 and all λ ∈ C+ such that Re λ ∈ Ik .
Step V For any fixed y ∈ X and λ as above, let us denote ψi := E |RV (λ, ai , y)|s .
Applying decoupling lemma to Inequality (3.49) and setting C1 := β s C0 , we get an
infinite system of inequalities
ψi ≤ C0 |R(λ, ai , y)|s + C1
R(λ, aj , ai )s ψj .
j :j =i
In the vector form, this system reads as follows:
ψi ≤ gi + (Aψ)i ,i = 1, 2, . . . ,
s
where ψ = (ψi ), g = (gi ) has entries gi = C0 |R(λ,
s where A is an
ai , y)| and
infinite matrix with nonnegative entries aij = C1 R(λ, aj , ai ) if i = j and 0
otherwise.
Iterating formally this infinite system of inequalities, we get
ψi ≤ gi + (Ag)i + A2 g + A3 g + . . . ≤ (I − A)−1 g .
i
i
i
In particular, this would yield the following inequality (one of the fundamental
points in the proof of (3.45)):
ψ ≤ 2 g
(3.50)
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
85
given A : L → L is a bounded linear operator acting in some Banach space L of
sequences such that
A ≤ 1/2.
For instance, choosing L = {ψ : ψ =
μi E |RV (λ, ai , y)|s ≤ 2C0
i
(3.51)
i μi |ψi | < ∞}, we obtain
μi |R(λ, ai , y)|s
(3.52)
i
given
A = sup
ψ=1 i
μi (Aψ)i ≤ 1/2.
(3.53)
For ψ such that ψ = 1, we have
μi (Aψ)i ≤
μi
aij ψj i
i
=
j
μj ψj μi aij /μj ≤ sup
μi aij /μj .
j
i
j
i
In particular, Inequality (3.53) holds whenever
⎛
sup ⎝
j
i:i =j
⎞
s
1
μi R(λ, aj , ai ) ⎠ /μj ≤
.
2C1
(3.54)
Finally, (3.26) together with (3.54) allows us to conclude that (3.51) holds, provided
⎛
sup ⎝
j
i:i =j
⎞
1
1
⎠ /μj ≤
μi
.
s
d(aj , ai )
2C1 C2
(3.55)
Step VI For λ as above and εj > 0 which we will choose later, consider events
Aj = {RV (λ, aj , y) > εj }.
Applying Chebyshev inequality, we will get, for each j = 1, 2, . . ., the following
inequality:
s
E RV (λ, aj , y)
.
P (Aj ) ≤
εjs
(3.56)
86
A. Bendikov et al.
Equations (3.56), (3.52), and (3.55) yield
P (Aj ) ≤
j
E RV (λ, aj , y)s
j
≤ 2C0
εjs
R(λ, aj , y)s
εjs
j
≤ 2C0 C2
j
1
d(aj , y)s εjs
provided εj are chosen such that
⎛
sup εjs ⎝
j
⎞
1
1
⎠≤
.
s
s
εi d(aj , ai )
2C1 C2
1
i:i =j
(3.57)
Let us choose s = 1/2 − δ and εj = 1/d(aj , y)r . Then, truncating, if necessary,
the potential V , i.e., passing to the potential V = V − V with V of finite rank
as explained in Step I, we can assume that the sequence εj is a strictly decreasing
sequence. By the ultrametric inequality, we have d(ai , aj ) = d(ai , y). Hence,
⎛
sup εjs ⎝
j
⎞
⎛
⎞
⎛
⎞
d(ai , y)rs
1
1
⎠ ≤ sup ⎝
⎠ + sup εjs ⎝
⎠
εis d(aj , ai )s
d(aj , ai )s
d(aj , ai )s
j
j
1
i:i =j
i:i<j
⎛
≤ sup ⎝
j
i:i<j
⎛
≤ sup ⎝
j
i:i<j
i:i>j
⎞
⎛
⎞
1
1
s
⎠ + sup ε ⎝
⎠
j
d(aj , ai )s
d(aj , ai )(1−r)s
j
i:i>j
⎞
⎛
⎞
1
1
⎠ + sup ⎝
⎠.
d(aj , ai )s
d(aj , ai )(1−r)s
j
i:i =j
Thus, truncating the potential V and then choosing 0 < δ < 1/2 − r/(1
− r),
we obtain Inequality (3.57). Moreover, thanks to
our
choice,
the
series
j εj
converges, hence of course converges the series j P (Aj ). Applying the BorelCantelli lemma, we conclude the following: For P −a.s. ω, there exists j0 (ω) such
that
RV (λ, aj , y) ≤ εj , forallj ≥ j0 (ω),
(3.58)
holds for all λ ∈ C+ such that Re λ ∈ Ik .
Step VII For λ as above, the function R(λ, x, y) = (L − λI)−1 δy (x) belongs to
L2 (x, m) and, by the homogeneity assumption, its norm R(λ, ·, y)2 does not
Hierarchical Schrödinger Type Operators: The Case of Locally Bounded Potentials
87
depend on y. Having this in mind, we write Inequality (3.47) (for the truncated
potential V )
⎞
RV (λ, aj , y)⎠
RV (λ, ·, y)2 ≤ R(λ, ·, y)2 ⎝1 + β
⎛
⎛
≤ R(λ, ·, y)2 ⎝1 + β
j
j ≥j0 (ω)
⎞
RV (λ, aj , y)⎠
εj + β
j <j0 (ω)
which clearly holds for all λ as above and for P −a.s. ω. As Im λ ↓ 0, we get finite
limit for P −a.s. ω and for each λ ∈ Ik , which does not belong to some exceptional
set Ik (ω) ⊂ Ik of Lebesgues measure zero (the exceptional set appears because we
pass to the boundary values of the Cauchy-Stieltjes integrals RV (λ, aj , y), j <
j0 (ω), as explained in Theorem 3.21). This is precisely what we claim in Eq. (3.48).
The proof is finished.
Acknowledgments We are grateful to the anonymous reviewer for valuable comments and
suggestions.
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Vaccine Effectiveness in a Case Study of
Italy
Vasiliki Bitsouni, Nikolaos Gialelis, and Ioannis G. Stratis
Abstract We present a compartmental mathematical model with demography for
the spread of the COVID-19 disease, considering also asymptomatic infectious
individuals. We compute the basic reproductive ratio of the model and study the
local and global stability for it. We solve the model numerically based on the case
of Italy. We propose a vaccination model, and we derive threshold conditions for
preventing infection spread in the case of imperfect vaccines.
Keywords Mathematical modelling of COVID-19 · SEIAR · SVEIAR ·
Asymptomatic cases · Endemic · Basic reproductive ratio · Stability analysis ·
Lyapunov function · Vaccine effectiveness · Epidemic prediction · Italy case
study · Numerical simulations
1 Introduction
In late 2019, the severe acute respiratory syndrome coronavirus 2 (SARS-CoV2) [1], a strain of coronavirus that causes coronavirus disease 2019 (COVID-19),
appeared in Wuhan, China, and rapidly spread across the globe. In January 2020, the
human-to-human transmission of SARS-CoV-2 was confirmed, during the COVID19 pandemic, and SARS-CoV-2 was designated a Public Health Emergency of
International Concern by the WHO, and in March 2020 the WHO declared it a
pandemic. Since 31 December 2019, and as of 23 July 2020, 15,455,848 cases of
V. Bitsouni
Department of Mathematics, National and Kapodistrian University of Athens, Athens, Greece
Applied Mathematics Laboratory, School of Science and Technology, Hellenic Open University,
Patras, Greece
e-mail: vbitsouni@math.uoa.gr
N. Gialelis · I. G. Stratis ()
Department of Mathematics, National and Kapodistrian University of Athens, Athens, Greece
e-mail: ngialelis@math.uoa.gr; istratis@math.uoa.gr
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_4
91
92
V. Bitsouni et al.
COVID-19 (in accordance with the applied case definitions and testing strategies in
the affected countries) are confirmed in more than 227 countries and 26 cruise ships
[2]. Currently, there are 5,415,828 active cases and 631,775 deaths.
As of spring 2020, Italy is one of the countries suffering the most with the
COVID-19 outbreak. One of the most critical facts about COVID-19, which affected
Italy too, is that a significant number of cases, mainly those of young age, have
been reported as asymptomatic [3], leading to the fast spread of the infection.
Fortunately, asymptomatic cases have a shorter duration of viral shedding and lower
viral load [4, 5]. However, the proportion of asymptomatic cases can range from 4%
to 80% [6], and, most of the time, they play a key role in infection transmission;
therefore, it is very important to model both symptomatic and asymptomatic cases.
The most crucial element of COVID-19 pandemic is demography. After the end
of the quarantine and lockdown, there have been many examples of countries
where the number of cases increased rapidly. Although demography is a very
important element in epidemics and especially in the case of COVID-19, only a
few models describe the dynamics of infection with SARS-CoV-2, considering the
demographic terms. To this end, here we develop an SEIAR model, considering
both symptomatic and asymptomatic cases and demographic terms, with a constant
influx of susceptible individuals. In this study we only consider horizontal infection
transmission.1
As the development of a vaccine against SARS-CoV-2 is an urgent demand,2 a
study of the epidemiological consequences that an imperfect potential vaccine can
have is needed, and, to the best of our knowledge, there is currently no relevant study
in the literature. Here, we develop a theoretical framework to assess the vaccine
effectiveness and the epidemiological consequences of a potential vaccine. Given
that the second phase might lead to more asymptomatic cases than the first phase, we
need to investigate the asymptomatic group for different coverage and efficacy. The
model derived in this study provides new insights on the effect of different vaccine
coverage and efficacy on infection spread in a population with demographics.
This study is organised as follows: In Sect. 2, we develop a new SEIAR model
for COVID-19. We derive the R0 of the model, and we study the local and global
stability of the steady states. Numerical results with a focus on data fitted to Italy
case are presented. In Sect. 3, we extend our model including also a vaccinated group
of individuals, we study the global stability of the steady states and we predict the
vaccine effectiveness. We conclude in Sect. 4 with a summary and discussion of
future directions.
1 Horizontal transmission is a transmission by direct contact between infected and susceptible
individuals or between disease vectors and susceptible individuals that are not in a parent-progeny
relationship. Vertical transmission is the passage of the disease-causing agent from the mother to
the baby during the period immediately before and after birth.
2 During the days of writing this paper, promising progress has been announced in this direction.
A Model for the Outbreak of COVID-19: Vaccine Effectiveness in a Case Study of Italy
93
2 Modelling Transmission Dynamics
2.1 An SEIAR Model for COVID-19
In this section we model the transmission of COVID-19, the disease caused by
SARS-CoV-2. To derive our model, we consider the people who have been in
contact with an infectious individual but remain uninfected for a latency period.
Moreover, a significant number of individuals being infected remain asymptomatic,
due to various factors such as age, health condition, etc. [3]. To this end, we
classify the total population (N) into five subclasses: susceptible (S), latent/exposed
(E), symptomatic infectious (I ), asymptomatic infectious (A) and recovered (R)
individuals; hence, we have that S + E + I + A + R = N. We can consider that
N is practically constant, since the time span of the epidemiological phenomenon
is relatively short and N is relatively large. We take into account the demographic
terms, and we consider the transmission to be exclusively horizontal. The SEIAR
model is governed by the following system of nonlinear ordinary differential
equations:
dS
= μN − βI SI − βA SA − μS,
dt
dE
= βI SI + βA SA − (k + μ) E,
dt
dI
= k (1 − q) E − (γI + μ) I,
dt
dA
= kqE − (γA + μ) A,
dt
dR
= γI I + γA A − μR,
dt
(1a)
(1b)
(1c)
(1d)
(1e)
with initial conditions:
(S (0) , E (0) , I (0) , A (0) , R (0)) = (S0 , E0 , I0 , A0 , R0 ) ∈ R+
0
5
,
(2)
where μ is the birth/death rate; βI and βA the transmission rates of I and A, respectively;3 k the incubation rate, i.e. the rate of latent individuals becoming infectious; q
the proportion of asymptomatic infectious individuals; and γI and γA the recovery
rates of infectious and asymptomatic infectious individuals, respectively. A flow
c I,A
, where c is the average number of close contacts of an individual with other
N
individuals and I,A is the probability of a contact to be effective in turning an S individual into
an I or A one, respectively.
3β
I,A
≡
94
V. Bitsouni et al.
Fig. 1 Flow diagram of the SEIAR model (1)
diagram of the model is illustrated in Fig. 1. A straightforward application of the
classical ODE theory yields that the above Cauchy problem is well posed.
2.2 The Basic Reproductive Ratio
The average number of secondary cases, arising from one infection when the entire
population is susceptible, is defined as the basic reproductive ratio and denoted by
R0 . Being the most important quantity on infectious disease epidemiology, the basic
reproductive ratio is a dimensionless quantity, which is often used to reflect how
infectious a disease is. To define R0 , we calculate the next-generation matrix of the
system (see, e.g. [7, 8]).
First, we linearise model (1) around the disease-free steady state, (N, 0, 0, 0, 0),
and we consider the infected states, i.e. E, I, A, to obtain the linearised infection
subsystem:
dE
= βI NI + βA NA − (k + μ) E,
dt
dI
= k (1 − q) E − (γI + μ) I,
dt
dA
= kqE − (γA + μ) A.
dt
(3a)
(3b)
(3c)
A Model for the Outbreak of COVID-19: Vaccine Effectiveness in a Case Study of Italy
95
Then, we set x = (E, I, A)tr , so that the system (3a)–(3c) can be written in the form
dx
= (T + ) x,
dt
where
⎡
⎤
0 βI N βA N
T = ⎣0 0
0 ⎦
0 0
0
is the transmission matrix and
⎤
− (k + μ)
0
0
⎦
= ⎣ k (1 − q) − (γI + μ)
0
kq
0
− (γA + μ)
⎡
is the transition matrix.
Then, R0 is defined as the dominant eigenvalue of matrix K = −T −1 as follows:
⎡
⎤
1
0
0
⎢
⎥
(k + μ)
0 βI N βA N
⎢
⎥
k (1 − q)
1
⎥ ⎢
⎢
⎥
−1
−
0
K = −T = − ⎣0 0
⎥
0 ⎦ · ⎢−
⎢ (k + μ) (γI + μ) (γI + μ)
⎥
⎣
⎦
0 0
0
1
kq
0
−
−
(k + μ) (γA + μ)
(γA + μ)
⎤
⎡
βI Nk (1 − q)
βA Nkq
βI N
βA N
+
⎢ (k + μ) (γI + μ) (k + μ) (γA + μ) (γI + μ) (γA + μ) ⎥
⎥,
=⎢
⎦
⎣
0
0
0
0
0
0
⎤
⎡
−
from which we obtain that
R0 =
βA Nkq
βI Nk (1 − q)
+
.
(k + μ) (γI + μ) (k + μ) (γA + μ)
(4)
96
V. Bitsouni et al.
2.3 Local Stability Analysis of the SEIAR Model
We proceed with the local stability analysis of the model. The Jacobian matrix of
system (1) is
⎡
⎤
−βI I − βA A − μ
0
−βI S
−βA S
0
⎢ β I +β A
− (k + μ)
βI S
βA S
0 ⎥
⎢
⎥
I
A
⎢
⎥
J=⎢
0
0 ⎥.
0
k (1 − q) − (γI + μ)
⎢
⎥
⎣
0
kq
0
− (γA + μ) 0 ⎦
γA
−μ
0
0
γI
(5)
Theorem 1 If R0 < 1, the disease-free steady state, (N, 0, 0, 0, 0), of system (1) is
locally stable.
Proof For the disease-free steady state (N, 0, 0, 0, 0), we obtain a double negative
eigenvalue, λ1 = −μ, and the characteristic equation of the reduced 3 × 3 matrix
λ3 + γI + γA + k + 3μ λ2 + (k + μ) (γI + γA + 2μ) + (γI + μ) (γA + μ)
− Nk (βI (1 − q) + βA q) λ + (k + μ) (γI + μ) (γA + μ)
− Nk βI (1 − q) (γA + μ) + βA q (γI + μ) = 0.
We prove the stability of this steady state using the Routh-Hurwitz criterion [9]. The
disease-free steady state is stable if and only if
γI + γA + k + 3μ > 0,
(6)
(k + μ) (γI + μ) (γA + μ) − βI Nk (1 − q) (γA + μ) − βA Nkq (γI + μ) > 0,
(7)
(γI + γA + k + 3μ) (k + μ) (γI + γA + 2μ) + (γI + μ) (γA + μ)
− Nk βI (1 − q) + βA q
− (k + μ) (γI + μ) (γA + μ)
+ Nk βI (1 − q) (γA + μ) + βA q (γI + μ) > 0.
(8)
Inequality (6) always holds. Inequality (7) can be equivalently written as
R0 =
βA Nkq
βI Nk (1 − q)
+
< 1.
(k + μ) (γI + μ) (k + μ) (γA + μ)
(9)
A Model for the Outbreak of COVID-19: Vaccine Effectiveness in a Case Study of Italy
97
By using Mathematica [10], we can confirm that inequality (8) holds for R0 < 1;
thus, the disease-free steady state is stable.
Since we incorporate the demographic terms, we are interested in exploring the
longer-term persistence and the endemic dynamics of the disease. Setting equal to
zero the right-hand side of system (1), we find a unique endemic steady state. Then,
we are interested in determining the conditions necessary for endemic steady-state
stability.
Theorem 2 If R0 > 1, the endemic steady state, (S ∗ , E ∗ , I ∗ , A∗ , R ∗ ), of system
(1) with
N
(γI + μ) (γA + μ) (k + μ)
=
,
k (βI (γA + μ) + q (βA (γI + μ) − βI (γA + μ)))
R0
Nμ
1
∗
1−
E =
,
R0
(k + μ)
1
Nk (1 − q) μ
1−
,
I∗ =
R0
(γI + μ) (k + μ)
1
Nkqμ
1−
,
A∗ =
R0
(γA + μ) (k + μ)
Nk qγA μ + γI (γA + μ − qμ) 1
1−
R∗ =
,
R0
(γI + μ) (γA + μ) (k + μ)
S∗ =
(10)
is locally stable.
Proof The characteristic equation of the Jacobian matrix (5) at the endemic steady
state is
λ4 + α3 λ3 + α2 λ2 + α1 λ + α0 = 0,
with
α3 = γI + γA + k + 3μ + μR0 > 0,
α2 = μR0 (γI + γA + k + 3μ) + (γI + γA + 2μ) (k + μ) + (γI + μ) (γA + μ)
−
N
k (βI (1 − q) + βA q) ,
R0
α1 = μR0 (γI + γA + 2μ) (k + μ) + (γI + μ) (γA + μ) (μR0 + k + μ)
−
N
k (βI (1 − q) (γA + μ + 1) + βA q (γI + μ + 1)) ,
R0
α0 = μR0 (k + μ) (γI + μ) (γA + μ) − μ
N
k (βI (1 − q) (γA + μ) + βA q (γI + μ)) .
R0
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V. Bitsouni et al.
From the Routh-Hurwitz criterion, the endemic steady (10) is locally stable if and
only if
α0 > 0, α1 > 0, α3 > 0 and α1 α2 α3 − α12 − α0 α32 > 0.
We always have that α3 > 0, whereas a0 > 0 is equivalent to R0 > 1. By using
Mathematica [10], we can confirm that the rest of the above relations hold for R0 >
1; thus, the endemic steady state is stable.
2.4 Global Stability Analysis of the SEIAR Model
Theorem 3 If R0 ≤ 1, then the disease-free steady state, (N, 0, 0, 0, 0), of system
(1) is globally asymptotically stable.
Proof We prove the global stability of the disease-free steady state (N, 0, 0, 0, 0)
by constructing a Lyapunov function. We consider the function V1 : R+ × R3 → R
with
NβA
S
NβI
I+
A.
V1 (S (t) , E (t) , I (t) , A (t)) = S − N − N ln
+E +
N
γI + μ
γA + μ
We take the derivative of V1 with respect to t:
NβI
NβA
N
V1 = S 1 −
+E +
I +
A
S
γI + μ
γA + μ
μN 2
βI Nk (1 − q)
βA Nkq
= 2μN − μS −
+ E (k + μ)
+
−1
S
(k + μ) (γI + μ) (k + μ) (γA + μ)
S
N
= −μN
+
− 2 + E (k + μ) (R0 − 1) .
N
S
From the arithmetic-geometric mean inequality, we have
1
2
S
N
+
N
S
;
≥
2
S
N
S N
=1⇒
+
− 2 ≥ 0.
N S
N
S
Thus, if R0 ≤ 1, then V1 ≤ 0 for all t ≥ 0 and (S, E, I, A) ∈ R+ × R3 sufficiently
close to (N, 0, 0, 0), and V1 (t) = 0 holds only for (S, E, I, A) = (N, 0, 0, 0).
Hence, the singleton {(N, 0, 0, 0)} is the largest invariant set for which V1 = 0.
Then, from LaSalle’s invariance principle [11], it follows that the disease-free steady
state is globally asymptotically stable.
Theorem 4 If R0 > 1, then the endemic steady state, (S ∗ , E ∗ , I ∗ , A∗ , R ∗ ), of
system (1) is globally asymptotically stable.
A Model for the Outbreak of COVID-19: Vaccine Effectiveness in a Case Study of Italy
Proof We consider the function V2 : R+
4
99
→ R with
S
E
V2 (S (t) , E (t) , I (t) , A (t)) = S − S ∗ − S ∗ ln ∗ + E − E ∗ − E ∗ ln ∗
S
E
βI S ∗
βA S ∗
I
A
+
I − I ∗ − I ∗ ln ∗ +
A − A∗ − A∗ ln ∗ .
γI + μ
I
γA + μ
A
We take the derivative of V2 with respect to t:
βI S ∗
βA S ∗
S∗
E∗
I∗
A∗
V2 = S 1 −
+E 1−
+
I 1−
+
A 1−
S
E
γI + μ
I
γA + μ
A
∗
S
E∗
= (μN − βI SI − βA SA − μS) 1 −
+ (βI SI + βA SA − (k + μ) E) 1 −
S
E
∗
∗
β S
I
+ I
(k (1 − q) E − (γI + μ) I ) 1 −
γI + μ
I
β S∗
A∗
.
+ A
(kqE − (γA + μ) A) 1 −
γA + μ
A
After using the relations
μN = βI S ∗ I ∗ + βA S ∗ A∗ + μS ∗
and
βI S ∗ I ∗ +βA S ∗ A∗ = (k + μ) E ∗ , k (1 − q) E ∗ = (γI + μ) I ∗ , kqE ∗ = (γA + μ) A∗ ,
βI S ∗ I ∗ E
βA S ∗ A ∗ E
and
, we have
I E∗
AE ∗
∗
S I E∗
I∗ E
S
S∗
∗
∗ ∗ S
− 2 − βI S I
+ ∗ ∗
+
V2 = − μS
+
−3
S∗
S
S
S I E
I E∗
∗
S A E∗
A∗ E
S
− βA S ∗ A ∗
−
3
.
+ ∗ ∗
+
S
S A E
A E∗
2
2
and by adding and subtracting the terms
From the arithmetic-geometric mean inequality, we have that
1
3
S∗
S I E∗
I∗ E
+ ∗ ∗
+
S
S I E
I E∗
;
≥
3
S∗ S I E∗ I ∗ E
=1
S S∗ I ∗ E I E∗
100
V. Bitsouni et al.
and
1
3
S A E∗
A∗ E
S∗
+ ∗ ∗
+
S
S A E
A E∗
;
≥
3
S ∗ S A E ∗ A∗ E
= 1.
S S ∗ A∗ E A E ∗
4
Hence, V2 ≤ 0 for all (S, E, I, A) ∈ R+ , and the equality holds only for the
endemic steady state (S ∗ , E ∗ , I ∗ , A∗ ). We conclude again from LaSalle’s invariance
principle that the endemic steady state is globally asymptotically stable.
2.5 Numerical Simulations for the SEIAR Model
We proceed to the estimation of the already known (discrete) epidemic curve of the
dR
(see, e.g.,
disease in Italy, as obtained from the data set [12], by (the continuous)
dt
[13] and [14]). We plot together the two functions in Fig. 2. The total population
of Italy is 60,456,999. Once the restriction of movement (quarantine) during the
manifestation of COVID-19 was applied, it limited the spread of the disease. To
this end, we follow the approach in [15], and we consider as the total population
N = 60,456,999/250.
Fig. 2 The number of confirmed cases per day in Italy until July 2020. The blue dots represent
dR
the data obtained from [12] and the red curve the graph of
, as obtained by (1). The parameters
dt
used here are as follows: βI = 2.55/N, βA = 1.275/N, k = 0.07, μ = 0.001, γI = 0.0625, γA =
0.083 and q = 0.425, and the initial conditions (ICs) are S0 = N − 200, I0 = 100, A0 = 100 and
E0 = R0 = 0
A Model for the Outbreak of COVID-19: Vaccine Effectiveness in a Case Study of Italy
101
Fig. 3 The dynamics of the proportion of the values of model (1). (a) For all subgroups. (b) For
the groups of both symptomatic and asymptomatic infectious individuals, I +A, towards the steady
state I ∗ +A∗ . The parameters used are the same as in Fig. 2 (see also Table 1). The initial conditions
are S0 = 1–0.0008, I0 = 0.0004, A0 = 0.0004 and E0 = R0 = 0
In Fig. 3 we show the dynamics of the proportion of the values of model (1) for
the set of parameters used in Fig. 2. We see in Fig. 3a that, for this set of parameters,
the solution of the system has an oscillatory behaviour towards the endemic steady
state. This can be more clear in Fig. 3b, where the proportion of the infectious
population (both symptomatic and asymptomatic) oscillates towards the proportion
of the steady state I ∗ + A∗ .
3 Modelling Transmission Dynamics of COVID-19 in a
Vaccinated Population
In this section, we consider the subclass of the vaccinated-with-a-prophylacticvaccine (V ) individuals. We set 0 ≤ p ≤ 1 for the vaccine coverage as well as
0 ≤ < 1 for the vaccine efficacy [16]. Then, the model becomes
dS
= (1 − p) μN − βI SI − βA SA − μS,
dt
dV
= pμN − (1 − ) (βI V I + βA V A) − μV ,
dt
dE
= (S + (1 − ) V ) (βI I + βA A) − (k + μ) E,
dt
dI
= k (1 − q) E − (γI + μ) I,
dt
dA
= kqE − (γA + μ) A,
dt
(11a)
(11b)
(11c)
(11d)
(11e)
102
V. Bitsouni et al.
dR
= γI I + γA A − μR,
dt
(11f)
along with the initial conditions:
(S (0) , V (0) , E (0) , I (0) , A (0) , R (0)) = (S0 , V0 , E0 , I0 , A0 , R0 ) ∈ R+
0
6
.
(12)
A flow diagram of the vaccination model is illustrated in Fig. 4. Following
the same steps as before and using the disease-free steady state of the model,
((1 − p) N, pN, 0, 0, 0, 0), we have that the basic reproductive ratio for the model
where vaccination is applied is
RV0 = (1 − p)
βI Nk (1 − q)
βA Nkq
+
(k + μ) (γI + μ) (k + μ) (γA + μ)
Fig. 4 Flow diagram of the SVEIAR model (11)
= (1 − p) R0 .
(13)
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103
The endemic steady state, (S ∗ , V ∗ , E ∗ , I ∗ , A∗ , R ∗ ), of model (11) is
(1 − p) Nk (1 − q) μ
1
1− V ,
1− V ,
(γI + μ) (k + μ)
R0
R0
(1 − p) Nk qγ μ + γ (γ + μ − qμ) A
I A
(1 − p) Nkqμ
1
1
1− V ,
1− V
.
(γA + μ) (k + μ)
(γI + μ) (γA + μ) (k + μ)
R
R
(1 − p) N pN
Nμ
, V,
(k + μ)
RV
R
0
0
1
0
0
We prove the global stability of the model following the same steps as before.
Theorem 5 If RV0 ≤ 1, then the disease-free steady state, ((1−p) N,pN, 0, 0, 0, 0),
of system (11) is globally asymptotically stable.
Proof We prove that the disease-free steady state of system (11) is globally
asymptotically stable by applying again the LaSalle’s invariance principle for the
2
Lyapunov function V1V : R+ × R3 → R, with
V1V (S (t) , V (t) , E (t) , I (t) , A (t)) = S − (1 − p) N − (1 − p) N ln
S
(1 − p) N
V
(1 − p) NβI
(1 − p) NβA
+ V − pN − pN ln
+E+
I+
A,
pN
γI + μ
γA + μ
and following the steps corresponding to the proof of Theorem 3.
Theorem 6 If RV0 > 1, then the endemic steady state, (S ∗ , V ∗ , E ∗ , I ∗ , A∗ , R ∗ ), of
system (11) is globally asymptotically stable.
Proof We prove that the endemic steady state of system (11) is globally asymptotically stable by applying again LaSalle’s invariance principle for the Lyapunov
5
function V2V : R+ → R, with
S
V
V2V (S (t) , V (t) , E (t) , I (t) , A (t)) = S − S ∗ − S ∗ ln ∗ + V − V ∗ − V ∗ ln ∗
S
V
β S ∗ + (1 − ) βI V ∗
E
I
I − I ∗ − I ∗ ln ∗
+ E − E ∗ − E ∗ ln ∗ + I
E
γI + μ
I
A
βA S ∗ + (1 − ) βA V ∗
A − A∗ − A∗ ln ∗ ,
+
γA + μ
A
and following the steps corresponding to the proof of Theorem 4.
104
V. Bitsouni et al.
3.1 Numerical Simulations for the SVEIAR Model
To assess the vaccine effectiveness, we focus on three important epidemiological
measures [17]: (1) the risk of infection spread, represented by RV0 , (2) the peak
prevalence of infection and (3) the time at which the peak prevalence occurs.
Relation (13) shows that the vaccine coverage, p, and vaccine efficacy, , act
multiplicatively on R0 . As the proportion of asymptomatic cases is still unknown, in
Fig. 5a, we present a contour plot of the dependence of RV0 on the vaccine coverage
and vaccine efficacy, for different proportion of asymptomatic cases. The coloured
curves represent the threshold RV0 = 1 in (13), . . . , between scenarios in which
the infection is expected to spread (represented by the area below the threshold;
RV0 > 1) or not spread (represented by the area above the threshold; RV0 < 1). The
plot indicates that the vaccine efficacy and coverage need to be greater for small
proportion of asymptomatic cases. As the number of symptomatic cases increases, a
more effective vaccine is needed. We see that even for a severe COVID-19 epidemic,
as in our case, the vaccine can prevent the infection spread if both the vaccine
efficacy and the vaccine coverage are high. Considering, however, that the data
reflect a period where the severity of COVID-19 was not yet known and the average
number of close contacts between individuals was very high due to occasions and
events, the transmission rate, βI , as obtained by the data is not the most appropriate
index to predict the vaccine effectiveness, as the situation has changed dramatically
and close contacts have been significantly reduced. Hence, in Fig. 5b, we present
a corresponding contour plot for a lower βI . We see that, in the case of a reduced
(a)
(b)
Fig. 5 Assessing the vaccine effectiveness: A contour plot showing the dependence of RV0 on the
vaccine efficacy, vaccine coverage and proportion of asymptomatic cases for (a) βI = 2.55 and
(b) βI = 0.2. The rest of the model parameters are given in Table 1
A Model for the Outbreak of COVID-19: Vaccine Effectiveness in a Case Study of Italy
105
Fig. 6 Assessing the vaccine effectiveness: The effect of vaccination on the prevalence of
infection, with ICs, S0 = V0 = 0.4996 and I0 = A0 = 0.0004, and for (a) βI = 2.55 and
(b) βI = 0.2. The rest of the model parameters are given in Table 1
transmission rate, the vaccine can prevent the infection spread, even for imperfect
vaccines and small vaccine coverage.
In Fig. 6 we see the effect of vaccine efficacy on the proportion of the infection
dynamics for high (Fig. 6a) and low (Fig. 6b) transmission rates. Higher vaccine
efficacy leads to milder but prolonged epidemics due to the slower rate of infection
transmission. Moreover, it causes later occurrence of the first infection incidence
and peak prevalence and a slower rate of postpeak prevalence decline.
4 Conclusions
We presented an ad hoc SEIAR model with horizontal transmission and demographic terms for the epidemic spread of COVID-19, and we extended the model
to include vaccination. The stability of both models is proved by implementing
suitable Lyapunov functions; the model is fitted to real data from the epidemic in
Italy. We studied the condition under which a vaccine can prevent disease spread.
We accessed the vaccine effectiveness, focusing on the risk of infection spread, the
peak prevalence of infection and the time at which the peak prevalence occurs.
Future work includes further investigation of the vaccine model, by incorporating
different vaccination strategies, and if possible the comparison with biological data.
An extension of the model will also include additional important factors of COVID19 spread, such as the population age, the geographical spread of the epidemics (see,
e.g. Refs [28–30] and other references therein) and the waning immunity gained by
the infected individuals, as well as vertical transmission and migration terms for the
infected individuals.
106
V. Bitsouni et al.
Table 1 Model parameters, values, units and relevant references
Param. Description
μ
Birth/death rate
βI
Transmission rate of
symptomatic infectious
individuals
βA
k
q
γI
γA
p
Transmission rate of
asymptomatic
infectious individuals
Incubation rate (rate of
latent individuals
becoming infectious)
Proportion of the
asymptomatic
infectious individuals
Recovery rate of the
symptomatic infectious
individuals
Recovery rate of the
asymptomatic
infectious individuals
Proportion of
vaccinated individuals
Vaccine efficacy
Value
Unit
Reference
0.001(2 · 10−5 –0.001) days−1
[18, 19]
2.55(0.14–2.55)
individuals−1 ·days−1 [5, 15, 20–22]
βI
2
individuals−1 ·days−1 [23]
0.071(0.071–0.2)
days−1
[24, 25]
0.425(0–1)
–
[3]
0.0625(0.02–0.0625)
days−1
[4, 26, 27]
0.083(0.083–0.33)
days−1
[4, 27]
0.5(0–1)
–
Estimated
0.5(0–1)
–
Estimated
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Rate of Convergence to the Poisson Law
of the Numbers of Cycles in the
Generalized Random Graphs
Sergey G. Bobkov, Maria A. Danshina, and Vladimir V. Ulyanov
√
Abstract Convergence of order O(1/ n) is obtained for the distance in total
variation between the Poisson distribution and the distribution of the number of fixed
size cycles in generalized random graphs with random vertex weights. The weights
are assumed to be independent identically distributed random variables which have
a power-law distribution. The proof is based on the Chen–Stein approach and on
the derived properties of the ratio of the sum of squares of random variables and the
sum of these variables. These properties can be applied to other asymptotic problems
related to generalized random graphs.
Keywords Generalized random graphs · Poisson law · Rate of convergence ·
Cycles
S. G. Bobkov
University of Minnesota, Minneapolis, MN, USA
National Research University Higher School of Economics, Moscow, Russia
e-mail: bobkov@math.umn.edu
M. A. Danshina
Moscow Center for Fundamental and Applied Mathematics, Lomonosov Moscow State
University, Moscow, Russia
e-mail: danschina.maria@yandex.ru
V. V. Ulyanov ()
National Research University Higher School of Economics, Moscow, Russia
Moscow Center for Fundamental and Applied Mathematics, Lomonosov Moscow State
University, Moscow, Russia
e-mail: vulyanov@cs.msu.ru
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_5
109
110
S. G. Bobkov et al.
1 Introduction
Complex networks attract increasing attention of researchers in various fields of
science. In the last years, numerous network models have been proposed. With
the uncertainty and the lack of regularity in real-world networks, these models
are usually random graphs. Random graphs were first defined by Paul Erdős and
Alfred Rényi in their 1959 paper “On Random Graphs” (see [10]) and independently
by Gilbert in [12]. The suggested models are closely related: there are n isolated
vertices, and every possible edge occurs independently with probability p : 0 <
p < 1. It is assumed that there are no self-loops. Later, the models were generalized.
A natural generalization of the Erdös–Rényi random graph is that the equal edge
probabilities are replaced by probabilities depending on the vertex weights. Vertices
with high weights are more likely to have more neighbors than vertices with small
weights. Vertices with extremely high weights could act as the hubs observed in
many real-world networks.
The following generalized random graph (GRG) model was first introduced by
Britton et al.; see [5]. Let V = {1, 2, .., n} be the set of vertices and Wi > 0 be the
weight of vertex i, 1 ≤ i ≤ n. The edge probability of the edge between any two
vertices i and j , for i = j , is equal to
pij =
Wi Wj
Ln + Wi Wj
(1)
and pii = 0 for all i ≤ n. Here, Ln = ni=1 Wi denotes the total weight of all
vertices. The weights Wi , i = 1, 2, ..., n can be taken to be deterministic or random.
If we take all Wi − s as the same constant Wi ≡ nλ/(n − λ) for some 0 ≤ λ < n,
it is easy to see that pij = λ/n for all 1 ≤ i < j ≤ n. That is, the ErdHos–Rényi
random graph with p = λ/n is a special case of the GRG.
There are many versions of the GRG, such as Poissonian random graph
(introduced by Norros and Reittu in [19] and studied by Bhamidi et al. [3]), rank1 inhomogeneous random graph (see [4]), random graph with given prescribed
degrees (see [8]), and Chung–Lu model of heterogeneous random graph (see [7]).
The Chung–Lu model is the closest to the model of generalized random graph. Two
vertices i and j are connected with probability pij = Wi Wj /Ln and independently
of other pairs of vertices, where W = (W1 , W2 , ..., Wn ) is a given sequence. It is
necessary to assume that Wi2 ≤ Ln , for all i. Under some common conditions (see
[15]), all of the abovementioned versions of the GRG are asymptotically equivalent,
meaning that all events have asymptotically equal probabilities. The updated review
on the results about these inhomogeneous random graphs can be seen in Chapter 6
in [21].
One of the problems that arise in real networks of various nature is the spread of
the virus. In [6], the authors proposed an approach called nonlinear dynamic system
(NLDS) for modeling such processes. Consider a network of n vertices represented
by an undirected graph G. Assume an infection rate β > 0 for each connected edge
Rate of Convergence to the Poisson Law of the Numbers of Cycles in GRG
111
that is connected to an infected vertex and a recovery rate of δ > 0 for each infected
individual. Define the epidemic threshold τ as a value such that
β/δ < τ ⇒ infection dies out over time
β/δ > τ ⇒ infection survives and becomes an epidemic.
τ is related to the adjacency matrix A of the graph. The matrix A = [aij ] is an
n × n symmetric matrix defined as aij = 1 if vertices i and j are connected by
an edge and aij = 0 otherwise. Define a walk of length k in G from vertex v0 to
vk to be an ordered sequence of vertices (v0 , v1 , ..., vk ), with vi ∈ V , such that vi
and vi+1 are connected for i = 0, 1, ..., k − 1. If v0 = vk , then the walk is closed.
A closed walk with no repeated vertices (with the exception of the first and last
vertices) is called a cycle. For example, triangles, quadrangles, and pentagons are
cycles of length three, four, and five, respectively. In the following, the cycle will be
denoted by the first k vertices, without specifying the vertex vk , which is the same
as v0 : (v0 , v1 , ..., vk−1 ).
In Theorem 1 in [6], it has been stated that τ is equal to 1/λ1 , where λ1 is the
largest eigenvalue of the adjacency matrix A. The following lower bound for λ1 (A)
was shown in [20]
λ1 (A) ≥
6) +
36)2 + 32e3 /n
,
4e
where n, e, and ) are the number of vertices, edges, and triangles in G, resp.
Moreover, using information about the cycle numbers of higher orders, one can
get more precise upper bounds for τ .
In [13], the central limit theorems were proved for the total number of edges
in GRG. There are also many results on asymptotic properties of the number of
triangles in homogeneous cases. For example, for the ErdHos–Rényi random graph,
the upper tails for the distribution of the triangle number had been studied in [2, 9,
14, 16]. Recently, in [18], it was shown for GRG model that asymptotic distribution
of the triangle number converges to a Poisson distribution under strong assumption
that the vertex weights are bounded random variables.
A lot of real-world networks such as social or computer networks in the Internet
(see, e.g. [11]) follow a so-called scale-free graph model; see Chapter 1 in [21]. In
Chapter 6, in [21], it was shown that when the vertex weights have approximately a
power-law distribution, the GRG model leads to scale-free random graph.
In the present paper, we prove
√ not only the convergence, but we get the
convergence rate of order O(1/ n) for the distance in total variation between the
Poisson distribution and the distribution of the number of fixed size cycles in GRG
with random vertex weights. The weights are assumed to be independent identically
distributed random variables which have a power-law distribution. The proof is
based on the Chen–Stein approach and on the derived properties of the ratio of the
sum of squares of random variables and the sum of these variables. These properties
can be applied to other asymptotic problems related to GRG.
112
S. G. Bobkov et al.
The main results are formulated in Sect. 2. For their proofs, see Sect. 4. Section 3
contains auxiliary lemmas, some of which are of independent interest.
2 Main Results
Let {1, 2, ..., n} be the set of vertices and Wi be a weight of vertex i : 1 ≤ i ≤ n.
The probability of the edge between vertices i and j is defined in (1). Let Wi ,
i = 1, 2, ..., n, be independent identically distributed random variables distributed
as a random variable W . For k ≥ 3, denote by I (k) the set of potential cycles of
length k. We have that the number of elements in I (k) is equal to (n)k /(2k), where
(n)k = n(n − 1)...(n − k + 1) is the number of ways to select k distinct vertices
in order, and the factor 1/(2k) appears since, for k > 2, a permutation of k vertices
corresponds to a choice of a cycle in I (k) together with a choice of any of two
orientations and k starting points. For example, all six cycles {1, 3, 4}, {3, 4, 1},
{4, 1, 3}, {4, 3, 1}, {1, 4, 3}, and {3, 1, 4} are, in fact, one cycle of length 3. For
α ∈ I (k), let Yα be the indicator that α occurs as a cycle in GRG. For example,
P(Y{1,3,4} = 1) = p13 p34 p41 .
For any integer-valued nonnegative random variables Y and Z, denote the total
variation distance between their distributions L(Y ) and L(Z) by
L(Y ) − L(Z) ≡ suph=1 |Eh(Y ) − Eh(Z)|,
(2)
where h is any real function defined
on {0, 1, 2, ...} and h ≡ supm≥0 |h(m)|.
For k ≥ 3, put Sn (k) = α∈I (k) Yα , that is, Sn (k) is the number of cycles of
length k. Let Zk be a random variable having Poisson distribution with parameter
λ(k) = (EW 2 /EW )k /(2k).
Theorem 1 For any k ≥ 3, one has
L(Sn (k)) − L(Zk ) = O(n−1/2 ),
(3)
P(W > x) = o(x −2k−1), as x → +∞.
(4)
provided that
Remark 1 Relation (3) holds under condition that W has power-law distribution.
The condition on the tail behavior of the distribution of W can be replaced by
stronger moment condition: the finiteness of expectation EW 2k+1 .
Remark 2 Recently in [18], the convergence in distribution of the number of
triangles Sn (3) in a generalized random graphs to the Poisson random variable
Z3 was proved by method of moments under assumption that the vertex weights
Wi -s are bounded random variables. In Theorem 1, we have used the Chen–
Stein approach; see, e.g., [1] and [2]. This allows us not only to extend the
Rate of Convergence to the Poisson Law of the Numbers of Cycles in GRG
113
Triangles number
Poisson (338.72)
100
80
60
40
20
0
280
300
320
340
360
380
400
Fig. 1 Histogram of the number of triangles in GRG with 2000 vertices. The distribution of vertex
weights Wi ∼ U ni(10, 15), forall i ≤ 2000. The number of realizations is 500
400
380
Ordered Values
360
340
320
300
280
280
300
320
340
360
380
400
Theoretical quantiles
Fig. 2 Q–Q plot for the number of triangles in GRG with 2000 vertices and the Poisson variable
P ois(338.72). Wi ∼ U ni(10, 15), forall i ≤ 2000. The number of realizations is 500
convergence result to cycles of any fixed length k but also to get the rate of
convergence. Moreover, we replace the assumption about the boundness of Wi -s
with the condition that Wi has a power-law distribution. As we noted in Introduction,
this condition better matches real-world networks.
Figures 1 and 2 illustrate the results of Theorem 1, with the example of the
number of triangles distribution.
114
S. G. Bobkov et al.
The next results are not directly connected with number of cycles in GRG. They
are an important part of the proof of Theorem 1. At the same time, the results are of
independent interest. They describe the asymptotic properties of ratio of a sum of
the squares of n i.i.d. random variables and a sum of these random variables. These
properties can be applied to other asymptotic problems related to GRG.
Given i.i.d. positive random variables X, X1 , . . . , Xn , define the statistics
Tn =
X12 + · · · + Xn2
.
X1 + · · · + Xn
Assume that X has a finite second moment, so that, by the law of large numbers,
with probability one
p
lim Tn =
n→∞
EX2 p
EX
for any p ≥ 1. Here, we describe the tail-type and moment-type conditions which
ensure that this convergence also holds on average.
Theorem 2 Given an integer p ≥ 2, the convergence
p
lim ETn = (EX2 /EX)p
n→∞
(5)
is equivalent to the tail condition
P{X ≥ x} = o(x −p−1 ) as x → ∞.
(6)
Moreover, if P{X ≥ x} = O(x −p−3/2) as x → ∞, then
ETn − (EX2 /EX)p = O(n−1/2)
p
(7)
The finiteness of the moment EXp+1 is sufficient for (5) to hold, while the finiteness
of the moments EXq is necessary for any real value 1 ≤ q < p + 1.
Let Mn = max1≤i≤n Xi . For p ≥ 2, define
(p)
Rn
p
= Tn Mn2 /(X1 + X2 + ... + Xn ).
(8)
(p)
By the law of large numbers, Rn → 0 as n → ∞ a.s., under mild moment
(p)
assumptions. The next theorem gives the order of convergence of ERn to zero
under tail-type and moment-type conditions.
Theorem 3 Given an integer p ≥ 2, if P(X ≥ x) = O(x −p−7/2 ) as x → +∞,
then
(p)
ERn
= O(n−1/2).
(9)
Rate of Convergence to the Poisson Law of the Numbers of Cycles in GRG
115
When p > 8 and EXp+4 is finite, the rate can be improved to
(p)
ERn
= O(n−(p−2)/(p+4) ).
(10)
Moreover, if E eεX < ∞ for some ε > 0, then
(p)
ERn
=O
(log n)2 n
(11)
.
3 Auxiliary Lemmas
Lemma 1 Let Sn = η1 + · · · + ηn be the sum of independent random variables
ηk ≥ 0 with finite second moment, such that ESn = n and Var(Sn ) = σ 2 n. Then,
for any 0 < λ < 1, one has
(1 − λ)2
n .
(12)
P{Sn ≤ λn} ≤ exp −
2 σ 2 + maxk (Eηk )2
Proof We use here the standard arguments. Fix a parameter t > 0. We have
E e−t Sn ≥ e−λt n P{Sn ≤ λn}.
Every function uk (t) = E e−t ξk is positive and convex and admits Taylor’s
expansion near zero up to the quadratic form, which implies that
uk (t) ≤ 1 − t Eξk +
t2
t2
Eξk2 ≤ exp − t Eξk + Eξk2 .
2
2
Multiplying these inequalities, we get
E e−t Sn ≤ exp − tn +
bt 2 ,
2
b=
n
Eξk2 .
k=1
The two bounds yield
P{Sn ≤ λn} ≤ exp − (1 − λ)nt + bt 2 /2 ,
and after optimization over t (in fact, t = 1−λ
b n), we arrive at the exponential bound
P{Sn ≤ λn} ≤ exp −
(1 − λ)2 2 n .
2b
116
S. G. Bobkov et al.
Note that
b = Var(Sn ) +
n
(Eξk )2 ≤ σ 2 + max (Eξk )2 n,
k
k=1
and (12) follows.
For further lemmas, we need additional notation.
Denote by F (x) = P{X ≤ x} (x ∈ R) the distribution function of the random
variable X, and put
εq (x) = x q (1 − F (x)),
x ≥ 0, q > 0.
Raising the sum Un = X12 + · · · + Xn2 to the power p with n ≥ 2p, we have
p
Un =
Xi21 . . . Xi2p ,
(13)
where the summation is performed over all collections of numbers i1 , . . . , ip ∈
{1, . . . , n}. For r = 1, . . . , p, we denoted by C(p, r) the collection of all tuples
γ = (γ1 , . . . , γr ) of positive integers such that γ1 + · · · + γr = p. For any γ ∈
C(p, r), there are n(n − 1) . . . (n − r + 1) sequences Xi1 , . . . , Xip with r distinct
terms that are repeated γ1 , . . . , γr times, resp. Therefore, by the i.i.d. assumption,
p
ETn =
p
n(n − 1) . . . (n − r + 1)
r=1
np
Eξn (γ ),
(14)
γ ∈C(p,r)
where
1
1
2γ
2γ
ξn (γ ) = X1 1 . . . Xr r /( Sr + Sn,r )p
n
n
and
Sr = X1 + · · · + Xr ,
Sn,r = Xr+1 + · · · + Xn .
In the following lemmas, without loss of generality, let EX = 1.
p
Lemma 2 For the boundedness of the sequence ETn , it is necessary that the
moment EXp be finite. Moreover, for the particular collection γ = (p) with r = 1,
we have
Eξn (γ ) ≥ 2−p np EXp 1{X≥n} .
(15)
Rate of Convergence to the Poisson Law of the Numbers of Cycles in GRG
117
2p
Proof Since ξn (γ ) = X1 /( n1 X1 + n1 Sn,1 )p , applying Jensen’s inequality, we get
2p
Eξn (γ ) ≥ EX1
=E
X1
( n1 X1 + n1 ESn,1 Sn,1 )p
X2p
p
( n1 X + n−1
n )
≥ 2−p np EXp 1{X≥n} .
In the sequel, we use the events
An,r = Sn,r ≤
n−r
2
and Bn,r = Sn,r >
n−r
.
2
(16)
By Lemma 1, whenever n ≥ 2p, for some constant c > 0 independent of n,
P(An,r ) ≤ e−c(n−r) ≤ e−cn/2 .
(17)
Lemma 3 If EXp is finite, then Eξn → (EX2 )p as n → ∞, where
1
1
ξn = X12 . . . Xp2 /( Sp + Sn,p )p .
n
n
2p
p
(18)
p
Proof Using X1 . . . Xp ≤ Sp , we have ξn ≤ Sp /( n1 Sn )p ≤ np Sp . Hence,
E ξn 1An,p ≤ np E Sp P(An,p ) = o(e−cn )
p
(19)
for some constant c > 0 independent of n. Here, we applied (17) with r = p and
p
Lemma 2 which ensures that E Sp < ∞. Further, ξn 1Bn,p ≤ 2p X12 . . . Xp2 .Hence,
the random variables ξn 1Bn,p have an integrable majorant. Since also ξn →
X12 . . . Xp2 (the law of large numbers) and 1Bn,p → 1 a.s. (implied by (17)), one
may apply the Lebesgue dominated convergence theorem, which gives Eξn 1Bn →
(EX2 )p . Together with (19), we get Eξn → (EX2 )p .
Lemma 4 If the moment EXp is finite, then for any γ = (γ1 , . . . , γr ) ∈ C(p, r),
2γ
E ξn (γ ) = 4p E
2γ
X1 1 . . . Xr r
( n1 Sr + 1)p
+ o(1).
118
S. G. Bobkov et al.
Proof Using an elementary bound X1 1 . . . Xr r ≤ (X1 + · · · + Xr )2γ1 +···+2γr =
2p
p
p
Sr and applying Jensen’s inequality, we see that ξn (γ ) ≤ np Sr ≤ np r p−1 (X1 +
p
· · · + Xr ). Hence,
2γ
Eξn (γ ) 1An,r ≤ n r
p p−1
r
2γ
EXk 1An,r = np r p EXp P(An,r ) = o(e−c n ).
p
k=1
(20)
On the other hand, on the set Bn,r , there is a point-wise bound
2γ
ξn (γ ) 1Bn,r ≤
2γ
X1 1 . . . Xr r
p
( n1 Sr + n−r
2n )
2γ
≤ 4p
2γ
X1 1 . . . Xr r
( n1 Sr + 1)p
(21)
.
Our task is reduced to the estimation of the expectation on the right-hand side of
(21). Let us first consider the shortest collection γ = (p) of length r = 1.
Lemma 5 Under the condition (6),
2p
E
X1
( n1 X1 + 1)p
= o(np−1 ).
(22)
In addition, if P{X ≥ x} = O(x −q ) for some real value q in the interval p < q <
2p, then
2p
E
X1
( n1 X1 + 1)p
= O(n2p−q ).
(23)
Proof We have
2p
E
X1
( n1 X1 + 1)p
2p
=E
X1
( n1 X1 + 1)p
2p
1{X1 ≥n} + E
X1
( n1 X1 + 1)p
1{X1 <n}
≤ np E Xp 1{X≥n} + E X2p 1{X<n} .
In view of (6), to derive (22), it remains to be bound to the last expectation by
o(np−1 ). Integrating by parts and assuming that x = n is the point of continuity of
F (x), we have, using εp+1 (x) → 0 as x → ∞,
n
E X2p 1{X<n} = −n2p (1 − F (n)) + 2p
x 2p−1 (1 − F (x)) dx
≤ 2p
n
0
0
x p−2 εp+1 (x) dx = o(np−1 ),
(24)
Rate of Convergence to the Poisson Law of the Numbers of Cycles in GRG
119
For the second assertion (23), we similarly have
n
2p
x 2p−1−q εq (x) dx = O(n2p−q ),
0
E X 1{X≥n} = O(n
p
p−q
)+p
∞
x p−q−1 εq (x) dx = O(np−q ).
n
Lemma 6 Let γ = (γ1 , . . . , γr ) ∈ C(p, r), 2 ≤ r ≤ p − 1. Under (6), we have
2γ
E
2γ
X1 1 . . . Xr r
( n1 Sr + 1)p
= o(np−r−1 log n).
(25)
Proof If all γi ≤ p/2, there is nothing to prove, since then
2γ
E
2γ
X1 1 . . . Xr r
( n1 Sr + 1)p
2γ
2γ
≤ E X1 1 . . . E Xr r ≤ (EXp )r .
In the other case, suppose for definiteness that γ1 is the largest number among
all γi s. Necessarily, γ1 > p/2 and γi < p/2 for all i ≥ 2. Since Sr < n implies
X1 < n, we similarly have
2γ
E
2γ
X1 1 . . . Xr r
( n1 Sr + 1)p
1{Sr <n} ≤ (EXp )r−1 E X2γ1 1{X<n} .
To bound the last expectation, note that r − 1 ≤ γ2 + · · · + γr < p/2, so that
p ≥ 2r − 1. Hence, if x = n is the point of continuity of F (x), similarly to (24), we
get
n
E X2γ1 1{X<n} ≤ 2γ1
x 2γ1 −p−2 εp+1 (x) dx.
(26)
0
But since γ1 ≤ p − r + 1,
n
n
2γ1 −p−2
x
εp+1 (x) dx ≤
x p−2r εp+1 (x) dx = o(np−2r+1 ),
1
(27)
1
if p ≥ 2r or p ≤ 2r − 2, which is even stronger than the rate o(np−r−1 ). In the
remaining case p = 2r − 1, the last integral is o(log n). This proves (25) for the part
of the expectation restricted to the set Sr < n, that is,
2γ
E
2γ
X1 1 . . . Xr r
( n1 Sr + 1)p
1{Sr <n} = o(np−r−1 log n).
(28)
120
S. G. Bobkov et al.
Note that the logarithmic term cannot be removed in the special situation where
p
3, r = 2, γ1 = 2, and γ2 = 1, in which case the last integral in (27) becomes
$ n= −1
x
ε4 (x) dx.
1
Turning to the expectation over the complementary set Sr ≥ n, introduce the
events
i = Xi ≥ max Xj , i = 1, . . . , r.
j =i
On every such set, Xi ≤ Sr ≤ rXi . In particular, Sr ≥ n implies Xi ≥ n/r. Hence,
together with (28), (25) would follow from the stronger assertion
2γ
E
2γ
X1 1 . . . Xr r
p
Xi
1{Xi ≥n}∩i = o(n−r−1 )
(29)
with an arbitrary index 1 ≤ i ≤ r.
Case 1 i ≥ 2. If we fix any values X1 = x1 and Xi = xi , then the expectation with
respect to Xj , j = i, in (29) will yield a bounded quantity (since the p-moment is
finite). Hence, (29) is simplified to
2γ
2γ −p
E X1 1 Xi i
1{Xi ≥n}∩{Xi ≥X1 } = o(n−r−1 ).
(30)
Here, the expectation over X1 may be carried out and estimated similarly to (26),
by replacing n with xi . Namely,
2γ
E X1 1 1{X1 ≤xi } ≤ 2γ1
xi
0
2γ −p
x 2γ1−p−2 εp+1 (x) dx = δ(xi ) xi 1
with some δ(xi ) → 0 as xi → ∞ (this assertion may be strengthened when 2γ1 −
p = 1). Hence, the expectation in (30) is bounded by
2γ +2γ1 −2p
E Xi i
2γ +2γ1 −2p
δ(Xi ) 1{Xi ≥n} ≤ δn E Xi i
1{Xi ≥n}
∞
= δn n2γi +2γ1 −2p (1 − F (n)) + ci δn
x 2γi +2γ1 −2p−1 (1 − F (x)) dx
= o(n2γi +2γ1 −3p−1 ) + ci δn
∞
n
x 2γi +2γ1 −3p−2 εp+1 (x) dx
n
= o(n2γi +2γ1 −3p−1 ),
where δn = supx≥n δ(x) → 0. To obtain (30), it remains
to check that 2γi + 2γ1 −
3p − 1 ≤ −r − 1. And indeed, since p = γi + γ1 + j =i,1 γj ≥ γi + γ1 + (r − 2),
the desired relation would follow from 2 (p − (r − 2)) − 3p − 1 ≤ −r − 1, that is,
p + r ≥ 4 (which is true).
Rate of Convergence to the Poisson Law of the Numbers of Cycles in GRG
121
Case 2 i = 1. If we fix any value X1 = x1 , the expectation with respect to Xj ,
j = 1, will yield a bounded quantity (since the p-moment is finite). Hence, (29) is
simplified to
E X2γ1 −p 1{X≥n} = o(n−r−1 ).
Here, the expectation may be estimated similarly. Namely,
E X2γ1 −p 1{X≥n} =
∞
x 2γ1−p dF (x)
n
2γ1 −2p−1
= o(n
)+
∞
x 2γ1−2p−2 εp+1 (x) dx = o(n2γ1 −2p−1 ).
n
It remains to be seen that 2γ1 − 2p − 1 ≤ −r − 1. Again, since γ1 ≤ p − (r − 1),
the latter would follow from 2(p − r + 1) − 2p − 1 ≤ −r − 1, which is the same
as r ≥ 2.
(p)
We now consider the lemmas which enable us to get a bound for ERn ; see (8).
Without loss of generality, let EX = 1 and n ≥ 2p.
Introduce additional notation: Mn,r = maxr<i≤n Xi and (1 ≤ r ≤ p).
Recall that there is the representation (13) but instead of (14) we write now
(p)
ERn
=
p
n(n − 1) . . . (n − r + 1)
np+1
r=1
E ηn (γ )Mn2 ,
(31)
γ ∈C(p,r)
where
2γ
ηn (γ ) =
2γ
X1 1 . . . Xr r
( n1 Sr + n1 Sn,r )p+1
.
In order to bound the expectations on the right-hand side of (31), we use again
the events An,r and Bn,r ; see (16). From elementary inequalities Mn ≤ Sn and
X1 1 . . . Xr r ≤ (X1 + · · · + Xr )2γ1 +···+2γr ≤ Sn ,
2γ
2γ
2p
p+1
it follows that ηn (γ )Mn2 ≤ np+1 Sn
p+1
≤ 2p np+1 (Sr
p+1
+ Sn,r ), implying
p+1
p+1
p+1
E ηn (γ ) Mn2 1An,r ≤ 2p np+1 E Sr P(An,r ) + ESr E Sn,r 1An,r .
(32)
Here, by Lemma 1 with λ = 1/2 and using n − r ≥ 12 n, we have
P(An,r ) ≤ exp −
1
n
,
16 b2
b 2 = EX2 .
(33)
122
S. G. Bobkov et al.
Also, assuming that the moment EXp+2 is finite and applying the Hölder inequality
with exponents (p + 2)/(p + 1) and p + 2, one may bound the last expectation in
(32) as
p+1
p+2
E Sn,r 1An,r ≤ E Sn,r
p+1
p+2
1
(P(An,r )) p+2 .
p+2
By Jensen inequality, E Sn,r ≤ r p+1 EXp+2 . Applying this in (32), the inequality
(33) yields an exponential bound
E ηn (γ ) Mn2 1An,r ≤ e−cn
(34)
with some constant c > 0 which does not depend on n.
As for the set Bn,r , we use on it a point-wise upper bound
2γ
2γ
ηn (γ ) ≤ 2p+1 X1 1 . . . Xr r /( n1 Sr + 1)p+1 . One may also use Mn ≤ Mr +Mn,r ≤
2 ).
Sr + Mn,r , implying, by Jensen’s inequality, Mn2 ≤ (r + 1) (X12 + · · · + Xr2 + Mn,r
It gives
E ηn (γ ) Mn2 1Bn,r ≤ 2p+1 (r + 1)
r
2γ
E
k=1
+2
p+1
(r + 1)
r
k=1
2γ
X1 1 . . . Xr r
( n1 Sr + 1)p+1
2γ
E
Xk2
2γ
X1 1 . . . Xr r
( n1 Sr + 1)p+1
2
EMn,r
(35)
2 may be replaced with EM 2 .
Without an essential loss, the last expectation EMn,r
n
The second last expectation was considered in Lemmas 5–6 under the condition (6),
which holds as long as the moment EXp+1 is finite. The third last expectation in
(35), due to an additional factor Xk2 , dominates the second last and needs further
consideration under stronger moment assumptions. Recalling (34) and returning to
(31), let us summarize in the following statement.
Lemma 7 If the moment EXp+2 is finite, then
(p)
c ERn
2γ
2γ
X1 1 . . . Xr r 2
E
Xk
1≤k≤r γ ∈C(p,r) np−r+1
( n1 Sr + 1)p+1
≤ e−cn + max
+
max
1
2γ
2γ
X1 1 . . . Xr r
E
EMn2
γ ∈C(p,r) np−r+1
( n1 Sr + 1)p+1
max
1
(36)
with some constant c > 0 which does not depend on n ≥ 2p.
In order to obtain polynomial bounds for the expectations in (36) under suitable
moment or tail assumptions, we need to develop corresponding analogs of Lemmas 5–6. We will consider separately the cases r = 1, r = p, and 2 ≤ r ≤ p − 1
Rate of Convergence to the Poisson Law of the Numbers of Cycles in GRG
123
under the tail condition
P{X ≥ x} = O(1/x p+α )
as x → ∞,
(37)
where α > 0 is a parameter. It implies that the moments EXq are finite for all q <
p + α and is fulfilled as long as the moment EXp+α is finite. Put ε(x) = x p+α (1 −
F (x)), where F denotes the distribution function of the random variable X.
Lemma 8 Under (37) with 1 < α ≤ p + 2,
2p
E
X1
= O(np−α+2 ).
(38)
Xk2 = O(np−α+2 log n).
(39)
( n1 X1 + 1)p+1
Moreover, for any index 1 ≤ k ≤ r,
2p
E
X1
( n1 X1 + 1)p+1
Proof The expectation in (38) is equal to and satisfies
2p
E
2p
X1
( n1 X1 + 1)p+1
1{X1 ≥n} + E
X1
( n1 X1 + 1)p+1
1{X1 <n}
≤ np+1 EXp−1 1{X≥n} + E X2p 1{X<n}
Similarly to (24), we get
EX
2p
1{X<n} ≤ 2p
n
x p−α−1 ε(x) dx = O(np−α ),
0
provided that α < p. In the case α = p, the last integral is bounded by O(log n). In
addition,
∞
p
p−α
E X 1{X≥n} = O(n
)+p
x p−α−1 εq (x) dx = O(np−α ).
n
This proves (38) for α ≤ p. If p < α ≤ p + 2, then (38) holds automatically, since
then 2p < p + α and therefore the expectation in (38) does not exceed the finite
2p
moment EX1 , while the right-hand side is bounded away from zero.
For the second assertion, one may assume that k = 1, in which case the
expectation in (39) is equal to and satisfies
2p+2
E
X1
( n1 X1 + 1)p+1
2p+2
=E
X1
( n1 X1 + 1)p+1
2p+2
1{X1 ≥n} + E
X1
( n1 X1 + 1)p+1
≤ np+1 EXp+1 1{X≥n} + E X2p+2 1{X<n} .
1{X1 <n}
124
S. G. Bobkov et al.
Here, similarly to the previous step, if α < p + 2,
E X2p+2 1{X<n} ≤ 2p
n
x p−α+1 ε(x) dx = O(np−α+2 ).
0
In the case α = p + 2, the last integral is bounded by O(log n). In addition,
EX
p+1
−α+1
1{X≥n} = O(n
)+p
∞
x −α ε(x) dx = O(n−α+1 ).
n
Lemma 9 If the moment EX4 is finite, then
E
X12 . . . Xp2
( n1 Sp + 1)p+1
= O(1).
Moreover, for any index 1 ≤ k ≤ p,
E
X12 . . . Xp2
( n1 Sp + 1)p+1
Xk2 = O(1).
This statement is clear. The last expectation does not exceed EX4 (EX2 )p−1
which is finite and does not depend on n.
Lemma 10 Let γ = (γ1 , . . . , γr ) ∈ C(p, r), 2 ≤ r ≤ p. Under the condition (37)
with 2 < α ≤ 4, for any index 1 ≤ k ≤ r,
2γ
E
2γ
X1 1 . . . Xr r
( n1 Sr + 1)p+1
Xk2 = O(np−r−α+4 ).
(40)
Proof One may reformulate (40) as the statement
2γ
E
2γ
X1 1 . . . Xr r
( n1 Sr + 1)p+1
= O(np−r−α+4 )
(41)
in which γ = (γ1 , . . . , γr ) ∈ C(p + 2, r). If all γi ≤ p+2
2 , there is nothing to prove,
since then
2γ
E
2γ
X1 1 . . . Xr r
( n1 Sr + 1)p+1
2γ
2γ
≤ E X1 1 . . . E Xr r ≤ (EXp+2 )r .
Rate of Convergence to the Poisson Law of the Numbers of Cycles in GRG
125
In the other case, we repeat the arguments used in the proof of Lemma 6. Suppose
for definiteness that γ1 is the largest number among all γi s. Necessarily, γ1 > p+2
2 ,
and therefore, γi < p+2
for
all
i
≥
2.
Since
S
<
n
⇒
X
<
n,
we
similarly
have
r
1
2
2γ
E
2γ
X1 1 . . . Xr r
( n1 Sr + 1)p+1
1{Sr <n} ≤ (EXp+2 )r−1 E X2γ1 1{X<n} .
(42)
To bound the last expectation, note that, if x = n is the point of continuity of
F (x),
n
2γ1
2γ1
E X 1{X<n} = −n (1 − F (n)) + 2γ1
x 2γ1 −1 (1 − F (x)) dx
≤ 2γ1
n
0
x 2γ1 −1−p−α ε(x) dx.
0
But since γ1 ≤ p − r + 3 (which follows from γ1 + γ2 + · · · + γr = p + 2 and
γi ≥ 1), we have
n
x
2γ1 −p−α−1
εp+α (x) dx ≤
1
n
x p−2r−α+5 ε(x) dx.
1
The last integral grows at the desired rate O(np−r−α+4 ) as the worst case, if and
only if p − 2r − α + 5 ≤ p − r − α + 3, that is, r ≥ 2 (which is true). Thus,
E X2γ1 1{X<n} = O(np−r−α+4 )
In view of (42), this proves (41) for the part of the expectation restricted to the set
Sr < n, that is,
2γ
E
2γ
X1 1 . . . Xr r
( n1 Sr + 1)p+1
1{Sr <n} = O(np−r−α+4 ).
(43)
Here, the worst situation is attained in the case r = 2, γ1 = p + 1, γ2 = 1.
Turning to the expectation over the complementary set Sr ≥ n, introduce the
events
i = Xi ≥ max Xj , i = 1, . . . , r.
j =i
On every such set, Xi ≤ Sr ≤ rXi . In particular, Sr ≥ n implies Xi ≥ n/r. Hence,
together with (43), (41) would follow from the inequality
2γ
E
2γ
X1 1 . . . Xr r
p+1
Xi
1{Xi ≥n}∩i = O(n−r−α+3 )
(44)
126
S. G. Bobkov et al.
with an arbitrary index 1 ≤ i ≤ r.
Case 1 i ≥ 2. If we fix any values X1 = x1 and Xi = xi , then the expectation
with respect to Xj , j = i, in (44) will yield a bounded quantity (since the (p + 2)
moment is finite). Hence, (44) is simplified to
2γ
2γ −p−1
E X1 1 Xi i
1{Xi ≥n}∩{Xi ≥X1 } = O(n−r−α+3 ).
(45)
Here, the expectation over X1 may be estimated similarly to the previous step, by
replacing n with xi . Recall that γ1 > p+2
2 and hence 2γ1 ≥ p + 3.
Case 1.1 2γ1 > p + α. Then, we have
2γ
E X1 1 1{X1 ≤xi } ≤ 2γ1
xi
0
2γ −p−α
x 2γ1 −1−p−α ε(x) dx ≤ Cxi 1
with some constant C > 0. Hence, up to a constant, the expectation in (44) is
bounded by
2γ +2γ1 −2p−α−1
E Xi i
1{Xi ≥n} =
∞
x 2γi +2γ1 −2p−α−1 dF (x)
n
= n2γi +2γ1 −2p−α−1 (1 − F (n))
∞
+ci
x 2γi +2γ1 −2p−α−2 (1 − F (x)) dx
n
2γi +2γ1 −3p−2α−1
= O(n
)
∞
+ci
x 2γi +2γ1 −3p−2α−2 ε(x) dx
n
2γi +2γ1 −3p−2α−1
= O(n
).
To obtain (45), it remains to check that 2γi + 2γ1 − 3p − 2α − 1 ≤ −r − α + 3.
And indeed, since
γj ≥ γi + γ1 + (r − 2),
p = γi + γ1 +
j =i,1
the desired relation would follow from 2 (p − (r − 2)) − 3p − 2α − 1 ≤ −r − α + 3,
that is, p + r ≥ α (which is true since α ≤ 4 while p, r ≥ 2).
Rate of Convergence to the Poisson Law of the Numbers of Cycles in GRG
127
2γ
Case 1.2 2γ1 ≤ p + α. Then, E X1 1 1{X1 ≤xi } ≤ E Xp+α which is bounded in xi ,
and the expectation in (44) does not exceed up to a constant
2γ +2γ1 −p−1
E Xi i
1{Xi ≥n} =
∞
x 2γi +2γ1 −p−1 dF (x)
n
= n2γi +2γ1 −p−1 (1 − F (n))
∞
+ci
x 2γi +2γ1 −p−2 (1 − F (x)) dx
n
2γi +2γ1 −2p−α−1
= O(n
)
∞
+ci
x 2γi +2γ1 −2p−α−2 ε(x) dx
n
2γi +2γ1 −2p−α−1
= O(n
).
To obtain (45), it remains to check that
2γi + 2γ1 − 2p − α − 1 ≤ −r − α + 3.
And indeed, by (45), the desired relation would follow from 2 (p − (r − 2)) − 2p −
α − 1 ≤ −r − α + 3, that is, r ≥ 0.
Case 2 i = 1. If we fix any value X1 = x1 in (44), the expectation with respect to
Xj , j = 1, yields a bounded quantity. Hence, (44) is simplified to
E X2γ1 −p−1 1{X≥n} = O(n−r−α+3 ).
We have
E X2γ1 −p−1 1{X≥n} =
∞
x 2γ1−p−1 dF (x)
n
= n2γ1 −p−1 (1 − F (n))
∞
+(2γ1 − p − 1)
x 2γ1 −p−2 (1 − F (x)) dx
n
= O(n2γ1 −2p−α−1 ) +
∞
x 2γ1 −2p−α−2 ε(x) dx
n
2γ1 −2p−α−1
= O(n
).
It remains to be seen that 2γ1 − 2p − α − 1 ≤ −r − α + 3, that is, 2γ1 + r ≤ 2p + 4.
But this follows from p + 2 = γ1 + · · · + γr ≥ γ1 + (r − 1) ≥ γ1 + 2r .
128
S. G. Bobkov et al.
4 Proofs of Main Results
Proof of Theorem 1 Fix k ≥ 3. In the following, we omit k in notation when it is
not necessary. So we write Sn = Sn (k), λ = λ(k), Z = Zk , and I = I (k). We say
that a random variable V has a mixed Poisson distribution with mixing distribution
F when, for every integer m ≥ 0,
m
− P(V = m) = E e
,
m!
where is a
random variable with distribution F.
Put = α∈I EW1 ,W2 ,...,Wn Yα .
We have for any real function h : {0, 1, 2, ...} → R
|Eh(Sn ) − Eh(Z)| ≤ E|EW1 ,...,Wn h(Sn ) − EW1 ,...,Wn h(V )| + |Eh(V ) − Eh(Z)|.
(46)
For each α ∈ I , define Bα ≡ {β ∈ I : α and β have at least one edge in common}.
Put
b1 =
pα pβ ,
α∈I β∈Bα
where pα = EW1 ,...,Wn Yα :
b2 =
pαβ ,
α∈I α =β∈Bα
where pαβ = EW1 ,...,Wn Yα Yβ .
Note that, for any α ∈ I and β ∈ I \ Bα , the cycles α and β may have joint
vertices but they do not have any edge in common. Therefore, for such α and
β, the random variables Yα and Yβ are conditionally independent given weights
W1 , ..., Wn . Thus, by Theorem 1 in [2], proved with the Chen–Stein method and
relations (2) and (46), we get
L(Sn (k)) − L(Z) E(b1 + b2 ) + |E − λ(k)|,
(47)
where we write here and in the following that An Bn or An Bn when there
exists a positive constant c not depending on n such that An ≤ cBn or An ≥ cBn .
For random variables b1 and b2 , we get, cf. (14), by the i.i.d. assumption and
simple inequality for positive c and d : 2cd ≤ c2 + d 2
E(b1 + b2 ) 2k p−1
n(n − 1) . . . (n − r + 1)
p=k+2 r=k
np
γ ∈C(p,r)
Eψn (γ ),
(48)
Rate of Convergence to the Poisson Law of the Numbers of Cycles in GRG
129
where
1
1
2γ
2γ
ψn (γ ) = W1 1 . . . Wr r /( Lr + Ln,r )p
n
n
and
Lr = W1 + · · · + Wr ,
Ln,r = Wr+1 + · · · + Wn .
For example, we have minimal values p = k + 2 and r = k + 1 for the cycles
α = (1, 2, . . . , k) and β = (1, 2, . . . , k − 1, k + 1). Then,
2
2
Wk2 Wk+1
/Lk+2
Epαβ EW14 W22 . . . Wk−1
n .
We have maximal values p = 2k and r = k for the cycle α = (1, 2, . . . , k). Then,
Epα2 ≤ EW14 . . . Wk4 /L2k
n .
Lemmas 4 and 6 and inequality (48) under condition (4) imply
log n
E(b1 + b2 ) = o
.
n
Now, we construct an upper bound for the last summand in (47).
It is clear that
(W12 + · · · + Wn2 )k
1
.
E
E ≤
2k
Lkn
(49)
(50)
On the other hand, note that for a positive a and positive sequence {xi }, i =
1, 2, . . . k, we have (see, e.g., Lemma 8 in [17])
k
<
i=1
1
1
≥ k −
a + xi
a
k
i=1 xi
.
a k+1
Therefore, by the i.i.d. assumption, we get
(W12 + · · · + Wn2 )k
1
E
E ≥
2k
Lkn
−c1
k−1
n(n − 1) . . . (n − r + 1)
nk
r=1
c2
−
n
γ ∈C(k+1,k)
2γ
2γ
W1 1 . . . Wr r
E
(Ln /n)k+1
where c1 and c2 do not depend on n.
Eψn (γ )
γ ∈C(k,r)
,
(51)
130
S. G. Bobkov et al.
Combining Lemmas 4 and 6 and relations (7), (47), (49), (50), and (51), we finish
the proof of Theorem 1.
Proof of Theorem 2 We split the proof of the theorem into several steps. Without
loss of generality, let EX = 1.
p
Necessity By Lemma 3, for the convergence ETn → (EX2 )p , it is necessary that
all summands in (14) with r < p should be vanishing at infinity. In particular,
for the shortest tuple γ with r = 1 as in Lemma 2, it should be required that
n1−p E ξn (γ ) → 0 as n → ∞. Hence, from the inequality (15), it follows that
EXp 1{X≥n} = o(1/n).
This relation may be simplified in terms of the tails of the distribution of X.
Indeed, E Xp 1{X≥n} ≥ np P{X ≥ n}, so that the property (6) is necessary for
p
the convergence ETn → (EX2 )p .
Sufficiency and Rate of Convergence First, note that the condition (6) ensures
that the moment EXp is finite. For the convergence part of Theorem 2, we apply
Lemmas 4–6, which imply that Eξn (γ ) = o(np−r ) for any collection γ =
(γ1 , . . . , γr ) with r < p. It remains to take into account Lemma 3 about the longest
tuple γ̃ = (1, . . . , 1) of length r = p and to recall the representation (14).
Turning to the rate of convergence, first, note that by Lemmas 4 and 6, for any
γ ∈ C(p, r) with 2 ≤ r ≤ p − 1,
log n n(n − 1) . . . (n − r + 1)
Eξ
(γ
)
=
o
n
np
n
(52)
For the shortest tuple γ = (p) with r = 1, we apply Lemma 5 with q = p + 32 and
3
thus assume that P{X ≥ x} = O(x −p− 2 ). Together with Lemma 4, this gives
1 n
Eξn (γ ) = O √ .
p
n
n
(53)
Note that with this tail hypothesis, necessarily, EXβ < ∞ for any β < p + 32 . Since
p ≥ 2, we have that the third moment EX3 is finite. Applying both (52) and (53) in
the representation (14) and using (20), we thus obtain that
√
p
ETn = Eξn 1Bn,p + O(1/ n),
(54)
with ξn defined in (18).
An asymptotic behavior of the last expectation in (54) remains to be studied.
Note that n1 Sn ≥ n1 Sn,p ≥ 12 on the set Bn,p as long as n ≥ 2p. Applying the
Taylor formula, we use an elementary inequality
|x −p − 1| ≤ p 2p+1 |x − 1| for
x ≥ 12 . In particular, on the set Bn,p , one has ( n1 Sn )−p − 1 ≤ p 2p+1 n1 Sn − 1.
Rate of Convergence to the Poisson Law of the Numbers of Cycles in GRG
131
This gives
ξn − X2 . . . X2 1B
p
1
1
1
p+1 2
2 −
S
S
≤
p
2
X
.
.
.
X
−
1
p
n,p
n,p
p
1
n
n
p 2p+1
1
≤ p 2p+1 X12 . . . Xp2 1 − Sn,p +
X12 . . . Xp2 Sp ,
n
n
so, taking the expected values,
Eξn 1B
n,p
1
− EX12 . . . Xp2 1Bn,p ≤ p 2p+1 (EX2 )p E 1 − Sn,p n
+
p 2p+1
(EX2 )p−1 EX3 .
n
In view of (17),
EX12 . . . Xp2 1Bn,p = EX12 . . . Xp2 + e−cn = (EX2 )p + o(e−cn ).
for some constant c > 0. Recalling (20), we thus get that
Eξn − (EX2 )p ≤ p 2p+1 (EX2 )p E 1 − 1 Sn,p n
+
p 2p+1
(EX2 )p−1 EX3 + o(e−cn ).
n
Finally,
1
1
1
p
E Sn,p − 1 = E |Sn,p − n| ≤ E |Sn,p − (n − p)| +
n
n
n
n
1=
p
p
1
≤
Var(Sn,p ) +
EX2 + .
≤ √
n
n
n
n
It remains to refer to (54).
Proof of Theorem 3 Let us apply Lemmas 8–10 in the inequality (36). Using the
bounds for the cases r = 1, r = p, and 2 ≤ r ≤ p − 1 and assuming that (37) is
fulfilled for an integer p ≥ 2 and a real number 2 < α ≤ 4, they imply that
(p)
ERn
≤ e−cn +
1
1 log n
1
1 log n +
+
+ 2 EMn2 ,
+
+
nα−2
n nα−3
nα−2
n
n
where the constant c > 0 does not depend on n. To simplify, we have to assume that
α ≥ 3 leading to
c ERn ≤
1
nα−3
+
log n
EMn2 .
n
(55)
132
S. G. Bobkov et al.
The last expectation in (55) may also be estimated in a polynomial way. Namely,
q
since, for any q ≥ 2, one has Mn2 ≤ (X12 + · · · + Xn )2/q , we get, by Jensen’s
inequality,
q
q
2
2
2
EMn2 ≤ (EX1 + · · · + EXn ) q = n q (EXq ) q .
Therefore, choosing 2 < q < p + α to be sufficiently close to p + α and using
α = 7/2, from (55), we obtain (9).
When EXp+4 is finite, we get (10).
At last, to prove (11), note that the finiteness of the exponential moment of X is
actually equivalent to the family of moment bounds (EXq )1/q ≤ cq, for q ≥ 1,
which for q ≥ 2 give
q
q
q
q
EMn2 ≤ E (X1 + · · · + Xn )2/q ≤ (EX1 + · · · + EXn )2/q ≤ (cq)2 n2/q .
Choosing here q to be of order log n, we arrive at EMn2 ≤ C (log n)2 with a constant
C independent of n. Applying this bound in (55) with α = 4, we then obtain the
much better rate as in (11).
Acknowledgments Theorem 1 has been obtained under support of the Ministry of Education and
Science of the Russian Federation as part of the program of the Moscow Center for Fundamental
and Applied Mathematics under the agreement No. 075-15-2019-1621. Theorems 2 and 3 were
proved within the framework of the HSE University Basic Research Program. Research of
S.Bobkov was supported by the NSF grant DMS-1855575.
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Combinatorial Identities with Binomial
Coefficients
Ya. M. Erusalimsky
Abstract The article is devoted to two methods for obtaining combinatorial identities: the consideration of combinatorial problems that allow different solutions and
when the analysis of the solution of a combinatorial problem leads to combinatorial
identities. Since some identities are known and we present only new proofs for them,
the article is mainly of a methodical nature. This applies primarily to the content of
Sect. 2.
In this paper, we obtained an additive representation for the factorial and some
regular identities for the natural powers of numbers. Some of them are raised to the
level of identities on sets of real and complex numbers.
1 Introduction
Combinatorics today—in the age of mass phenomena and mass processes—is
experiencing a rapid stage of its development, including its applications. Theory
of probability, theory of reliability, theory of information, and coding theory are the
main “clients” and “consumers” of its results.
An important and beautiful section of combinatorics is combinatorial identities,
one of the main sources of which is the famous numerical triangle of Pascal [1].
While dealing with various problems of discrete mathematics, including graphs
with reachability constraints [2], the author has repeatedly encountered a situation
where a specific combinatorial problem leads to the new combinatorial identities
(see, e.g., [3, 4]).
In this paper, we will demonstrate how combinatorial problems generate combinatorial identities.
Ya. M. Erusalimsky ()
Southern Federal University, Rostov-on-Don, Russia
e-mail: ymerusalimskiy@sfedu.ru
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_6
135
136
Ya. M. Erusalimsky
Since some identities are known and we present only new proofs for them, the
article is mainly of a methodical nature. This applies primarily to the content of
Sect. 2.
2 Combinatorial Problem That Allows Different Solutions
Consider the task: “In a store that sells bread of n (n ≥ 2) types, how many variants
are there for buying two loaves of bread?”
As you know, tasks about purchases can be solved using a general scheme, which
leads us to the concept of a combination with repetitions (see, e.g., [5]). Denote by
xi , 1 ≤ i ≤ n, xi ∈ Z+ , the quantity of loaves of the i-th type included in the
purchase. Our task is to calculate the quantity of solutions to the equation:
Use the standard equation environment to typeset your equations, e.g.,
x1 + x2 + . . . + xn = 2, xi ∈ Z+ .
(1)
Using substitution yi = xi + 1, 1 ≤ i ≤ n, we proceed to find the quantity of
solutions to the equation:
y1 + y2 + . . . + yn = n + 2, yi ∈ N.
(2)
To calculate the quantity of solutions to this equation, we use the following
reasoning. Fix n + 2 different points on the straight line and choose from the n + 1
intervals between the n − 1 intervals; in each of the chosen intervals, we draw a
vertical dash. Assuming x1 is equal to the number of points before the first dash, x2
is equal to the number of points between the first and second dash, and so on, we get
the solution of Eq. (2). This means that the number of solutions to this equation is
equal to the number of variants tochoosean n− 1 interval
from the available n + 1
n+1
n+1
intervals for placing dashes, i.e.,
=
.
n−1
2
Now we will solve the same task in a different way. The set of all variants for
buying two loaves of bread will be divided into two disjoint subsets A and B. The
first set contains such variants of buying, each of which consists of different types
of loaves, while the second contains such methods of buying, each of which has the
same type of loaves. It is clear that
n
n
|A| =
, |B| = n =
.
2
1
(3)
Combinatorial Identities with Binomial Coefficients
137
Thus, we have obtained a combinatorial identity (which is a partial case of Pascal’s
identity):
n+1
n
n
=
+
2
2
1
(n ≥ 2).
(4)
Let us change our task a little: “In a store that sells bread of n (n ≥ 3) types, how
many variants of buying three loaves of bread?”
It is clear that by applying a solution scheme that reduces the problem to counting
the quantity of solutions to thecorresponding
we get that the quantity of
equation,
n+2
n+2
variants to buy in this case is
=
. We divide now the set of all
n−1
3
variants of buying three loaves of bread into three disjoint subsets A, B, and C,
taking to the first set such variants of buying, each of which contains different types
of loaves; to the second set such variants of buying, each of which contains two
loaves of one type and another loaf of another type; and to the third set such variants
of buying, each of which contains all three loaves of one type. It is clear that
n
n
n
|A| =
, |B| = 2 ·
, |C| =
.
3
2
1
(5)
From the previously obtained and (5), we have a combinatorial identity:
n
n
n
n+2
+2·
+
.
=
3
2
1
3
(n ≥ 3)
(6)
Using similar reasoning, in the case of buying four loaves of bread, we get
n+3
n
n
n
n
=
+3·
+3·
+
.
4
4
3
2
1
(n ≥ 4)
(7)
By analogy, we get
n
n
n
n
n
n+4
=
+4·
+6·
+4·
+
. (n ≥ 5)
5
5
4
3
2
1
(8)
In general case, we have a beautiful combinatorial identity, in the right part of which
there are binomial coefficients of the (k − 1)-th degree:
k−1 n+k−1
k−1
n
=
·
,
k
i
k−i
n ≥ k.
(9)
i=0
As is known (see, e.g., [1]), similar reasonings when considering the problem of the
number of choices of k people from a set consisting of n men and m women leads
138
Ya. M. Erusalimsky
to Vandermonde identities:
k n
m
n+m
·
,
=
r
k−r
k
(0 ≥ k ≥ n + m).
i=0
Let us consider another combinatorial task: “How many variants are there to put
n the same (identical) coins in three different pockets so that each pocket is not
empty?”
Obviously, the problem has a solution only if n ≥ 3. Denote by x1 , x2 , andx3
the quantity of coins in the first, second, and third pockets, respectively. Then, the
problem was reduced to counting the quantity of solutions to the equation.
x1 + x2 + x3 = n,
xi ∈ N.
(10)
n−1
Obviously, the number of solutions to Eq. (10) is equal to
.
2
Now, let us solve the same problem differently. First of all, we find the total
quantity of coin layouts by pocket, and then, we delete from this the quantity of
layouts where at least one pocket is empty. The total quantity of coin layouts is
equal to the number of solutions to the equation
x1 + x2 + x3 = n,
xi ∈ Z+ ,
(11)
n+2
and it is equal to
.
2
The set of coin layouts in which at least one of the pockets is empty is divided
into two disjoint subsets A and B, taking to the first such layouts, in which only one
pocket is empty, and to the second such layouts in which two pockets are empty. It
is clear that
n−1
n−1
|A| = 3 ·
, |B| = 3 = 3 ·
.
(12)
1
0
From (11) and (12) we get
n+2
n−1
n−1
=
−3·
− 3 or
2
2
1
n+2
n−1
n−1
=
+3·
+1 .
2
2
1
(13)
Similarly, in the case of four pockets (n ≥ 4), we obtain
n+3
n−1
4
n−1
4
n−1
4
n−1
=
+
·
+
·
+
·
.
3
3
1
2
2
1
3
0
(14)
Combinatorial Identities with Binomial Coefficients
139
In the case of k pockets (n ≥ k), we obtain
n+k−1
n−1
k
n−1
k
n−1
k
n−1
=
+
+
·
+ ... +
·
.
k−1
k−1
1
k−2
2
k−3
k−1
0
(15)
Note that identity (15) is similar to identity (9) but does not match it.
3 Graph-Lattice and Combinatorial Identities
The graph-lattice has vertices at points with integer coordinates located in the first
quadrant of the Cartesian plane. Two oriented arcs come out of each vertex of the
graph-lattice: a horizontal arc, leading to the nearest right vertex, and a vertical arc,
leading to the nearest upper vertex (see Fig. 1).
Consider the problem of the quantity of ways (oriented paths) leading from the
initial vertex O(0; 0) to an arbitrary vertex A (k; m), k, m = 0. Here, (k; m) are the
Cartesian coordinates of vertex A. The set of such paths is denoted by μ(O, A).
It is clear that all ways leading from the vertex O to the vertex A on the graphlattice are polylines containing k horizontal arcs and m vertical arcs, located in a
rectangle with vertices at the specified vertices O and A and sides parallel to the
coordinate axes. Let’s match each such way with an n-bit binary number (n = k+m)
containing m units (and n − m zeros), according to the following rule: if at the i-th
step the way passes along a horizontal arc, then the i-th position in this number is
filled with zero, and if at the i-th step the path passes along a vertical arc, then the ith position is filled with one. This encoding is one to one. Therefore, the number of
Fig. 1 Graph-lattice
140
Ya. M. Erusalimsky
ways leading from vertex O to vertex A on the graph-lattice is equal to the quantity
of n-bit binary numbers containing exactly m ones (and n − m zeros). It is clear
that their quantityis equal
to the number of m-element subsets in the set
containing
n
n
n elements, i.e.,
. We have obtained that |μ (O, A(k; m))| =
, where
m
m
n = k + m. This means that if we write, near each vertex, the quantity of paths
leading to this vertex from the initial vertex, we will get a Pascal triangle.
Let’s fix the vertex B (k1 , m1 ) (k1 ≤ k, m1 ≤ m) in the rectangle generated
by the vertices O and A (k; m). Let’s denote the set of the ways leading from
the vertex O to the vertexA and passing through the vertex B, via μ(O, B, A).
Obviously, there is the next equality
n1
n − n1
|μ(O, B, A)| =
·
.
m1
m − m1
(16)
Here, n = k + m, n1 = k1 + m1 .
Definition A cut of the set of μ(O, A)—the ways, leading from the vertex O to the
vertex A—is a set of the vertices of a graph-lattice whose removal “breaks” all the
ways leading from O to A.
Obviously, these are subsets of the vertices of the rectangle generated by the
vertices O and A that lie in this rectangle on straight lines inclined to the positive
direction of the abscissa axis at an angle of 3π/4 (see, e.g., Fig. 2; the vertices of
one of these cuts are highlighted in gray). The vertices of such cut are characterized
by the fact that the sum of the coordinates for each of them is the same (in Fig. 2,
the sum of the coordinates of the vertices of the cutting set is 4). Such a cut will be
denoted by ρs (O, A), where the subscript is the value of the sum of the coordinates
of each of the vertices of the cut. The value s will be called the characteristic of the
cut. The characteristic of the cut is such that 1 ≤ s ≤ n − 1. Such sets are minimal
embedding sets in the set of cuts.
Fig. 2 The cut whose
characteristic is equal to 4
Combinatorial Identities with Binomial Coefficients
141
The following is true:
Statement The sets μ (O, x, A(k; m)) for any two different vertices of the cut
ρs (O, A) do not intersect. The union of all these sets is equal to the set of all the
ways on the graph-lattice leading from O to A, i.e.,
μ (O, x, A) ∩ μ (O, y, A) = ∅,
>
μ (O, A) =
∀x∀y x = y; x, y ∈ ρs (O, A).
(17)
1 ≤ s ≤ n − 1, n = k + m.
(18)
μ (O, x, A) ,
x∈ρs (O,A)
From (18), it follows that, for any s (1 ≤ s ≤ n − 1), the next equality holds:
n
=
m
|μ(O, x, A)| .
(19)
x∈ρs (O,A)
If we substitute in (19) the values for |μ(O, x, A)| from (17), then we obtain the
well-known Vandermonde identity.
In [1, 3, 4] we obtained some combinatorial identities using the ways enumerating
problem on a graph-lattice generated by Pascal’s triangle.
4 Combinatorial Problem: The Analysis of the Solution
Which Generates Combinatorial Identities
Now, using the combinatorial task of the quantity of surjective mappings, we get a
whole spectrum of “beautiful” combinatorial identities.
The formula for the quantity of surjective mappings acting from X to Y is wellknown (see [5]):
n−1
n
X
m
i+1
(−1)
·
· (n − i)m , |X| = m, |Y | = n.
surY = n −
i
(20)
i=1
Analysis of the proof of formula (20) shows that no assumptions are made about the
relationship between m = |X| and n = |Y |. It is clear that sur Y X = ∅ for any m
which satisfied the next condition 1 ≤ m ≤ n − 1. Thus, we proved the identities:
nm =
n−1
n
(−1)i+1 ·
· (n − i)m ,
i
1 ≤ m ≤ n − 1.
(21)
i=1
The author calls the identities (21) the regular representations of powers of n.
142
Ya. M. Erusalimsky
In the case of n = m, the set of surjective mappings coincides with the set of
bijective mappings, so from (21) for m = n, we obtain the identity:
n! = nn −
n−1
(−1)i−1 ·
i=1
n
· (n − i)n ,
i
n ∈ N.
(22)
We called Equality (22) an additive representation for n!. We consider the
polynomials En,m (x), defined by the equality:
n
n
i
En,m (x) =
(−1) ·
· (x − i)m .
i
(23)
i=0
Find the derivatives of the polynomial En,n (x). It is clear that
(1)
En,n
(x) = n · En,n−1 (x)
(2)
En,n
(x) = n · (n − 1) · En,n−2 (x)
...
(k)
(x) = n · (n − 1) · · · · · (n − k + 1) · En,n−k (x)
En,n
(n)
En,n
(x) = 0.
From the Formulas (21) and (22), we obtain that
En,n (n) = n!
(24)
(1)
(2)
(n)
En,n
(n) = En,n
(n) = · · · = En,n
(n) = 0.
(25)
From (24) and (25) by the uniqueness theorem for polynomials, it follows that
En,n (x) = n!
En,m (x) = 0
∀x ∈ R(C),
(26)
∀x ∈ R (C), 0 ≤ m ≤ n − 1.
(27)
By isolating the first summand in (27), we obtain the following identities:
n
n
i+1
(−1)
·
x =
· (x − i)m , 1 ≤ m ≤ n − 1, ∀x ∈ R (C).
i
m
(28)
i=1
The obtained identities (26) are so unusual that you want to “touch them with your
hands.” This is what we will do now.
Combinatorial Identities with Binomial Coefficients
143
Let n = 3, x = 0, 1, 2, 7, 12 .
3! = 6 = 1 · (0)3 − 3 · (−1)3 + 3 · (−2)3 − 1 · (−3)3 = 3 − 24 + 27 (x = 0)
3! = 6 = 1 · (1)3 − 3 · (0)3 + 3 · (−1)3 − 1 · (−2)3 = 1 − 3 + 8 (x = 1)
3! = 6 = 1 · (2)3 − 3 · (1)3 + 3 · (0)3 − 1 · (−1)3 = 8 − 3 + 1 (x = 2)
3! = 6 = 73 − 3 · 63 + 3 · 53 − 1 · 43 = 343 − 648 + 375 − 64. (x = 7)
3
1
1 3
3 3
5 3
3! = 6 =
−3· −
+3· −
−1· −
=
2
2
2
2
48
1
1 3 81 125
+
=
=6 x=
.
= + −
8 8
8
8
8
2
Let us now consider (28) for n = 4, m = 1; 2; 3, x = 2; 5.
2 = 4 · (1) − 6 · (0) + 4 · (−1) − 1 · (−2) (x = 2, m = 1);
22 = 4 · (1)2 − 6 · (0)2 + 4 · (−1)2 − 1 · (−2)2 =
= 4 + 4 − 4 == 4 (x = 2, m = 2);
23 = 4 · (1)3 − 6 · (0)3 + 4 · (−1)3 − 1 · (−2)3 =
= 4 − 4 + 8 = 8 (x = 2, m = 3);
5 = 4 · (4) − 6 · (3) + 4 · (2) − 1 · (1) =
= 16 − 18 + 8 − 1 = 5 (x = 5, m = 1);
52 = 4 · (4)2 − 6 · (3)2 + 4 · (2)2 − 1 · (1)2 =
= 64 − 54 + 16 − 1 = 25 (x = 5, m = 2);
53 = 4 · (4)3 − 6 · (3)3 + 4 · (2)3 − 1 · (1)3 =
= 256 − 162 + 32 − 1 = 125 (x = 5, m = 3).
It also follows from (28) that for any n, k ∈ N, x ∈ R(C), the next identity is valid:
n+k
i+1 n + k
(−1)
x =
· (x − i)n
i
n
(29)
i=1
Here is an example of the “work” of Formula (29) for n = 2, k = 1; 2; 3, x = 3.
32 = 3 · (2)2 − 3 · (1)2 + 1 · (0)2 = 12 − 3 + 0 = 9
32 = 4 · (2)2 − 6 · (1)2 + 4 · (0)2 − 1 · (−1)2 = 16 − 6 + 0 − 1 = 9
144
Ya. M. Erusalimsky
32 = 9 = 5 · (2)2 − 10 · (1)2 + 10 · (0)2 − 5 · (−1)2 + 1 · (−2)2 =
= 20 − 10 + 0 − 5 + 4 = 9.
Let us return to identity (28). Replace the variable in it by x = dx (d = 0):
x m
d
=
n
(−1)i+1 ·
i=1
n
m
x
1 n
n
−i
·
= m·
· (x − d · i)m .
(−1)i+1 ·
i
i
d
d
i=1
m
By dividing both parts into
xm =
n
(−1)i+1 ·
i=1
1
d
, we obtain
n
· (x − d · i)m ∀x ∈ R(C), 1 ≤ m ≤ n − 1, d = 0.
i
(30)
For m = n we obtain the following identity:
d · n! =
n
n
i=0
n
· (x − d · i)n ∀x ∈ R (C) , d = 0.
(−1) ·
i
i
(31)
Identities (30) look very “nice” for integer numbers:
zm =
n
i=1
(−1)i+1 ·
n
· (z + k · i)m ∀z ∈ Z, k (= 0) ∈ Z, 1 ≤ m ≤ n − 1.
i
(32)
Indeed, take, for example, z = 2, k = 10, n = 4 :
2 = 4 · 12 − 6 · 22 + 4 · 32 − 1 · 42
22 = 4 · 122 − 6 · 222 + 4 · 322 − 1 · 422 = 576 − 2904 + 4096 − 1764 = 4
23 = 4 · 123 − 6 · 223 + 4 · 323 − 1 · 423 = 6912 − 63888 + 131072 − 74088 = 8.
Let it be now z = 5, k = −2, n = 4:
5 = 4 · 3 − 6 · 1 + 4 · (−1) − 1 · (−3)
52 = 4 · 32 − 6 ·2 +4 · (−1)2 − 1 · (−3)2 = 36 − 6 + 4 − 9 = 25
53 = 4 · 33 − 6 · 13 + 4 · (−1)3 − 1 · (−3)3 = 108 − 6 − 4 + 27 = 125.
Combinatorial Identities with Binomial Coefficients
145
Remark 1 To get (32) from (30), we put d = −k.
By increasing n in (32), we extend the set of identities and, hence, the powers
of the number z, represented by these identities. By changing k, we shift from the
point z, i.e., we change the set of integers, the powers of which are present in the
right part of the identities (32).
Identities (32) mean that any integer z and its integer powers up to n − 1
inclusive are expressed by a linear combination of n integers or its integer
powers
with an arbitrary integer, with coefficients
up
to
n − 1 inclusive starting
n
n
n
, respectively. These integers are forming an
;−
; · · · ; (−1)n+1 ·
n
1
2
arithmetic progression, whose difference is equal to the difference of the first of
them and the number z.
Therefore, the greater the distance from z, which is the first number on the right
side in (32), the “rarer” these numbers are located.
In conclusion, we write another integer identity that is a consequence of identity
(32):
zn =
n
i=1
(−1)i+1 ·
n+l
· (z + k · i)n ∀z ∈ Z, k (= 0) ∈ Z, l ∈ N.
i
(33)
Remark 2 The identities (30)–(32) generalize the identities (21) and (22). However,
the author is not aware of their proof, which does not use the polynomial identities
(26) and (27), which we proved in this paper using the uniqueness theorem for the
polynomials and combinatorial identities (21) and (22).
Remark 3 The combinatorial identities (21) and (22) are present in [6], but they are
obtained in another method and are not reduced to the polynomial identities (26)
and (27), which are valid on sets of real and complex numbers.
Remark 4 Finite differences for an arbitrary function are defined on a set of
integers, given by the following formulas:
(f )(x) := f (x) − f (x − 1),
(2 f )(x) := ((f )(x)) − 2f (x − 1) + f (x − 2),
(3 f )(x) := ((2 f )(x)) = f (x) − 3f (x − 1) + 3f (x − 2) − f (x − 3), . . . ,
n
n
(n f )(x) := (δ n−1 f (x)) =
(−1)n ·
· f (x − i), . . . .
i
i=0
146
Ya. M. Erusalimsky
They are also the “source” of polynomial identities such as obtained above, since
the following statements hold for them:
1. For any polynomial pn (x) = an x n + an−1 x n−1 + . . . + a0 (an = 0) and m >
n(m pn )(x) = 0.
2. For any polynomial pn (x) = an x n +an−1 x n−1 +· · · +a0 (an = 0) (n pn )(x) =
n! · an .
In particular, identity (22) can be obtained from (2) by assuming pn (x) = x n .
Acknowledgement The author is grateful to the reviewers for their useful comments.
References
1. Erusalimskiy, I.M.: Pascal’s Triangle: Combinatorics and Random Walks, 2nd edn. University
Book, Moscow (2020), 121p.
2. Erusalimskiy, I.M.: Graphs with non-standard reachability: tasks, applications. In: Erusalimskiy,
I.M., Skorokhodov, V.A., Petrosyan, A.G., Kuzminova, M.V. (eds.). Southern Federal University, Rostov-on-don (2009), 195 p
3. Erusalimskiy, I.M.: Graph lattice: random walk and combinatorial identities. Bol. Soc. Mat.
Mex. 22, 329–335 (2016). https://doi.org/10.1007/s40590-016-0115-9
4. Erusalimskiy, I.M.: 2 and 3 ways on a graph and combinatorial identities (Izvestia vuzov. SeveroKavkazskii region. Ser. Est. nauk, 2017) 1(193), 25–30 (2017)
5. Erusalimskiy, I.M.: Discrete mathematics. Theory and practice. Textbook: Spb: LAN’ (2018),
476 pp.
6. Riordan, J.: Combinatorial Identities. Wiley, New York–London–Sidney (1968)
Random Tempered Distributions on
Locally Compact Separable Abelian
Groups
Manuel L. Esquível and Nadezhda P. Krasii
Abstract By means of sequences of random variables of controlled growth, that
is, either rapidly decreasing or slowly increasing, we define a space of random
tempered generalized functions—of Schwartz type—on a locally compact abelian
separable group, and we characterize the set of those generalized functions that have
a mean. We show that this approach covers both the case of the torus and the real
line case and also allows us to define tempered random distributions over the Cantor
set by means of the associated Cantor group.
Keywords Generalised functions on infinite-dimensional spaces · Generalised
stochastic processes · Analysis on separable locally compact abelian groups ·
Cantor group
1 Introduction
A characterization of random Schwartz distribution having a mean—in the torus
and in the real line—by means of the growth of its generalized random coefficients
with respect to a basis of an associated L2 space was given in [10] and in [9].
The extension of these results to other locally compact groups seems to require a
convenient definition of a Schwartz distribution—or generalized function—over the
group. There is a substantial work in this direction. Let us briefly review some of
the sources that could help us to achieve our goal. The approach in [34] requires the
existence of one parameter subgroups and so may be considered in particular cases.
M. L. Esquível ()
FCT Nova & CMA, UNL, Caparica, Portugal
e-mail: mle@fct.unl.pt
N. P. Krasii
Don State Technical University, Rostov-on-Don, Russian Federation
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_7
147
148
M. L. Esquível and N. P. Krasii
In the work [4], there is a general definition of a smooth function over the group that
leads to a smooth structure in the sense of[2]; this author considers that:
. . . Bruhat space is the most natural, convenient and useful for applications. . .
and manages to provide a quite general theory of derivatives on locally compact
groups. We observe that by means of an axiomatic construction, given in [5], in the
particular case of parametrizable groups, it is possible to recover, by isomorphism,
Bruhat’s space.
In [21], for a regular representation acting on L2 (G), the author gives an explicit
construction of a space that can be thought as a generalization of the indefinitely
derivable functions with compact support over G setting the stage for defining
general distributions ver. G.
The work [32] bears some similarities with [34]. By considering the set of all
continuous homomorphisms of the real line onto the compact group G and, then,
the linear span of elements of irreducible matrix representations of G as the a
locally convex space of test functions, the author manages to define the Fourier
transform. In the case of compact semisimple Lie groups, the Fourier transforms are
characterized by growth.
In [1], a notion of derivative is defined such that a function f ∈ L1 (G) is
derivable with respect to h, an homomorphism from G to the real line, if also
h(f ) ∈ L1 (G).
We choose to proceed differently from the previously referred works. Aiming
at characterizing random tempered distributions on the group by some growth
conditions of an associated sequence of coefficients, we consider an approach akin
to the white noise approach having its roots in the theory of rigged Hilbert spaces
(see [13, p. 106–114] and [14, p. 63]). By means of spaces of sequences, we define
S as a space of functions on the group and S as a space of linear forms over S,
such that
S ⊂ L2 (G) ⊂ S ,
and that there is a duality between S and S. Then, by using sequences of
coefficients that are random variables satisfying growth conditions, we have the
desired random tempered Schwartz distributions. For this purpose, we require the
space L2 (G) to have a denumerable base; this happens, for instance, when μ is a
Radon measure on G being locally compact, metrizable, and a countable union of
compact sets (see [30, p. 219]). In some particular cases, we may define a notion
of derivative that allows to recover the torus case—for which there is a basis of
characters—and the Euclidean case, for which there is the denumerable base of
Hermite functions; the defined derivative operator also applies to the Cantor group.
To get a perspective of the degree of generality we achieve with our method, we
first recall two structure results. The first is a result on the structure of Lp separable
spaces for finite measure space (G, μ) (see [3] or [28, p. 128]) .
Random Tempered Distributions on Separable LCAG
149
Theorem 1 If (G, μ) is a finite measure space and L2 (G) is separable, then L2 (G)
is isometric and order isomorphic to exactly one of the following three spaces:
• The complex sequence space l 2 (Γ, C) with #Γ ≤ ℵ0
• The direct sum l 2 (Γ, C) ⊕ L2 ([0, 1]) with #Γ ≤ ℵ0
• The classical space L2 ([0, 1])
The structure of separable measure space L2 (G) is linked to the existence of atoms
in the measure space (G, μ); for instance, if this space has no atoms, then L2 (G) is
isometrically isomorphic to L2 ([0, 1]) (see [30, p. 224]). In adding a group structure,
more can be said, so in the non-necessarily abelian case, Proposition 5.11 in [35, p.
50] shows that L2 (G) is separable if and only if the dual group is countable and also
if and only if the group G is metrizable.
The second result is a structure result on locally compact abelian (LCA) groups
that we quote from [16, p. 297].
Theorem 2 Every LCA group is a union of an increasing directed family of open
subgroups which are isomorphic to groups of the form
Rm × Zn × Tk × F ,
(1)
with F a finite group and m, n, k ∈ N.
This result shows that a first step to study random tempered distributions on a more
general LCA group would be to study these distributions on fundamental groups
described in Formula (1) and that a second step would be to extend the study to
unions of directed families of elementary groups.
The study of random Fourier series on compact and locally compact groups has
been pursued in several directions. For instance, in [31], the authors study the LCA
case, and they prove a central limit theorem and in the compact non-abelian case,
they study random Fourier series of continuous functions (with Gaussian distributed
coefficients) and special sets of convergence.
2 Random Tempered Generalized Functions on Some LCA
Groups
We briefly summarize some needed facts about locally compact abelian groups
following the classical references [16, 18, 29, 33, 37] and [17]. Let (G, %) be a locally
compact abelian group, let μ be its normalized Haar measure, and let us suppose
that L2 (G), which is a separable Hilbert space with the inner product given for,
φ, ψ ∈ L2 (G) by,
φ(g)ψ(g)dμ(g) .
< φ, ψ >=
G
150
M. L. Esquível and N. P. Krasii
We observe that if G is abelian and compact and the Haar measure is normalized,
then the characters on G form a complete orthonormal family in L2 (G) (see [22,
p.193] or [30, p. 224]). Moreover, assuming that L2 (G) is separable immediately
entails—by the Gramm-Schmidt process (see [19, p. 242] or [23, p. 94])—that there
exists (en )n≥1 a denumerable complete orthonormal set of L2 (G) generating a dense
subspace of L2 (G). Let (en )n≥1 be a denumerable orthonormal set of L2 (G). Then
(see [22, p. 29]), the following are equivalent:
1. The family (en )n≥1 is complete.
2
2. For every φ ∈ L2 (G), we have that φL2 = +∞
n=1 |< φ, en >| .
+∞
3. For every φ ∈ L2 (G), we have that φ = n=1 < φ, en > en .
Moreover, if the family (en )n≥1 is a complete orthonormal set of L2 (G), then, for
all φ, ψ ∈ L2 (G), we have the Parseval duality formula (see [23, p. 98]),
< φ, ψ >=
+∞
(2)
< φ, en > < ψ, en > ,
n=1
which is given in [19, p. 245] or in [36, p. 85], even for non-necessarily denumerable
complete orthonormal subsets of L2 (G) . Examples of groups G, of interest to us
here, that admit a denumerable complete orthonormal set of L2 (G) are as follows:
• The one-dimensional Euclidean space with the R with the set Hermite functions
(see [19, p. 243–245]) or [40, p. 290] and also Sect. 3.1)
• The torus, that is, the set T = {|z| = 1}, with the orthonormal complete family
(eint )n∈Z (see Sect. 3.2)
• The Cantor group with the characters as a complete orthonormal family (see
Sect. 3.3)
2.1 Classical Tempered Distributions in LCA Separable Groups
Motivated by the known results on the order of magnitude of Fourier coefficients of
functions (see [22, p. 24]) and of distributions (see [25, p. 16]), we next recall the
definition of spaces of sequences (see [26, p. 57]) and some of its most important
properties.
Definition 1 (Sequence Spaces)
1. The space of rapidly decreasing complex number sequences is given by the
following:
rd = (rn )n≥1 ∈ C
N0
: ∀p ≥ 1 ∃cp > 0 ∀n ≥ 1 |rn | ≤
cp
(1 + n)p
.
Random Tempered Distributions on Separable LCAG
151
2. The space of slowly increasing complex number sequences is given by the
following:
si = (sn )n≥1 ∈ CN0 : ∃q ≥ 1 ∃cq > 0 ∀n ≥ 1 |sn | ≤ cq (1 + n)q
.
We now have the important isomorphism theorem between the spaces si and rd
(see [26, p. 59]).
Theorem 3 The space rd with the family of seminorms (ρk )k∈N – with ρk (r) =
+∞
k
n=1 (1+n) |rn | for r = (rn )n≥1 ∈ rd—is a locally convex Hausdorff space having
a dual isomorphic to the space si. The
duality may be written for r = (rn )n≥1 ∈ rd
and s = (sn )n≥1 ∈ si, by < r, s >= +∞
n=1 rn sn .
We may now define on G an analogue of Schwartz space of tempered functions on
the real line.
Definition 2 The space S of rapidly decreasing functions on the group G is given
by the following:
2
S = φ ∈ L (G) : ∃(an )n≥1 ∈ rd, φ =L2
+∞
a n en
n=1
+∞
For such a φ = +∞
n=1 an en ∈ S, we write φ =
n=1 rn (φ)en to emphasize the
growth properties of the coefficients.
Remark 1 For φ ∈ S and any g ∈ G, we have that
+∞
+∞
|φ(g)| = |rn (φ)| < +∞ ,
rn (φ)en (g) ≤
n=1
n=1
by reason of |en | = 1 and (rn (φ))n≥1 ∈ rd. Moreover, we have unicity of the
representation;
in fact, ifwe suppose that φ ∈ S admits two representations φ =
+∞ a
b
r
(φ)e
and
φ = +∞
n
n=1 n
n=1 rn (φ)en , due to the speed of convergence of the series
of the representations, we have that for every character ek ,
0 =< φ, ek > − < ψ, ek > =
+∞
ran (φ)< en , ek >
n=1
−
+∞
rbn (φ)< en , ek > = ran (φ) − rbn (φ) ,
n=1
due to, both the Parseval duality formula (2), and, by the orthogonality relations,
< en , ek >= δnk , with δnk the Kronecker’s delta.
152
M. L. Esquível and N. P. Krasii
By defining an obvious isomorphism between S and the sequence space rd, we
now have the following result.
Theorem 4 With
family of seminorms (σk )k∈N —given for any φ ∈
the increasing
k |r (φ)|—the space S is a Hausdorff locally convex
S by σk (φ) = +∞
(1
+
n)
n
n=1
topological vector space.
Let us now consider a space of Schwartz generalized functions—continuous
linear forms over S (see [27, p. 51])—over the group G which is isomorphic to
the sequence space si.
Definition 3 Let S denote the dual space of S, that is, the space of continuous
linear maps from S to C, more precisely
S = {T ∈ L(S, C) : ∃k > 0 ∃c > 0 ∀φ ∈ S |< T , φ >| ≤ cρk (φ)}
(3)
with L(S, C) the space of linear maps from S to C.
Theorem 5 S is a locally convex topological vector space and admits a description given by
S =
+∞
bn en : (bn )n≥1 ∈ si
n=1
emphasize the slow
meaning that, if for T ∈ S we write T = +∞
n=1 sn (T )en —to growth properties of the coefficients—the image of T on φ = +∞
n=1 rn (φ)en ∈ S
+∞
is given by the duality formula < T , φ >= n=1 sn (T )rn (φ) that encompasses
Formula (2).
Proof This is a consequence of Theorem 3.
2.2 Random Tempered Distributions
We first introduce a space of random Schwartz distributions over G by means of
series of L2 (G) basis functions having as coefficients random variables satisfying
a certain growth condition. Our approach of a random distribution is akin to the
definition of a generalized random process in [13, p. 242]. The results presented here
generalize those presented in [9]—where the particular case of periodic distributions
was studied—and in [10] for the case of the real numbers.
Let (Ω, A, P) be a complete probability space. Let M denote the space of random
variables taking complex values and MN the space of sequences of elements of
M indexed by the integers N. For A, an element of M, let E[|A|] represent the
expectation of A. Let Cm be the space of sequences (cn )n∈N0 ∈ MN such that for
Random Tempered Distributions on Separable LCAG
153
A ∈ M with A ≥ 0 and E[A] < +∞, we have the following:
∀n ∈ N0 |cn | ≤ A(1 + n)k a. s. on Ω .
(4)
Condition given in Formula (4) is the analogue—for random variables—to the
defining condition of slowly increasing sequences in Definition 1; the constant in
the right-hand side is, in the present case, a random variable.
This space can be described by the equivalent conditions given in the next
theorem that we recall from [8].
Theorem 6 For (cn )n∈N0 ∈ MN , the following are equivalent:
1. (cn )n∈N0 ∈ Cm .
2. There exists an integrable random variable A and a real bounded random
variable K, defined on Ω, such that
∀n ∈ N0 |cn | ≤ A (1 + n)K a. s. on Ω
3. The sequence (E[|cn |])n∈N0 is a sequence of slow growth or, in an equivalent
rephrasing,
(∃a > 0) (∃k ∈ N0 ) (∀n ∈ N0 ) E[|cn |] ≤ a (1 + n)k
Remark 2 (Coefficients of a Random Tempered Distribution) Associated with every
sequence of random variables (cn )n∈N0 in Cm , there is a random Schwartz distribution T ∈ S on G, defined for a φ ∈ S, represented by (rn (φ))n∈N0 , by
< T , φ >=
+∞
cn rn (φ) .
(5)
n=0
The distribution T is well defined because, by the second condition of the
Theorem 6, the sequence (cn )n∈N0 is almost surely a sequence of slow growth.
Besides, the sequence (rn (φ))n∈N0 , representing φ, is a rapidly decreasing sequence,
and so, the series, in the left-hand side of Formula (5), converges almost surely.
Let us mention some properties of the random Schwartz distributions just defined.
Theorem 7 (On the Uniqueness of the Series Representation in Formula (5))
Let T be given, as in Formula (5), by a sequence (cn )n∈N0 . Then, the following are
equivalent:
1. T = 0 a. s. on Ω .
2. ∀n ∈ N0 cm = 0 a. s. on Ω .
The proof is similar to the one in [8] and we omit it. As a consequence, for every
random Schwartz distribution, defined this way, there is, up to equality almost
surely, a unique sequence of random variables such that Formula (5) is verified
154
M. L. Esquível and N. P. Krasii
almost surely. Such a sequence will be denoted by the following:
(sn (T ))n∈N0 .
An obvious corollary of this theorem gives a necessary and sufficient condition for
equality of two random Schwartz distributions obtained by the way just explained.
Theorem 8 (On the Uniqueness of the Series Representation in Formula (5))
Let T and U be two random Schwartz distributions obtained by Formula (5) from
the sequences (sn (T ))n∈N0 and (sn (U ))n∈N0 , respectively. Then, the following are
equivalent:
1. T = U a. s. on Ω .
2. ∀n ∈ N0 sn (T ) = sn (U ) a. s. on Ω .
In this case also, the proof is simple and similar to the one in [8], and so, we omit it.
Due to the hypotheses made on a sequence, in Cm , a random Schwartz distribution given, by way of Formula (5), has a mean in the sense of the next definition (see
also [13, p. 246]).
Definition 4 Let T be a random Schwartz distribution associated to a complex
sequence (sn (T ))m∈N , by Formula (5). Then, T admits T̄ as a mean if and only
if:
1. ∀φ ∈ S
2. ∀φ ∈ S
< T , φ >∈ M ∩ L1 (Ω) .
E[< T , φ >] =< T̄ , φ > .
A random Schwartz distribution, built with a sequence (cm )m∈N0 ∈ Cm by
Formula (5), does admit a mean. This mean has a representation as series in the
complete orthonormal set of L2 (G) denoted by (en )n≥1 having, as coefficients,
the sequence of expectations (E[cm ])m∈N0 . This simple result follows from an
application of Lebesgue dominated convergence theorem as we will show next.
Theorem 9 If T = +∞
n=0 sn (T ) en is a random Schwartz distribution such that the
sequence (sn (T ))n∈N0 is inthe class Cm , then T admits the nonrandom Schwartz
distribution given by T̄ = +∞
n=0 E[sn (T )] en as a mean.
Proof For every φ ∈ S, the sequence rn (φ))n∈N0 is a rapidly decreasing sequence.
By Theorem 6, the sequence (E[|sn (T )|])n∈N0 is a slow growth sequence, and so,
the series
+∞
n=0
E[|sn (T )|] |rn (φ)| ,
Random Tempered Distributions on Separable LCAG
155
converges almost surely. Now, by the Lebesgue dominated convergence theorem,
< T̄ , φ >=
+∞
E [|sn (T )|] rn (φ) = E
n=0
+ +∞
,
sn (T ) rn (φ) = E [< T , φ >] ,
m=0
where the last equality results from the duality formula (5).
We next prove a converse of Theorem 9 that holds, in a sense, that we now
proceed to explain. Let T be a measurable map from (Ω, A) into S , which we
consider endowed with the Kolmogoroff σ -algebra associated to the dual countably
Hilbertian (or Schwartz) topology on S (for the case of the real line, see [20, p.
6–16]). Then, for almost every ω ∈ Ω, the Hermite coefficients of T (ω), which we
denote, as usual, by
sn (T )(ω) =< T (ω), en > ,
for n ∈ N0 , are all well defined, and furthermore, we can consider (sn (T ))n∈N0 as
a well-defined sequence of random variables. For a general T , no growth condition
on the sequence (sn (T ))n∈N0 is verified so as to ensure that this sequence is in
the class Cm . For instance, consider a sequence in which only a finite number of
terms are nonzero and having one of these as a non-integrable random variable.
Another example, in the real line, is given by a sequence of random variables taking
small values with a big probability and big values with small probability such as
(sn (T ))n∈N0 verifying
P[sn (T ) = nen ] =
1
1
, P[sn (T ) = 0] = 1 − 2 .
n2
n
n
As the series +∞
n=1 P[sn (T ) = ne ] converges, then, by Borel-Cantelli lemma, T
is almost surely given by a finite sum of terms. That is, for almost all ω ∈ Ω, there
exists a N ∈ N0 , N = N(ω) such that
T (ω) =
N
sn (T )(ω) en .
n=1
Now, as we have
∀n ∈ N0 E[sn (T )] =
en
,
n
which does not define a sequence of slow growth, the sequence sn (T ))n∈N0 cannot
be in the class Cm , as a result of Theorem 6.
The next theorem shows that the first condition in Definition 4 is a sufficient
condition on T for the sequence (sn (T ))n∈N0 to be in the class Cm .
156
M. L. Esquível and N. P. Krasii
Theorem 10 Let T be defined (almost surely) in S, by a sequence of random
variables (sn (T ))n∈N0 , such that
∀φ ∈ S
+∞
< T , φ >=
sn (T ) rn (φ) a. s. on Ω .
n=1
Then, if
∀φ ∈ S
< T , φ >∈ L1 (Ω) ,
the sequence (sn (T ))n∈N0 is in the class Cm .
Proof The proof is an adaptation of the proof given to a similar result for Schwartz
distributions in the real numbers (see [10]). Let us show first, using the closed graph
theorem, that the map ΛT defined for every test function φ ∈ S by ΛT (φ) =<
T , φ > is continuous from S into L1 (Ω). As T is, almost surely, a random tempered
distribution, if (φl )l∈N0 is a sequence of test functions converging to zero inS, then
the sequence of random variables (Ul )l∈N0 , defined for l ∈ N, by
Ul =< T , φl > ,
converges a.s. to zero. So, this sequence (Ul )l∈N0 converges also in probability to
zero. Now, suppose that (Ul )l∈N0 converges to U in L1 (Ω); then, the sequence
converges also in probability to U , and so, U = 0. By the closed graph theorem
(see [38, p. 51]), the map ΛT is continuous. Taking in account the topologies of S
and L1 (Ω), the continuity of ΛT can be expressed in the following way:
∃k ∈ N0 , ∃ck > 0 ∀φ ∈ S || < T , φ > ||L1 (Ω) ≤ ck sup (1 + |n|)k |rn (φ)| .
n∈N0
(6)
As a second step, we use the sequence of Rademacher functions [41, p. 212] defined
in [0, 1] by
∀t ∈ [0, 1] ∀n ∈ N radn (t)(t) = σ (sin(2n+1 πt)) ,
where σ (t) denotes the sign of t defined by the following:
σ (t) =
|t |
t
ift = 0
0
otherwise .
(7)
And, we also consider rd, the space of rapidly decreasing complex sequences in
Definition 1 with the topology induced by the family of seminorms (ρk )k∈N in
Random Tempered Distributions on Separable LCAG
157
Theorem 3 which, we recall, are defined for r = (rn )n≥1 ∈ rd by the following:
ρk (r) =
+∞
(1 + n)k |rn | .
n=1
Then, for every t ∈ [0, 1], the map from rd to rd which associates to each sequence
r = (rn )n≥1 ∈ rd, the sequence w = (wn )n∈N0 defined by
∀n ∈ N0 wn = radn−1 (t) rn ,
is a homeomorphism such that
∀k ∈ N ρk (w) = ρk (r) .
As a consequence of this observation and of the expression of the continuity of ΛT ,
given by (6) and of Parseval formula, we have that
(∃k ∈ N, ck > 0) (∀φ ∈ S), (∀t ∈ [0, 1])
,
++∞
E radn−1 (t)(t)sn (T )rn (φ) ≤ ck sup (1 + n)k |rn (φ)| .
n∈N0
(8)
n=1
In the third step of the proof, we will show that the left-hand side of the inequality
in (8) can be replaced by the following expression:
⎡
12 ⎤
+∞
|sn (T )|2 |rn (φ)|2 ⎦ .
E⎣
n=1
In order to do as stated, we observe that, as the sequence (rn (φ))n∈N is rapidly
decreasing and, almost surely, (sn (T ))n∈N is a sequence of slow growth, we have
almost surely
+∞
|sn (T )|2 |rn (φ)|2 < +∞ .
n=1
Now, by the standard inequality for Rademacher functions, namely, KhintchineKahane inequality (see [41, p. 213]), we have almost surely for some constant c
+∞
n=1
12
|sn (T )| |rn (φ)|
2
2
1 +∞
≤ c
radn−1 (t) sn (T )rn (φ) dt .
0 n=1
158
M. L. Esquível and N. P. Krasii
To conclude as desired, it is enough to apply Fubini theorem, to get for k, ck , and φ
as in (8)
⎡
12 ⎤
+∞
|sn (T )|2 |rn (φ)|2 ⎦ ≤ c ck sup (1 + n)k |rn (φ)| .
E⎣
(9)
n∈N0
n=1
In the fourth step, we remark that the left-hand side of the inequality, in (9), can be
replaced by
++∞
,
√
E
αn |sn (T )| |rn (φ)| ,
n=1
where (αn )n∈N0 is an arbitrary sequence of strictly positive numbers such that
n=+∞
αn = 1. This statement results from the fact that, for almost every ω ∈ Ω,
n=1
the expression
+∞
|sn (T )|2 |rn (φ)|2 =
n=1
+∞
αn |sn (T )|2
n=1
|rn (φ)|2
,
αn
can be considered as an integral over N0 , of the function defined by
|sn (T )|2
∀n ∈ N0
|rn (φ)|2
,
αn
with respect to the measure over (N0 , P(N0 )) that puts a√mass αn on each integer n.
Applying Jensen inequality, with the convex function − x on the interval [0, +∞[,
we have that a.s.
+∞
12
|sn (T )|2 |rn (φ)|2
≥
n=1
+∞
n=1
|rn (φ)|
αn |sn (T )| √
.
αn
As a consequence, for k, ck , and φ as in (8),
E
++∞
√
,
αn |sn (T )| |rn (φ)|
≤ c ck sup (1 + n)k |rn (φ)| .
(10)
n∈N0
n=1
√
This expression shows that the sequence (E[|sn (T )| αn ])n∈N0 is of slow growth
at infinity. In order to conclude now, it is enough to consider, for instance, the
sequence (αn )n∈N0 defined by αn = 1/n2 A with A = π 2 /6. This sequence
satisfies the hypothesis
made in the fourth step, and it is clear that if with this
√ sequence, E |sn (T )| αn n∈N , is a sequence of slowly increasing at infinity, then
0
Random Tempered Distributions on Separable LCAG
159
(E [|sn (T )|])n∈N0 is also slowly increasing, thus showing that the random Schwartz
distribution T is in the class Cm .
As a consequence of this theorem, we can now formulate the result which gives a
characterization of random Schwartz distributions having a mean or a first moment.
Theorem 11 Let T be a measurable random Schwartz distribution. Then, T has a
first moment if and only if the sequence (sn (T ))n∈N0 of its coefficients defined in
Remark 2 is in the class Cm .
Proof That the last condition is sufficient was already shown in Theorem 9. The
condition is necessary as a consequence of Theorem 10.
Remark 3 This last result can also be read as a characterization of the stochastic
processes with a first moment which have as trajectories tempered distributions. In
fact, let X be a generalized stochastic process (see [15, p. 115]). This will mean for
us, according to the reference quoted, that X = (Xφ )φ∈S is a family of random
variables on a probability space (Ω, A, P) such that, for all ω ∈ Ω, the map from
S into C, given by X. (ω), is a tempered distribution. We can then consider that
X has, as trajectories, tempered distributions. Obviously, such an object defines a
random distribution, in the sense, we have been using, and moreover, the sequence
(sn (X))m∈N is a sequence of random variables. As a consequence, Theorem 11 can
be applied giving the characterization mentioned.
3 Derivatives of Tempered Generalized Functions on Some
LCA Groups
The main purpose of this section is to show how to recover previous results on
random Schwartz tempered distributions of the real line and the torus with the
formalism here presented. Moreover, we will show that we can define Schwartz
tempered distributions—and consequently random Schwartz tempered distributions
also—over the Cantor group, and so, we may extend the theory to a nonclassical
setting. Let (en )n∈N be a complete orthonormal sequence in L2 (G). Usually,
distribution theory is associated to the possibility of extending the usual notion of
derivative to wider spaces. We next define a notion of derivative that covers the
classical derivation operators and allow the definition of new ones.
Definition 5 An operator D : D ⊂ L2 (G) ,→ L2 (G) defined on a vectorial space
domain S ⊆ D is a derivative operator on L2 (G) if the following conditions are
verified:
1. The elements of (en )n∈N are eigenvectors of D with real eigenvalues—that is,
D(en ) = λn en and λn ∈ R—and the operator D acts on the basis elements with
controlled growth, that is,
∃α ∈]0, +∞[ , ∀n ∈ N |D(en )| ≤ (1 + n)α |en | .
(11)
160
M. L. Esquível and N. P. Krasii
2. The operator is linear over the domain D:
∀φ1 , φ2 ∈ D ∀λ1 , λ2 ∈ C D(λ1 φ1 + λ2 φ2 ) = λ1 D(φ1 ) + λ2 D(φ2 ) .
Remark 4 The restriction α ∈]0, +∞[ in Expression (11) allows for derivatives of
fractional order which are important in applications.
Theorem 12 A derivative operator D on D ⊆ L2 (G) admits an extension to S
that verifies, for all T ∈ S ,
∀φ ∈ S , < D(T ), φ >=< T , D(φ) > .
Proof If T is represented by (sn (T ))n∈N and φ is represented by (rn (φ))n∈N , as
D(T ) is represented by (λn sn (T ))n∈N , we have, by Theorem 5, that
< D(T ), φ >=
+∞
λn sn (T )rn (φ) =< T , D(φ) > .
n=1
as D(φ) is represented by (λn rn (φ))n∈N which is a rapidly decreasing sequence. We consider next three instances of LCA separable groups and of derivative
operators in these groups. For simplicity, we presented only the unidimensional
torus and the real line. The presentation of the multivariate case is straightforward
at the price of a more cumbersome notation.
3.1 The Real Line
We consider the additive LCA group (R, λ) with λ the Lebesgue measure, that
is, our chosen Haar measure. We follow [39, p. 260–263] for a brief summary of
some needed results. For L2 (R), there is an orthonormal complete sequence of
Schwartz functions and the Hermite functions. The Hermite polynomials on R are
the polynomials Hm (x), for m ∈ N, defined by the following equations:
√
1
m
d m −2πx 2
2
m
e
=
(−1)
m! 2m− 4 π 2 Hm (x) e−2πx .
dx m
Associated with these polynomials are the Hermite functions given, for m ∈ N, by
Hm (x) = Hm (x)e−πx .
2
We may immediately observe that the family (Hm )m∈N is an orthonormal set of
functions in L2 (R) and that it is complete by recovering the representation for the
Random Tempered Distributions on Separable LCAG
161
elements of L2 (R), as a series of Hermite functions by means of the special form of
the Fourier transform of the Hermite functions.
Theorem 13 The Fourier transform of a Hermite function is given by F(Hm )(ξ ) =
(−1)m Hm (ξ ). For every φ ∈ L2 (R), the following equality in the L2 sense holds:
φ=
+∞
am (φ) Hm ,
m=0
where the coefficients (am (φ))m∈N are given by
am (φ) =
φ(x)Hm (x) dλ(x) ,
R
and the following Plancherel type result holds:
+∞
|am (φ)| =
|φ(x)|2 dλ(x) .
2
m=0
R
The Hermite functions are products of polynomials by an exponential function. By
using the usual derivation, we can define natural operators given by
(τ+ f )(x) = (
d
f )(x) + 2πxf (x)
dx
(τ− f )(x) = −(
d
f )(x) + 2πxf (x) .
dx
(12)
for sufficiently regular functions f . The recurrence relations that hold among
the Hermite functions imply the next result about the action of τ+ , τ− , and the
composition τ+ ◦ τ− operating on the Hermite functions.
Theorem 14 For every m ≥ 1,
√
τ+ (Hm ) = 2 πm Hm−1 , τ− (Hm ) = 2 π(m + 1) Hm+1 , (τ+ ◦ τ− ) (Hm )
= 4π(1 + m)Hm .
(13)
It is now clear that D ≡ τ+ ◦ τ− satisfies the conditions of Definition 5 and so is
a derivative operator over S by Theorem 12. Let us now quote a classical result of
Laurent Schwartz (see [39, p. 262]).
Theorem 15 A necessary and sufficient condition for T being a classical tempered
distribution, that is, a continuous form over the space of rapidly decreasing
functions over the line, is that the sequence (am (T ))m∈N is a slowly increasing
sequence.
162
M. L. Esquível and N. P. Krasii
Theorem 15 shows that the classical tempered distributions on the line are exactly
the elements of S .
3.2 The Torus
We follow [22, p. 1–30] for the notations for the model of the torus that we adopt
(an alternative would be to follow [39, p. 224–231]). We consider the torus to be the
quotient group T = R/2πZ where 2πZ is the group of integral multiples of 2π. The
Haar measure on T is the normalized Lebesgue measure, and so, we have that
L2 (T) = f : T ,→ C :
1
2π
|f (t)|2 dt < +∞
,
T
as functions over T can be identified with periodic function over R of period 2π. The
family (eint )n∈Z is a complete orthonormal system in L2 (T) which is denumerable.
Following the usual notations, we have with
0 = 1
∀n ∈ Z , φ(n)
2π
φ(t)eint dt
T
0(n)eint in L2 (T) and that given any sequence of complex numbers
that φ = n∈Z φ
(an )n∈Z such that n∈Z |an |2 < +∞, there exists φ ∈ L2 (T) such that 0
φ (n) = an
for all n ∈ Z. Now, the exponential functions eint , for n ∈ Z, are also characters,
and so, by defining the operator D : S ,→ S by its operation on the characters of
the L2 (T) basis by
D(eint ) = −i
d int
e = neint .
dt
we have that the operator D satisfies Definition 5. Again, Theorem I in [39, p. 225]
shows—for the multivariate case—that there is a topological isomorphism between
the Schwartz distributions over the torus and the space of the slowly increasing
sequences, and so we, again, recover the classical case.
3.3 The Cantor Group
This last example shows that the approach here presented is more general than the
classical examples of the torus and the real line. A brief introduction to the Cantor
group can be read in [22]. The connection with the Cantor set can be seen in [19, pp.
13,70]. A study of some other properties of this group can be found in [7]. Let us
briefly summarize some basic facts. By definition, we have that C := {0, 1}N0 , and
Random Tempered Distributions on Separable LCAG
163
so, as {0, 1}, with the given that the discrete topology is a compact group, we have
that C is compact by Tychonoff’s theorem and it is also a group with coordinate-wise
addition modulo 2. The map Φ that to c = (cn )n∈N0 ∈ C associates
Φ(c) = 2
+∞
cn
n=1
3n
,
is a homeomorphism onto the middle-thirds Cantor set on the line. The group C is
metrizable as for c = (cn )n∈N0 , c = (cn )n∈N0 ∈ C, and the distance d(c, c ) =
2− inf{n∈N0 :cn =cn } that defines the topology of C. A canonical Haar measure on C
is the Φ pulled back measure of the Lebesgue-Stieltjes measure on the Cantor set
defined by the devil’s staircase Cantor function (see [6]), and it is also the product
of the normalized Haar measures over each of the factors {0, 1} of C. That is, if
we consider over each factor {0, 1} of C := {0, 1}N0 the probability μn such that
μn ({0}) = 1/2 = μn ({1}), then a normalized Haar measure over C is given by
⊗n∈N0 μn . As a consequence, we have that each c = (cn )n∈N0 ∈ C is a sequence of
independent random variables with Bernoulli distribution of parameter 1/2.
To describe the dual group, we consider, for each k ∈ N, the map ωk : C ,→ {z ∈
C : |z| = 1} such that ωk (c) = 2ck+1 − 1 ∈ {−1, 1} for c = (cn )n∈N0 ∈ C. Now,
for each finite choice of distinct integers 0 ≤ j1 < j2 < · · · < jk , we define, with
n = 2j1 + 2j2 + · · · + 2jk ,
Γn = Γ2j1 +2j2 +···+2jk := ωj1 ◦ ωj1 ◦ · · · ◦ ωjk .
It is clear that for c = (cn )n∈N0 ∈ C,
Γn (c) = (2cj1 +1 − 1)(2cj2 +1 − 1) · · · (2cjk +1 − 1)
= (−1)
cjm +1 =0,m=1,...,k 1
= (−1)
+∞
n=1 cn dn
,
and so, (Γn )n∈N0 is a sequence of characters of the group C. All characters are of
this form, and (Γn )n∈N0 is an orthonormal family (see [11, 376–380]). We observe
that the dual group 0
C of C is the set of sequences d = (dn )n∈N ∈ {0, 1}N such
that only a finite number of the terms are equal to one. As C is compact, the set of
characters 0
C are a complete orthonormal family in L2 (C). Now, we can apply the
proposed construction of tempered distributions (in Sect. 2) over C. Let us define
the operator D : S ,→ S by the way it evaluates on the characters, that is,
∀n ≥ 1 , D(Γn ) = nΓn .
164
M. L. Esquível and N. P. Krasii
It is clear that this operator satisfies the conditions of Definition 5 as we have that
2
L (C) = φ =
+∞
a n Γn :
n=1
= T =
+∞
+∞
|an | < +∞ ⊂ S
2
n=1
sn Γn : (sn )n≥1 ∈ si
n=1
and so, as (nsn )n≥1 ∈ si, we have that
D(T ) =
+∞
nsn Γn ∈ S .
n=1
We have thus defined Schwartz distributions over C—a set homeomorphic to the
Cantor set—despite the fact that a usual theory of differentiation over this group
encounters natural difficulties (see [24] for a discretization approach). The proposed
theory of random Schwartz distributions over C is now straightforward, and in future
work, we will develop some aspects of this theory that can be applied to modeling
on fractals (see [12] for a problem in optics) but on a random environment.
Acknowledgments This work was partially supported, for the first author, through the project of
the Centro de Matemática e Aplicações, UID/MAT/00297/2020, financed by the Fundação para a
Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) and, for the second
author, through Russian Foundation for Basic Research (RFBR) (grant no. 19-01-00451).
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On Solutions of Stochastic Equations
with Current and Osmotic Velocities
Yuri E. Gliklikh
Abstract The main aim of this paper is to collect the results connected with
existence of solutions of equations with current and osmotic velocities, published in
several articles, and supply all constructions and results with complete proofs. Some
new properties of both types of equations and their interrelation are described.
Keywords Nelson’s mean derivatives · Current velocities · Osmotic velocities ·
Equations with current and osmotic velocities · Existence of solutions
1 Introduction
Current and osmotic velocities are symmetric and antisymmetric, respectively,
Nelson’s mean derivatives (see [1–3] and the details below). Note that classical
Nelson’s mean derivatives give some information about the drift of a diffusion
process but no information about the diffusion coefficient. In order to make it
possible to recover a process from its mean derivatives, in [4] (see also [5]), we
introduced an additional mean derivative called quadratic.
Note that current velocities are natural analogs of ordinary physical velocities of
deterministic processes and osmotic velocities in some sense measure how fast “the
randomness” of a process growths up. Thus, the investigation of equations given in
terms of current and quadratic velocities as well as osmotic and quadratic velocities
is important for applications to mathematical physics and other branches of science.
The main aim of this paper is to collect the results connected with existence
of solutions of equations with current and osmotic velocities, published in several
articles (mainly in [6–8] and some others) and supply all constructions and results
with complete proofs. Some new properties of both types of equations and their
interrelation are described.
Y. E. Gliklikh ()
Voronezh State University, Voronezh, Russia
e-mail: yeg@math.vsu.ru
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_8
167
168
Y. E. Gliklikh
We deal with equations in Euclidean space Rn . By L(Rn , Rn ), we denote the
space of n × n matrices. By S+ , we denote the set of symmetric positive definite
matrices. Its closure, the set of symmetric positive semi-definite matrices, we denote
by S̄+ . The symbol ∂x∂ i denotes both the partial derivative (i.e., the differential
operator) and the ith vector of basis in Rn . For a vector field v and a function f ,
the symbol vf denotes the derivative of f in the direction of v. Everywhere below,
we use Einstein’s summation convention, i.e., the existence of repeated upper and
lower indices means the sum with respect to such index from 1 to n.
2 Preliminaries on Mean Derivatives
Here, we describe the notions and facts from the theory of mean derivatives and
close to it that is necessary for our purpose. More details can be found in [3, 5].
Consider a stochastic process ξ(t) in Rn , t ∈ [0, T ] given on a certain probability
space (Ω, F , P) such that ξ(t) is an L1 random element for all t.
ξ
Denote by symbol Nt the σ -sub-algebra in F generated by pre-images of Borel
n
sets in R under the mapping ξ(t) : Ω → Rn .
ξ
ξ
The conditional expectation with respect to Nt is denoted by Et (·).
Ordinary mathematical expectation is denoted by E.
Recall that, strictly speaking, almost surely (a.s.) the sample paths of ξ(t) are not
differentiable for almost all t. Hence, the “classical” derivatives in time of ξ(t) exist
only in the sense of distributions. In order to avoid using the distributions, following
Nelson (see., e.g., [1–3]), we give the next:
Definition 1
(i) The forward mean derivative Dξ(t) of ξ(t) at time instant t is an L1 random
element of the form
ξ ξ(t + Δt) − ξ(t)
Dξ(t) = lim Et
(1)
Δt →+0
Δt
where the limit is supposed to exist in L1 (Ω, F , P) and Δt → +0 means that
Δt tends to 0 and Δt > 0.
(ii) The backward mean derivative D∗ ξ(t) of ξ(t) at time instant t is an L1 random
element
ξ ξ(t) − ξ(t − Δt)
(2)
D∗ ξ(t) = lim Et
Δt →+0
Δt
where the conditions and notation are the same as in (i).
Note that usually, Dξ(t) = D∗ ξ(t), but if, say, ξ(t) a.s. has smooth sample paths,
those derivatives evidently coincide.
On Solutions of Stochastic Equations with Current and Osmotic Velocities
169
From the properties of conditional expectation (see [9] ), it follows that Dξ(t)
and D∗ ξ(t) can be represented via superpositions of ξ(t) and Borel measurable
vector fields (regressions)
Y 0 (t, x) = lim E
Δt →+0
Y∗0 (t, x) = lim E
Δt →+0
ξ(t + Δt) − ξ(t)
| ξ(t) = x
Δt
ξ(t) − ξ(t − Δt)
| ξ(t) = x
Δt
(3)
on Rn . This means that Dξ(t) = Y 0 (t, ξ(t)) and D∗ ξ(t) = Y∗0 (t, ξ(t)).
Definition 2 The derivative DS = 12 (D + D∗ ) is called symmetric mean derivative.
The derivative DA = 12 (D − D∗ ) is called the antisymmetric mean derivative.
Consider the vector fields
v ξ (t, x) =
1 0
Y (t, x) + Y∗0 (t, x)
2
uξ (t, x) =
1 0
Y (t, x) − Y∗0 (t, x)v).
2
and
Definition 3 v ξ (t) = v ξ (t, ξ(t)) = DS ξ(t) is called the current velocity of ξ(t);
uξ (t) = uξ (t, ξ(t)) = DA ξ(t) is called the osmotic velocity of ξ(t).
For stochastic processes, the current velocity is a direct analog of ordinary
(“physical”) velocity of deterministic processes (see, e.g., [1–3, 5]). The osmotic
velocity measures how fast “the randomness” is growing up.
Following [4] (see also [5]), we introduce the differential characteristic D2 of
ξ(t) by the formula
D2 ξ(t) = lim
)t →+0
ξ
Et
(ξ(t + )t) − ξ(t)) ⊗ (ξ(t + )t) − ξ(t))
,
)t
(4)
where the limit is supposed to exist in L1 (Ω, F , P). On the basis of properties of
Itô integrals, it is shown that after taking the limit, D2 ξ(t) becomes a symmetric
positive semi-definite matrix function on [0, T ] × Rn .
Definition 4 D2 is called the quadratic mean derivative.
Below, it is convenient to use the following definition of diffusion process.
170
Y. E. Gliklikh
Definition 5 By diffusion process, we call a strong solution of a stochastic differential equation in Itô form
ξ(t) = ξ0 +
t
a(s, ξ(s))ds +
t
0
A(s, ξ(s))dw(s)
(5)
0
where a(t, x) and A(t, x) are C ∞ -smooth mappings from [0, T ] × Rn , to Rn ,
to L(Rn , Rn ), respectively.
Theorem 1 ([4, 5]) For diffusion process (5), the forward mean derivative Dξ(t)
exists and takes the form Dξ(t) = a(t, ξ(t)).
Theorem 2 ([4, 5]) For diffusion process (5), the quadratic mean derivative D2 ξ(t)
exists and takes the form D2 ξ(t) = α(t, ξ(t)) where α = AA∗ is the diffusion
coefficient.
3 The Formulae for Current and Osmotic Velocities
Let a C ∞ -smooth nonautonomous (2, 0)-tensor field with values in S̄+ (n) be
given on Rn . In other words, it is a C ∞ -smooth nonautonomous field of symmetric
positive definite matrices α(t, x) = (α ij (t, x)).
Since all the matrices are nondegenerate and the field is C ∞ -smooth, there exists
unique C ∞ -smooth field of symmetric positive definite inverse matrices
(αij (t, x)) ((0, 2)-tensor field). The latter field generates a new time-dependent
Riemannian metric αt (·, ·) = αij (t, x)dx i ⊗ dx j on Rn . Note that αt (·, ·) can be
also considered as the restriction on the level surface t = const of the Riemannian
metric ᾱ(·, ·) on R×Rn with the matrix of metric tensor at the point (t, x) ∈ R×Rn
in the form
1
0
.
(ᾱkl (t, x)) =
0 (αij (t, x))
Note that the determinants of the matrices (ᾱkl (t, x)) and (αij (t, x)) coincide. It is
well-known that the volume form of metric αt (·, ·) on the level surface t = const is
described as
=
Λαt = det(αij (t, x))dx 1 ∧ dx 2 ∧ · · · ∧ dx n ,
and so the volume form of ᾱ(·, ·) is described as
Λᾱ =
=
det(αij (t, x))dt ∧ dx 1 ∧ dx 2 ∧ · · · ∧ dx n = dt ∧ Λαt .
For simplicity for the latter form, we shall use the notation dt ∧ Λαt .
On Solutions of Stochastic Equations with Current and Osmotic Velocities
171
Below, we deal with C ∞ -smooth fields of nondegenerate linear operators
A(t, x) : R × Rn → Rn and t ∈ R, x ∈ Rn (i.e., with C ∞ -smooth
nonautonomous (1, 1)-tensor fields on Rn ). For example, let ξ(t) be a diffusion
process (5) where A(t, x) is C ∞ -smooth and nondegenerate. Then, its diffusion
coefficient A(t, x)A∗(t, x) is a C ∞ -smooth field of symmetric positive definite
matrices α(t, x) = (α ij (t, x)) ((2, 0)-tensor field on Rn ), i.e., on its basis, we can
construct the Riemannian metric αt (·, ·) as above.
Note that the transitions from the field (α ij (t, x)) as above to the field A(t, x) is
also possible.
Lemma 1 ([4, 5]) Let α(t, x) be jointly continuous (measurable, smooth) mapping from [0, T ] × Rn to S+ (n). Then, there exists jointly continuous (measurable
and smooth, respectively) mapping A(t, x) from [0, T ] × Rn to L(Rn , Rn ) such that
for all t ∈ R and x ∈ Rn the equality A(t, x)A∗ (t, x) = α(t, x) holds.
Denote by ρ̂(t, x) the probabilistic density of the random element ξ(t) with
respect to the Lebesgue measure on R × Rn , i.e., with respect to Euclidean volume
form ΛE = dt ∧ dx 1 ∧ · · · ∧ dx n . This means that for every [0, T ] ⊂ R and every
bounded continuoua function f : [0, T ] × Rn → R, the following relation holds:
T
E(f (t, ξ(t))) dt =
0
0
Ω
=
f (t, ξ(t))dP dt
T [0,T ]×Rn
f (t, x)ρ̂(t, x)dt ∧ ΛE .
Theorem 3 ([10, Theorem 1.1] or [11, Formula (2.8)]) Let ξ(t) be a diffusion
process with the diffusion coefficient (a ij ) and density ρ̂. Then, under the assumption that ρ̂ is smooth and nowhere equals to zero,
u(t, x) =
∂
ij
1 ∂x j (α (x)ρ̂(t, x)) ∂
.
2
ρ̂(t, x)
∂x i
(6)
Proof Here, we suggest the proof, alternative to [10, Theorem 1.1] and [11, formula
(2.8)].
Take an arbitrary smooth function f (t, x) with compact support on Rn . Note that
ξ
f (t, ξ(t)) is Nt -measurable. Hence,
t
ξ
E f (ξ(t))Et
t −Δt
A(t, ξ(t))dw(t)
Since f (ξ(t − Δt)) and
= E f (ξ(t))
$t
t
t −Δt
A(t, ξ(t))dw(t) .
t −Δt A(t, ξ(t))dw(t) are independent and
t
E
t −Δt
A(t, ξ(t))dw(t) = 0,
172
Y. E. Gliklikh
we obtain
t
E(f (ξ(t))
t −Δt
A(t, ξ(t))dw(t)
t
= E (f (ξ(t)) − f (ξ(t − Δt)))
t −Δt
A(t, ξ(t))dw(t)
.
$t
By the Ito formula, f (ξ(t)) − f (ξ(t − Δt)) = t −Δt (df · a(s, ξ(s)))ds +
$
$
t
1 t
2 t −Δt tr f (ξ(s))ds + t −Δt (df · A(s, ξ(s)))dw(s) (by ·, we denote the action
of 1-forms on vectors). Thus,
t
E f (ξ(t))
=E
˜
t
t −Δt
t −Δt
A(t, ξ(t))dw(t)
(df · a(s, ξ(s)))A(s, ξ(s))dsdw(s)
1 t
tr f (ξ(s))(A(s, ξ(s)), A(s, ξ(s)))dsdw(s)
2 t −Δt
t
+
(df · A(s, ξ(s)))A(s, ξ(s))ds .
+
t −Δt
The first two integrals in the right-hand side are equal to zero. The calculations in
coordinates yield the equality (df · A)A = df · (AA∗ ).
On the other hand,
T
E f (t, ξ(t))uξ (t, ξ(t)) dt
0
1
=−
2
T
E f (t, ξ(t)) lim
0
Δt →+0
$t
ξ t −Δt A(s, ξ(s))dw(s)
) dt
Et (
Δt
1 T
1 T
E(df · AA∗ )dt = −
df · AA∗ · ρ̂ ξ Λαt dt
2 0
2 0
Rn
T 1
=
f · d(AA∗ · ρ̂ ξ )Λαt dt
2 0
Rn
d(AA∗ · ρ̂ ξ ) ξ
1 T
1 T d(AA∗ · ρ̂ ξ ) dt
f
ρ̂
Λ
E f
=
dt
=
αt
2 0
ρ̂ ξ
2 0
ρ̂ ξ
Rn
∂
ij ξ
1 T j (α ρ̂ ) ∂
=
E f (t, ξ(t)) ∂x ξ
dt.
2 0
ρ̂
∂x i
=−
On Solutions of Stochastic Equations with Current and Osmotic Velocities
173
Since it is valid for an arbitrary function f as above, this means that uξ =
∂
ij ξ
j (α ρ̂ ) ∂
1 d(AA∗ ·ρ̂ ξ )
= 12 ∂x ρ̂ ξ
.
2
ρ̂ ξ
∂x i
ij
∂
Remark 1 Denote by Ξ (t, x) the vector field with coordinate presentation ∂α
.
∂x j ∂x i
1
1
ξ
ξ
One can easily derive from (6) that u (t, x) = 2 Gradt log ρ (t, x) + 2 Ξ (t, x)
where Gradt denotes the gradient with respect to the metric αt (·, ·). Indeed,
∂
(α ij ρ ξ (t,x))
∂
∂x j
ρ ξ (t,x)
∂x i
∂α ij ∂
∂ρ ξ
j
∂ρ ξ
j
ij
∂
∂
∂
= 12 α ij ∂x
+ 12 ∂α
, where α ij ∂x
= Gradt log ρ ξ and
ρ ξ ∂x i
∂x j ∂x i
ρ ξ ∂x i
= Ξ . Take into account that here, the gradient is found only with respect to
∂x j ∂x i
the space variables, i.e., here, t is a parameter.
Denote by symbol ρ ξ (t, x) the probability distribution density of the random
element ξ(t) with respect to the volume form dt ∧ Λαt on [0, T ] × Rn , i.e., for every
continuous bounded, function f : [0, T ] × Rn → R the relation
T
E(f (t, ξ(t)))dt =
0
=
f (t, ξ(t))dP dt
T 0
Ω
ξ
[0,T ]
Rn
f (t, x)ρ (t, x)Λαt dt
holds.
√
Note that by the construction, ρ̂(t, x) = ρ(t, x) det (at (x)).
Lemma 2 For v ξ (t, x) and ρ ξ (t, x) of process (5), the following relation of
continuity equation type in the form
∂ρ ξ (t, x)
= − Divt (ρ ξ (t, x)v ξ (t, x)),
∂t
(7)
holds where Divt denotes the divergence with respect to Riemannian metric αt (·, ·)
on the level surface t = const.
Proof Denote by symbol ΛE the Euclidean volume form dx 1 ∧ · · · ∧ dx n on Rn .
Thus, Λαt = det(αij )ΛE .
Recall that Divt (ρ ξ v ξ ) = d((ρ ξ v ξ ) Λαt ), where is the interior product of
vector (ρ ξ v ξ ) and n-form Λαt and the exterior differential d is considered on the
level surface t = const, i.e., it contains
differentiation only in space variables x i .
n
ξ
ξ
Then, (ρ v ) Λαt = det(αij ) i=1 (ρ ξ v ξ )i dx 1 ∧· · ·∧dx i−1 ∧dx i+1 ∧· · ·∧dx n ,
and so,
Divt (ρ ξ v ξ ) =
(ρ ξ v ξ )i ∂ det(αij ) ∂(ρ ξ v ξ )i
+
.
∂x i
∂x i
det(αij )
(8)
174
Y. E. Gliklikh
Specify a smooth real function f (t, x) with compact support and two numbers
0 ≤ s < t ≤ T . By symbol df , we denote the differential with respect to space
∂f
i
coordinates x i : df = ∂x
i dx . By calculations in coordinates, we obtain
[s,t ]×Rn
[s,t ]×Rn
df · (ρ ξ v ξ (τ, ξ(τ ))) dτ ∧ Λαt =
=
df · (ρ ξ v ξ (τ, ξ(τ ))) det(αij ) dτ ∧ ΛE =
,
∂(ρ ξ v ξ )i
ξ ξ i ∂ det(αij )
det(αij )
+ (ρ v )
f (τ, x)
dτ ∧ ΛE =
−
∂x i
∂x i
[s,t ]×Rn
+
,
∂(ρ ξ v ξ )i
(ρ ξ v ξ )i ∂ det(αij ) =
f (τ, x)
−
+
det(αij ) ·
∂x i
∂x i
det(αij )
[s,t ]×Rn
+
=
· dτ ∧ ΛE =
+
,
∂(ρ ξ v ξ )i
(ρ ξ v ξ )i ∂ det(αij )
f (t, x)
dτ ∧ Λαt =
+
−
∂x i
∂x i
det(αij )
[s,t ]×Rn
f (τ, x) Div(ρ ξ v ξ ) dτ ∧ Λαt .
−
[s,t ]×Rn
By the Itô formula,
E f (t, ξ(t)) − f (s, ξ(s)) =
E
t ∂f
s
∂τ
dτ +
t
df · Y 0 (τ, ξ(τ ))dτ +
s
1
2
t
tr f (A, A)dτ
s
and by the backward Itô formula,
E f (t, ξ(t)) − f (s, ξ(s)) =
E
t ∂f
s
∂τ
dτ +
t
s
df · Y∗0 (τ, ξ(τ ))dτ −
1
2
t
tr f (A, A)dτ .
s
Hence,
E f (t, ξ(t)) − f (s, ξ(s)) = E
t ∂f
s
∂τ
dτ +
t
s
df · v ξ (τ, ξ(τ ))dτ .
On Solutions of Stochastic Equations with Current and Osmotic Velocities
175
But
E
t ∂f
s
dτ +
∂τ
∂f
t
df · v ξ (τ, ξ(τ ))dτ =
s
ρ ξ + [df · (ρ ξ v ξ (τ, ξ(τ )))] dτ ∧ Λαt =
∂τ
∂ρ ξ ∂ f (τ, x)
(f (τ, x)ρ ξ dτ ∧ Λαt −
dτ ∧ Λαt
∂τ
[s,t ]×Rn ∂τ
[s,t ]×Rn
−
f (τ, x) Div(ρ ξ v ξ ) dτ ∧ Λαt =
[s,t ]×Rn
[s,t ]×Rn
E f (t, ξ(t)) − f (s, ξ(s)) −
−
[s,t ]×Rn
[s,t ]×Rn
f (τ, x)
∂ρ ξ dτ ∧ Λαt
∂τ
f (τ, x) Div(ρ ξ v ξ ) dτ ∧ Λαt .
$
$
ξ
ξ ξ
Thus, [s,t ]×Rn f (τ, x) ∂ρ
∂τ dτ ∧ Λαt + [s,t ]×Rn f (τ, x) Div(ρ v ) dτ ∧ Λαt =
ξ
0. Since this is valid for every function f (t, x) as above, we obtain that ∂ρ
∂τ =
ξ
ξ
− Divt (ρ v ).
Lemma 3 The following formula takes place:
Divt (ρ ξ v ξ ) = αt (v ξ , Gradt ρ ξ ) + ρ Divt v ξ
(9)
where Gradt denotes the gradient with respect to Riemannian metric αt (·, ·) on the
level surface t = const.
Proof In complete analogy with the derivation of formula (8), we obtain
Divt (v ξ ) =
ξ ξ i
ξ i
(v ξ )i ∂ det(αij ) ∂(v ξ )i
+
.
∂x i
∂x i
det(αij )
(10)
ξ
)
Recall that ∂(ρ∂xvi ) = ρ ξ ∂(v
+ (v ξ )i ∂ρ
. Then, from formula (8), taking into
∂x i
∂x i
ξ
account also formula (10), we derive that Divt (ρ ξ v ξ ) = ρ ξ Divt v ξ + (v ξ )i ∂ρ
,
∂x i
ξ
where (v ξ )i ∂ρ
is the derivative of density ρ ξ in the direction of vector field v ξ
∂x i
on the level surface. But by the definition of gradient Gradt , the latter derivative is
equal to αt (v ξ , Gradt ρ ξ ).
In the particular case of autonomous α(x), it generates the unique Riemannian
metric α(·, ·) on Rn with the corresponding Div and Grad.
176
Y. E. Gliklikh
Corollary 1 If α(x) is autonomous, relation (7) takes the form
∂ρ ξ (t, x)
= − Div(ρ ξ (t, x)v ξ (t, x))
∂t
(11)
and formula (9) takes the form
Div(ρ ξ v ξ ) = α(v ξ , Grad ρ ξ ) + ρ Div v ξ
(12)
4 Equations with Current Velocities
As it is said above, the current velocity is a direct analog of ordinary velocity of
deterministic physical processes. Thus, the case of equations with current velocities
is apparently the most natural from the physical point of view.
Let v : R × Rn → Rn and α : R × Rn → S̄+ (n) be Borel measurable mappings.
The system of the form
DS ξ(t) = v(t, ξ(t))
D2 ξ(t) = α(t, ξ(t))
(13)
is called the first-order equation with current velocities.
Definition 6 We say that (13) has a solution on the interval [0, T ], if there exists a
probability space (Ω, F , P) and a stochastic process ξ(t) given on it for t ∈ [0, T ],
such that it satisfies (13).
Theorem 4 Let v : [0, T ] × Rn → Rn and α : [0, T ] × Rn → S+ (n) be jointly
C ∞ -smooth in all variables (in particular, this means that for every t, the field
α determines the time-dependent Riemannian metric αt (·, ·) on Rn , introduced in
Sect. 3). Let also the following estimates take place for a certain K > 0:
v(t, x) < K(1 + x),
(14)
tr α(t, x) < K(1 + x2 )
(15)
|Ξ (t, x)| < K(1 + x).
(16)
Let ξ0 be a random element with values in Rn whose probability distribution density
ρ0 with respect to the volume form Λα0 of metric α0 (·, ·) on Rn (see Sect. 3) is
C ∞ -smooth and nowhere equal to zero. Then, for the initial condition ξ(0) = ξ0 ,
Eq. (13) has a solution well posed on the entire interval t ∈ [0, T ] that is unique as
a diffusion process.
On Solutions of Stochastic Equations with Current and Osmotic Velocities
177
Proof Since v(t, x) is C ∞ -smooth and estimate (14) is valid for it, its flow gt is
well-defined on the entire interval t ∈ [0, T ]. By symbol gt (x), we denote the orbit
of this flow (i.e., the solution of equation x (t) = v(t, x)) with initial condition
g0 (x) = x. Since v(t, x) is C ∞ -smooth, its flow is C ∞ -smooth as well.
By the use of Lemma 3, we transform equation of continuity (7) to the form
∂ρ(t, x)
= −αt (v, Gradt ρ) − ρ Divt v.
∂t
(17)
Suppose that ρ(t, x) is nowhere equal to zero in [0, T ] × Rn . Then, we can divide
(17) by ρ and so transform it to the form
∂p
= −αt (v, Gradt p) − Divt v
∂t
(18)
where p = log ρ. Introduce p0 = log ρ0 .
Show that the solution of Eq. (18) with initial condition p(0) = p0 is described
by the formula
p(t, x) = p0 (g−t (x)) −
t
0
(Divτ v)(s, gs (g−t (x)) ds.
(19)
Introduce the product [0, T ] × Rn , and consider function p0 as given on the level
surface t = 0. Consider the vector field (1, v(t, x)) on [0, T ] × Rn . The orbits of its
flow ĝt , starting at the points of surface t = 0, take the form ĝt (0, x) = (t, gt (x)),
and the flow ĝt is C ∞ -smooth as well as gt . Consider on [0, T ]×Rn the Riemannian
metric ᾱ(·, ·) (see Sect. 3). Note that for every (t, x), the point ĝ−t (t, x) belongs
to the level surface t = 0, where the function p0 is given. Thus, on the one hand,
(1, v)p(t, x) (the derivative of function p(t, x) in the direction of vector field (1, v))
equals − Divt v(t, x) by definition. But on the other hand, one can easily calculate
that (1, v)p(t, x) = ∂t∂ p(t, x) + αt (v(t, x), Gradt p(t, x)). So, (18) is satisfied.
We emphasize that ρ = ep is indeed nowhere equal to zero, and so, our
arguments are correct.
Recall that the process having density ρ and diffusion coefficient α has the
osmotic velocity u, uniquely and independently of v calculated by formula (6).
Hence, the process having current velocity v and diffusion coefficient α (and
consequently density ρ) has the drift (forward mean derivative) a(t, x) = v(t, x) +
√
u(t, x) = v(t, x)+ 12 Gradt p+ 12 Ξ = v(t, x)+Gradt log ρ + 12 Ξ (see Remark 1),
uniquely determined by this formula. Thus, the process having current velocity
v and diffusion coefficient α is a solution of the following stochastic differential
equation:
ξ(t) = ξ0 +
t
0
a(s, ξ(s))ds +
t
A(s, ξ(s))dw(s),
0
(20)
178
Y. E. Gliklikh
where C ∞ -smooth field A(t, x) such that A(t, x)A∗ (t, x) = α(t, x) exists by
Lemma 1. Since the coefficients of this equation are C ∞ -smooth and estimates (14),
(15), and (16) are fulfilled, from results of [12], it follows that this equation has a
unique strong solution that is well defined on the entire interval [0, T ].
Lemma 4 Let v(t, x), α(x), and Λα be the same as in Theorem 4 but α(x) be
autonomous. Let ρ(t, x) be the smooth nowhere equal to zero probability density
whose existence is proved in Theorem 4. Then, the flow of vector field (1, v(t, x)) on
R×Rn preserves the volume form ρ(t, x)dt ∧Λα , i.e., L(1,v(t,x))ρ(t, x)dt ∧Λα = 0,
where L(1,v(t,x))ρ(t, x)dt ∧ Λα is the Lie derivative of the form ρ(t, x)dt ∧ Λα in
the direction of vector field (1, v(t, x)).
Proof Introduce p(t, x) = log ρ(t, x), i.e., ρ(t, x) = ep(t,x). Recall that ρ(t, x)dt∧
Λα = ρ(t, x) det(aij )(x)dt ∧ ΛE . Since ρ(t, x) det(aij )(x)dt ∧ ΛE is a volume
form,
=
=
L(1,v)ρ(t, x) det(aij )(x)dt ∧ ΛE = d (1, v) ρ(t, x) det(aij )(x)dt ∧ ΛE ,
where denotes the inner product of the vector (1, v) and the form
ρ(t, x) det(aij )(x)dt ∧ ΛE (see [13]). From the latter formula, one can easily
derive that
=
L(1,v)ρ(t, x) det(aij )(x)dt ∧ ΛE =
=
=
∂ρ
+ vρ + ρ div v
det(aij )(x)dt ∧ ΛE + v det(aij )(x) ρdt ∧ ΛE =
∂t
(21)
=
=
∂p
+ vp + div v + v log det(aij )(x) ρ det(aij )(x)dt ∧ ΛE .
∂t
But div v+v log det(aij )(x) = Div v, and vp is the derivative of p in the direction
of v that by definition equals α(Grad p, v), where Grad is the gradient with respect
to the metric α(·, ·). Thus,
=
L(1,v)ρ(t, x) det(aij )(x)dt ∧ ΛE =
=
∂p
+ Div v + α(Grad p, v) ρ(t, x) det(aij )(x)dt ∧ ΛE .
∂t
(22)
On the other hand, it is easy to calculate that Div(ρv) = ρ v + det(aij ) +
v ρ + ρ div v. The continuity equation (11) is transformed to the form ∂ρ
∂t =
On Solutions of Stochastic Equations with Current and Osmotic Velocities
179
−α(Grad ρ, v) − ρ Div v, i.e., after dividing by ρ to the form
∂p
= −α(Grad p, v) − Div v.
∂t
On substituting this relation for ∂p
∂t to (22), we obtain that
=
L(1,v)ρ(t, x) det(aij )(x)dt ∧ ΛE = 0.
Corollary 2 Let α(x) and Λα be the same as in Lemma 4 and ρ(t, x) be a certain
smooth nowhere equal to zero probability density. Let a smooth vector field v(t, x)
on Rn be such that L(1,v) dt ∧ Λα = 0 (i.e., (1, v) preserves the form dt ∧ Λα on
R × Rn ). Then, ρ(t, x) and v(t, x) satisfy continuity-type equation (11).
Proof Introduce p = log ρ. Use the arguments as at the end of the proof
of Lemma 4, applying them in opposite direction. This allows us to derive
from L(1,v)dt ∧ Λα = L(1,v)ρ(t, x) det(aij )(x)dt ∧ ΛE = 0 that ∂p
∂t =
−α(Grad p, v) − Div v. Replace in the latter relation p by log ρ. Recall that by
definition, α(Grad log ρ, v) = v log ρ (the derivative of log ρ in the direction of v).
∂p
So, α(Grad log ρ, v) = vρ
ρ , and since vρ = α(Grad ρ, v), ∂t = −α(Grad p, v) −
∂ρ
Div v transforms into ∂tρ = − ρ1 α(Grad ρ, v) − Div v. After multiplying by ρ and
taking into account (12), we obtain (11).
5 Equations with Osmotic Velocities
Consider C ∞ -vector field u(t, x) and C ∞ -smooth (2, 0)-tensor field of symmetric
positive definite matrices α(x) = (α ij )(x) ON Rn . Since the field (α ij )(x) is C ∞ smooth and positive definite, the field of inverse matrices (αij ) exists, and it can
be considered as a Riemannian metric on Rn (see Sect. 3). To avoid some technical
difficulties, we suppose that u(t, x), α(x) and Ξ (x) (see Remark 1) are uniformly
bounded with respect to the corresponding norms.
Definition 7 The system
DA ξ(t) = u(t, ξ(t))
D2 ξ(t) = α(ξ(t))
is called the first-order equation with osmotic velocities.
By technical reasons, here, we consider autonomous field α(x).
(23)
180
Y. E. Gliklikh
Definition 8 System (23) has a solution on the interval [0, T ], if there exists a
probability space (Ω, F , P) and a stochastic process ξ(t) given on it for t ∈ [0, T ]
and having values in Rn , such that a.s. (23) holds.
Taking into account formula (6) and Remark 1, one can easily see that (23) can
have solutions not for all right-hand sides and initial values since u and α should
be connected by formula u(t, x) = 12 (Ξ (x) + Grad log ρ̂(t, x)) with probabilistic
density ρ̂(t, x) with respect to the Lebesgue measure on Rn for any specified t
(see (6)). In particular, this relation should hold for the initial density ρ0 (smooth
and nowhere equal to zero), initial value of the osmotic velocity u(0, x), and the
quadratic velocity α(x).
Lemma 5 For any smooth u(t, x) and smooth positive definite α(x) for any
specified t, there exists a positive function ρ̂(t, x), given up to a multiplicative
constant, such that (6) holds.
Proof Introduce p̂(t ) (x) = log ρ̂(t, x) (hence, ρ̂(t, x) = ep̂(t) (x) ). Now, we can
transform (6) into the form u(t, x) = 12 Grad p̂(t )(x) + 12 Ξ (x), where Ξ (x) is
known since its coordinates consist of the first partial derivatives of the known
smooth field α(x). Recall that Grad is given only by space variables and t
here is only a parameter. Note that u(t, x) is known from the hypothesis. So,
Grad p̂(t, x) = 2u(t, x)−Ξ (x). Denote the ith coordinate of vector 2u(t, x)−Ξ (x)
by (2u(t, x) − Ξ (x))i and the j th coordinate of 1-form d p̂(t ) (differential of p̂(t ) ;
here, d is the exterior differential) by (d p̂(t ))j . Then, (d p̂(t ))j = αij (2u(t, x) −
Ξ (x))i , and so, d p̂(t ) is known. Emphasize that d p̂(t ) is given for any specified t
and that d p̂(t ) is smooth in t by construction.
By the standard method, it is possible to recover p̂(t ) from d p̂(t ) on Rn up to an
additive constant. For this, specify a certain value of p̂(t ) at a certain point x ∈ Rn
for a certain specified t. For simplicity, let x be the origin in Rn , and let the value of
p̂(t ) at x be zero. Construct the value of p̂(t ) (x) at every x ∈ Rn for any specified t
by the following
way. Let σ (s) be a curve connecting the origin with x. Determine
$
p̂(t )(x) = σ d p̂(t ) . Take another curve σ1 (s) connecting the origin with x. Change
the orientation on σ1 „ and consider the union of σ and σ1 as the boundary ∂Θ of
an arbitrary specified two-dimensional
sub-manifold Θ in Rn . Then,
?
$$
? by the Stokes
theorem (see, e.g., [14]), ∂Θ d p̂(t ) = Θ dd p̂(t ). Since d 2 = 0, ∂Θ dp(t) = 0,
and so, the constructed value of p̂(t ) (x) does not depend on the choice of curve σ . In
particular, it is possible to construct p̂(t, x) by using the abovementioned integrals
along the straight rays starting from the origin.
By the construction, ρ̂ = ep̂ is positive for all (t, x). Since p̂ is defined up to an
additive constant, ρ̂ is defined up to a multiplicative constant.
Note that among the functions ρ̂(t, x) constructed in Lemma 5 for any$specified t,
there should exist a probabilistic density on Rn , i.e., a function such that Rn ρ̂ΛE =
1. Since the functions
ρ̂(t, x) are constructed up to multiplicative constant, if for any
$
t the integral Rn ρ̂ΛE has finite value for some function, it has finite values for all
other functions. In this case, there exists a function, for which the integral equals
to 1. But if the integral has infinite value (the integral does not exist) for a certain
On Solutions of Stochastic Equations with Current and Osmotic Velocities
181
function, it does not exist for all others. In this case, the problem of existence of
solution to Eq. (23) is not well posed. We can formulate a condition, under which
this problem is well posed, and moreover, it is possible to prove the existence of
solution.
Condition 1 Among the function ρ̂(t, x), whose existence is shown in Lemma 5 up
to a multiplicative constant, there exists a function that is a probabilistic density
with respect to ΛE on Rn for every t ∈ [0, T ], and this function is C ∞ -smooth
jointly in all variables.
If we know ρ̂(t, x) from Condition 1, we can find the corresponding density
ρ(t, x) with respect to Λα by formula ρ̂(t, x) = ρ(t, x) det(αij (x)). In particular,
with the same system of arguments as above, the fields u(0, x) and α(x) determine
unique ρ̂0 (see formula (6)) and so unique ρ0 that is the initial density of the only
initial value ξ0 under which the solution can exist.
Remark 2 Note that density ρ does not determine the process uniquely; for unique
determination of the process, we need the measure on the space of sample paths.
For given ρ and (α ij ), the osmotic velocity u(t, x) is uniquely defined by formula
(6). From Corollary 2, one can easily see that any vector field v(t, x) such that
L(1,v(t,x))ρ(t, x)dt ∧ Λα = 0 is a current velocity of a certain process with density
ρ and diffusion coefficient (α ij ), since for this v and given ρ equality (11) holds.
Theorem 5 Let the vector field u(t, x) be smooth and uniformly bounded and
α(x) be the smooth and uniformly bounded autonomous field of symmetric positive
definite matrices. Let also Condition 1 be fulfilled. Then, for the initial condition
ξ(0) = ξ0 , there exists a solution of Eq. (23), and this solution is not unique.
Proof Since Condition 1 is fulfilled, the density ρ̂(t, x) with respect to ΛE is found,
and so, the density ρ(t, x) with respect to dt ∧ Λα is found as well by formula
ρ̂(t, x) = ρ(t, x) det(αij (x)). The function ρ(t, x) allows us to find at least one
vector field v(t, x) such that it is a current velocity of a certain possible solutions
that we are looking for. We will find it by using the fact that by Remark 2, this field
should satisfy the relation L(1,v)ρ(t, x)dt ∧ Λα = 0. Recall that ρ(t, x)dt ∧ Λα =
ρ(t, x) det(aij )(x)dt ∧ ΛE . By formula (21),
=
L(1,v)ρ(t, x) det(aij )(x)dt ∧ ΛE =
=
=
∂p
+ vp + div v + v log det(aij )(x) ρ det(aij )(x)dt ∧ ΛE
∂t
and if ∂p
∂t + vp + div v + v log det(aij )(x) = 0, by Remark 2, v(t, x) is the current
velocity of a certain solution.
We look for such a vector field among the vector fields that have only one nonzero
coordinate (say, the first one) while all other coordinates are equal to zero. In this
182
Y. E. Gliklikh
case, the latter formula takes the form
∂ log det(aij )(x)
∂p
∂p
∂v 1
+ v1 1 + 1 + v1
= 0.
(24)
∂t
∂x
∂x
∂x 1
√
∂ log det(aij )(x)
∂p
Note that in (24), the functions ∂p
,
,
and
are known. So, (24)
∂t ∂x 1
∂x 1
for every specified t can be considered as a first-order ordinary differential equation
with respect to v 1 . Determine the initial condition v01 for this equation as a smooth
real valued function on the subspace x 1 = 0 in Rn , and use this initial condition for
all t. The obtained solution v 1 (t, x 1 ) depends on the point in the subspace x 1 = 0,
from which it started, and the value of the initial condition at this point, as on the
parameters. Set the other coordinates equal to zero. As a result, the vector field
v(t, x) on Rn is constructed that is the current velocity of a certain solution of (23).
Now, find the forward mean derivative by the formula a(t, x) = v(t, x)+u(t, x).
It follows from Lemma 1 and from the hypothesis of Theorem that there exists
a smooth and uniformly bounded A(x) such that A(x)A∗ (x) = α(x). Then,
from the general theory of equation with forward mean derivatives, it follows that
ξ(t), having the density ρ(t, x), as above, must satisfy the following stochastic
differential equation in Itô form
ξ(t) = ξ0 +
t
0
a(s, ξ(s))ds +
t
A(s, ξ(s))dw(s).
(25)
0
From the hypothesis and the results of [12], it follows that (25) with the initial
condition ξ0 has a unique strong solution ξ(t), well defined on t ∈ [0, T ]. The fact
that DA ξ(t) = u(t, ξ(t)) and D2 ξ(t) = α(ξ(t)) follows from the construction.
It is clear that the current velocity of the type constructed above can be found
among the vector fields having another nonzero coordinate. That is why this solution
is not unique.
Acknowledgement The research is supported by Russian Foundation for Basic Research (RFBR)
Grant No. 18-01-00048
References
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6. Azarina, S.V., Gliklikh, Yu.E.: On existence of solutions to stochastic differential equations
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Stochastic Methods in Investigation
of Modern Networks
Vladimir A. Gorlov and Alla V. Makarova
Abstract There are lots of methods to analyze complex network systems; one
of them is to use the stochastic analysis. This approach is more accurate than
others, and with its help, we can get the best solution of optimization (Bander
and White, Transport Sci 36(2):218–230, 2002; Bertsekas, Dynamic programming
and optimal control. Athena Scientific, Belmont, 1995). In the current article, we
use Kolmogorov equation to find out the optimal parameters of the system in
various conditions. It also helps us to predict the productivity of the networks
and to build the self-defined networks with the ability to rebuilt or restore their
operation automatically. To solve this problem, it is necessary to describe all
possible conditions for the system with automatic rebuilding or restoring and
without these characteristics. The important part of the research is to implement the
graph theory and on its base to build the mathematical model of the whole system.
Then, the methods of stochastic analysis using Kolmogorov equation solve the
problem of the system elements productivity. At the end of this approach, we obtain
the accurate probabilities for each condition. The significance of the method in such
problems lies in the fact that we can create on its base the automatic algorithm for
complex networks with the vast amount of conditions.
Keywords SDN · Kolmogorov equations · Markov process · Efficiency ·
Network · Differential equation · Probability mathematical modelling · Optimal
control
1 SDN Network Without Reservation
The research is supported by the RFBR (Russian Foundation for Basic Research)
grant 18-01-00048.
Let’s consider the SDN (Self-Defined Network) network based on principles
and architecture used for building those kind of networks. Also, let the network
V. A. Gorlov · A. V. Makarova ()
N.E. Zhukovsky and Y.A. Gagarin Air Force Academy, Voronezh, Russian Federation
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_9
185
186
V. A. Gorlov and A. V. Makarova
consists of one server and two switchers. As well known, in SDN, the switchers
have limited option list, and all control functions, routing and other operations, lie
on SDN controller. SDN controller is a technically complex device for the network,
so it’s important to keep it in online mode, and we need to minimize the errors and
the faults. Reservation is one of the methods that can help us to solve the problem
of the faults and unstable processing of SDN network. To estimate the efficiency
of reservation, we need to build the mathematical models for both systems without
reservation and with it [1, 2].
In previous study [6], we made some assumptions: If one route is offline, the
system stays stable; the SDN network is unworkable if SDN controller or all
switchers are offline; at the same time, only one server is fault; the restore time
of controller is more than the restore time of switchers. We also defined the list
of conditions for the network: the system is online; one switcher is fault; two
switchers are fault; restoring switcher if the server is online; restoring two switchers
if the server is online; server is offline; server and switcher are offline; server
and two switchers are offline; restoring controller; controller from set of servers
is unworkable.
1.1 Transfers Between Conditions in SDN
To realize the processing of SDN network, it’s necessary to describe the system
conditions and how it transfers from one condition to another and what it means. To
answer this question, we built the spreadsheet with conditions and description (look
at Fig. 1).
1.2 Modelling of SDN Without Reservation
Furthermore, if we combine all these condition in one whole, we obtain the state
graph for the system without reservation (Fig. 2).
If we notice that we have the system with reservation, there is one new state
where the system transfers to condition 10—SDN controller from the set of servers
is unworkable. So, now, we can build a new state graph for the system with
reservation (Fig. 3).
Further using the state graph for the system without reservation, we can write the
equations that will help us to define the probabilities for each state in considered
Markov process [3, 4, 7–9].
Stochastic Methods in Investigation of Modern Networks
Transfers
1
1
λsw
λsw
187
Description
2
Switcher failure or two
switchers failures, failure rate is
equal
3
λsr
1
2
3
λsr
λsr
2
3
6
7
8
λ0
λ0
λ0
7
λsw
6
λsw
Transfer in restore condition
5
λ0
λ0
6
Server restore, switcher
restore in background
λ0
9
μm
9
If we have server or switcher
failure and happens server or
switcher failure
6
4
9
8
Server failure
6
μ
1
5
Failures removal
μm
9
μ
1
4
Fig. 1 Transfers between conditions in SDN with description
188
V. A. Gorlov and A. V. Makarova
Fig. 2 State graph for SDN without reservation
Then, we obtain the following system of Kolmogorov differential equations:
⎧
dP1 (t )
⎪
= −(2λsr )P1 + μP3 + μP4 + μP8 ,
⎪
⎪ dt
⎪
⎪
dP
(t
)
2
⎪
⎪ dt = λsw P1 − (λ0 + λsr )P2 ,
⎪
⎪
⎪
⎪ dP3 (t ) = λsw P1 + (λ0 + λsr )P3 ,
⎪
dt
⎪
⎪
⎪
dP4 (t )
⎪
⎪
⎨ dt = λ0 P2 − μP4 ,
dP5 (t )
(1)
= λ0 P3 − μP5 ,
dt
⎪
⎪
dP
(t
)
⎪
6
⎪ dt = λsr P1 − (2λsw + λ0 )P6 ,
⎪
⎪
⎪
⎪
dP7 (t )
⎪
= λsr P2 + λsw P6 − λ0 P7 ,
⎪
dt
⎪
⎪
⎪
dP8 (t )
⎪
= λsr P3 + λsw P6 − λ0 P8 ,
⎪
⎪
dt
⎪
⎩ dP9 (t )
= λ0 P6 + λ0 P7 − λ0 P8 − μP9 .
dt
The solution of this system could be obtained by well-known algorithm for
differential equations systems. However, we can simplify the calculation if we notice
Stochastic Methods in Investigation of Modern Networks
189
Fig. 3 State graph for SDN with reservation
that in this network, there is a stationary Markov process, so then, dPi = 0, i.e.,
probabilities aren’t changing through time.
Now, we overwrite the left side of the equations to zero and put in the system
normalization condition:
n
Pi = 1.
i=0
Considering all these remarks, we obtain the following:
⎧
⎪
−(2λsr )P1 + μP3 + μP4 + μP8 = 0,
⎪
⎪
⎪
⎪
⎪
λsw P1 − (λ0 + λsr )P2 = 0,
⎪
⎪
⎪
⎪
⎪
λsw P1 + (λ0 + λsr )P3 = 0,
⎪
⎪
⎪
⎪
⎪
⎪
⎨λ0 P2 − μP4 = 0,
λ0 P3 − μP5 = 0,
⎪
⎪
⎪
⎪
λsr P1 − (2λsw + λ0 )P6 = 0,
⎪
⎪
⎪
⎪
⎪
λsr P2 + λsw P6 − λ0 P7 = 0,
⎪
⎪
⎪
⎪
⎪
λsr P3 + λsw P6 − λ0 P8 = 0,
⎪
⎪
⎪
⎩
λ0 P6 + λ0 P7 − λ0 P8 − μP9 = 0.
(2)
190
V. A. Gorlov and A. V. Makarova
As we see from the system of differential equations written above, it can be
simplified, because some of probabilities are equal.
So, we have the following:
⎧
⎪
λsw P1 − (λ0 + λsr )P2 = 0,
⎪
⎪
⎪
⎪
⎪
λ0 P2 − μP4 = 0,
⎪
⎪
⎪
⎨λ P − (2λ + λ )P = 0,
sr 1
sw
0 6
⎪λsr P2 + λsw P6 − λ0 P7 = 0,
⎪
⎪
⎪
⎪
⎪
⎪
λ0 P6 + λ0 P7 − λ0 P8 − μP9 = 0,
⎪
⎪
⎩
P1 + 2P2 + 2P4 + P6 + 2P7 + P9 = 1.
(3)
To find out the solution, we involve Mathcad where we define matrix X and vector
Y . Furthermore, if we have the intensity points, we obtain the accurate probabilities
for each state, illustrated on Fig. 4.
2 Modelling of SDN Network with Reservation
In previous section, we have considered the SDN network without reservation and
built the mathematical model for this configuration.
Using all links and elements of graph theory from Fig. 1, we can obtain the
state graph for system without reservation [6] and state graph for the system with
reservation (Fig. 3). Notice that the system with reservation has one more condition:
SDN controller from the set of servers is unworkable.
Further using the algorithm from [6] and the state graph for system with
reservation from Sect. 1, we can write the equations that will help us to define the
probabilities for each state in considered Markov process [3, 8, 9].
Therefore, we obtain the following system of Kolmogorov differential equations:
⎧
dP (t )
⎪
⎪ dt1 = −(2λsw + λsr )P1 + μP3 + μP4 + μP8 ,
⎪
⎪
⎪
⎪ dP2 (t ) = λsw P1 − (λ0 + λsr )P2 ,
⎪
⎪
dt
⎪
⎪
dP3 (t )
⎪
⎪
= λsw P1 − (λ0 + λsr )P3 ,
⎪
dt
⎪
⎪
dP
(t
)
⎪
4
⎪
= λ0 P2 − μP4 ,
⎪
dt
⎪
⎪
⎨ dP5 (t ) = λ P − μP ,
0 3
5
dt
dP6 (t )
⎪
= λsr P1 − (2λsw + λ0 + λsr )P6 ,
⎪
dt
⎪
⎪
⎪
dP7 (t )
⎪
⎪
= λsr P2 + λsw P6 − (λ0 + λsr )P7 ,
⎪
dt
⎪
⎪
dP8 (t )
⎪
⎪
= λsr P3 + λsw P6 − (λ0 + λsr )P8 ,
⎪
dt
⎪
⎪
⎪ dP9 (t ) = λ P + λ P − λ P − μP + λ P 0,
⎪
0 6
0 7
0 8
9
0 1
⎪
dt
⎪
⎪
⎩ dP1 0(t ) = λ P + λ P − λ P − λ P 0.
sr 6
sr 7
sr 8
0 1
dt
(4)
3.87140704320
0608*10-6
7.74281408640
1218*10-6
7.74281408640
1218*10-6
1.17642861530
2545*10-6
9.10916313551
40848*10-12
9.10916313551
40848*10-12
3.52934050088
6445*10-6
1.9357132928
23544*10-6
7.7428531712
94175*10-6
7.7428531712
94175*10-6
5.8821920816
87397*10-7
2.2773060177
35746*10-12
2.2773060177
35746*10-12
3.5293425766
84651*10-6
1
1
1
3.87140704320
0608*10-6
1.9357132928
23544*10-6
4
1
3.5293363492
97348*10-6
3.6436165059
83808*10-11
3.6436165059
83808*10-11
2.3528180272
01446*10-6
7.7427359176
77193*10-6
7.7427359176
77193*10-6
7.7427359176
77192*10-6
7.7427359176
77192*10-6
0.9999631468
2908
Fig. 4 Current probabilities for the states in SDN without reservation
9
8
7
6
5
4
3
2
1
2
0.99997206577
0406
1
0.9999765253
00732
0.5
0.9999787
55080811
9.6785908
92341742
*10-7
9.6785908
92341742
*10-7
7.7428727
13873395
*10-6
7.7428727
13873395
*10-6
2.9511082
92252383
*10-7
5.6932840
87269202
*10-13
5.6932840
87269202
*10-13
3.5293436
1458467*
10-6
5
1
3.5293342735
06462*10-6
5.6931127069
03878*10-11
5.6931127069
03878*10-11
2.9409980323
3458*10-6
7.7426968338
46117*10-6
7.7426968338
46117*10-6
9.6783710423
07647*10-6
9.6783710423
07647*10-6
0.9999586874
1808
10
1
3.5293238945
88615*10-6
2.2771689178
6334*10-10
2.2771689178
6334*10-10
5.8817510571
97453*10-6
7.7425014199
99912*10-6
7.7425014199
99912*10-6
1.9356253559
99978*10-5
1.9356253559
99978*10-5
0.9999363909
59674
15
1
3.5293135157
31758*10-6
5.1234587037
96127*10-10
5.1234587037
96127*10-10
8.8222590975
88638*10-6
7.7423060150
01899*10-6
7.7423060150
01899*10-6
2.9033647556
25712*10-5
2.9033647556
25712*10-5
0.9999140954
95552
20
1
3.52930331369
3589*10-6
9.10806640335
57052*10-10
9.10806640335
57052*10-10
1.17625221765
0559*10-5
7.74211061885
1476*10-6
7.74211061885
1476*10-6
3.87105530942
5739&10-5
3.87105530942
5739&10-5
0.99989180102
5647
24
1
3.52929483394
3107*10-6
1.31152647103
2283*10-9
1.31152647103
2283*10-9
1.41145562828
3036*10-5
7.74195430830
1015*10-6
7.74195430830
1015*10-6
4.64517258498
0608*10-5
4.64517258498
0608*10-5
0.99987396616
5514
Stochastic Methods in Investigation of Modern Networks
191
192
V. A. Gorlov and A. V. Makarova
The solution of this system could be obtained by well-known algorithm for
differential equations systems. However, we can simplify the calculation if we notice
that in this network, there is a stationary Markov process, so then, dPi = 0, i.e.,
probabilities aren’t changing through time.
Now, we overwrite the left side of the equations to zero and put in the system
normalization condition:
n
Pi = 1.
i=0
Thus, we obtain the following:
⎧
⎪
−(2λsr )P1 + μP3 + μP4 + μP8 = 0,
⎪
⎪
⎪
⎪
⎪
λsw P1 − (λ0 + λsr )P2 = 0,
⎪
⎪
⎪
⎪
⎪
⎪
λsw P1 + (λ0 + λsr )P3 = 0,
⎪
⎪
⎪
⎪
⎪
λ0 P2 − μP4 = 0,
⎪
⎪
⎪
⎨λ P − μP = 0,
0 3
5
⎪
λsr P1 − (2λsw + λ0 + λsr )P6 = 0,
⎪
⎪
⎪
⎪
⎪
⎪
λsr P2 + λsw P6 − (λ0 + λsr )P7 = 0,
⎪
⎪
⎪
⎪
⎪
λsr P3 + λsw P6 − (λ0 + λsr )P8 = 0,
⎪
⎪
⎪
⎪
⎪
⎪
⎪λsr P6 + λsr P7 − λ0 P8 − μP9 + λ0P1 0 = 0,
⎪
⎩λ P + λ P + λ P − λ0P 0 = 0.
sr 6
sr 7
sr 8
1
(5)
The last system of differential equations can be simplified, because some of
probabilities are equal.
Thereby, we have the following:
⎧
⎪
λsw P1 − (λ0 + λsr )P2 = 0,
⎪
⎪
⎪
⎪
⎪
λ0 P2 − μP4 = 0,
⎪
⎪
⎪
⎪
⎪
⎪
⎨λsr P1 − (2λsw + λ0 + λsr )P6 = 0,
λsr P2 + λsw P6 − (λ0 + λsr )P7 = 0,
⎪
⎪
⎪
⎪
λ0 P6 + 2λ0 P7 − μP9 + λ0 P1 0 = 0,
⎪
⎪
⎪
⎪
⎪
λ
⎪
sw P6 + 2λsr P7 − λ0 P1 0 = 0,
⎪
⎪
⎩
P1 + 2P2 + 2P4 + P6 + 2P7 + P9 + P1 0 = 1.
(6)
To find out the solution, we involve Mathcad where we define matrix X and vector
Y . Moreover, if we have the intensity points, we obtain the accurate probabilities for
each state, illustrated on Fig. 5.
3.87140704320
0608*10-6
3.87140704320
0608*10-6
7.74281408640
1216*10-6
7.74281408640
1216*10-6
1.17642723128
1225*10-6
9.10914706001
2178*10-12
9.10914706001
2178*10-12
3.52934050088
6445*10-6
1.38405347008
8641*10-12
1.9357132928
23544*10-6
1.9357132928
23544*10-6
7.7428531712
94174*10-6
7.7428531712
94174*10-6
5.8821886215
8984*10-7
2.2773040084
12807*10-12
2.2773040084
12807*10-12
3.5293425766
84652*10-6
3.4601377456
88239*10-13
1
1
1
0.99997206577
0406
0.9999765253
00732
0.9999787
55080811
9.6785908
92341742
*10-7
9.6785908
92341742
*10-7
7.7428727
13873395
*10-6
7.7428727
13873395
*10-6
2.9411074
27222461
*10-7
5.6932815
75550584
*10-13
5.6932815
75550584
*10-13
3.5293436
14584669
*10-6
8.6503494
52310621
*10-14
1
5.5362008548
95788*10-12
3.5293363492
297349*10-6
3.6436036463
00351*10-11
3.6436036463
00351*10-11
2.3528124912
57784*10-6
7.7427359176
77192*10-6
7.7427359176
77192*10-6
7.7427359176
77192*10-6
7.7427359176
77192*10-6
0.9999586874
1808
4
Fig. 5 Current probabilities for the states in SDN with reservation
10
9
8
7
6
5
4
3
2
1
2
1
0.5
1
8.6503036596
61937*10-11
3.5293238945
88615*10-6
5.6930875905
533343*10-11
5.6930875905
33343*10-11
2.9409893825
33247*10-6
7.7426968338
46119*10-6
7.7426968338
46119*10-6
9.6783710423
07647*10-6
9.6783710423
07647*10-6
0.9999586874
1808
5
1
3.4601011117
47004*10-11
3.5293238945
88615*10-6
2.2771488257
27233*10-10
2.2771488257
27233*10-10
5.8817164602
04763*10-6
7.7425014199
99912*10-6
7.7425014199
99912*10-6
1.9356253549
99978*10-5
1.9356253549
99978*10-5
0.9999363909
59674
10
1
7.7851817095
69678*10-10
3.5293135157
31758*10--6
5.1233908959
74683*10-10
5.1233908959
74683*10-10
8.8221812593
33107*10-6
7.7423060150
01899*10-6
7.7423060150
01899*10-6
2.9033647556
25712*10-5
2.9033647556
25712*10-5
0.9999140954
95552
15
1
1.38402416321
9997*10-10
3.52930313693
589*10-6
9.10790568139
968*10-10
9.10790568139
968*10-10
1.17623838062
337*10-5
7.74211061885
1476*10-6
7.74211061885
1476*10-6
3.87105530942
5739&10-5
3.87105530942
5739&10-5
0.99989180102
5647
20
1
1.99298541702
0808*10-10
3.52929483394
3107*10-6
1.31149869927
8956*10-9
1.31149869927
8956*10-9
1.41143570398
3217*10-5
7.74195430830
1014*10-6
7.74195430830
1014*10-6
4.64517258498
0608*10-5
4.64517258498
0608*10-5
0.99987396616
5514
24
Stochastic Methods in Investigation of Modern Networks
193
194
V. A. Gorlov and A. V. Makarova
Thus, every complex network can be simulated by mathematical modelling
and applying the stochastic analysis to investigation of efficiency and reliability
networks. As a result, we obtain accurate characteristics.
3 Reservation Effect Estimation
To estimate the efficiency, let’s calculate the availability factors for the model with
and without reservation applying the following formula:
Kr (t) =
n
Pi (t),
i=1
where n is the number of processing conditions and Pi (t) is the probability of
processing condition.
As we can see from Figs. 1, 2, and 3, there are five conditions for the system
without reservation and eight conditions for the system with reservation. So, we
obtain the availability factor for the system without reservation:
Kr (t) =
5
Pi (t)
i=1
and
Kr (t) =
8
Pi (t)
i=1
for the system with reservation.
Let’s simplify the last equations because some probabilities are equal. Thus, for
the system without reservation, we got the following:
Kr (t) = P1 + 2P2 + 2P4
for the system with reservation:
Kr (t) = P1 + 2P2 + 2P4 + P6 + 2P7 .
Using Mathcad, we obtain the availability factors for all failure detection rates.
Analyzing the results, we can say that, firstly, the system with reservation is more
stable at the same numerical values, because the availability factor is more, and,
secondly, the reservation process is less dependent on failure detection rates.
Stochastic Methods in Investigation of Modern Networks
195
References
1. Bander, J.L., White, C.C.: A heuristic search approach for a nonstationary stochastic shortest
path problem with terminal cost. Transport. Sci. 36(2), 218–230 (2002)
2. Bertsekas, D.P.: Dynamic Programming and Optimal Control. Athena Scientific, Belmont
(1995)
3. Gihman, I.I., Skorohod, A.V.: Theory of Stochastic Processes, vol. 3. Springer, New York, NY
(1979)
4. Gliklikh, Yu.E.: Global and Stochastic Analysis with Applications to Mathematical Physics.
Springer, London (2011)
5. Gorlov, V.A.: Anisotropic diffusion in anisotropic Stepanov spaces. In: Bulletin of the South
Ural State University. Mathematical Modelling, Programming and Computer Software (Bulletin
SUSU MMCS), vol. 12(3), pp. 153–160 (2019)
6. Gorlov, V.A., Makarova, A.V.: Stochastic analysis in modelling and efficiency estimation of
modern networks. Global and Stochastic Analysis Vol. 7(2) (July–December, 2020), pp.131–
137
7. Makarova, A.V.: Stochastic inclusions with forward mean derivatives having decomposable
right-hand sides. Bull. South Ural State Univ. Ser. Math. Modell. Programm. Comput. Softw.
(Bulletin SUSU MMCS) 12(2), 143–149 (2019)
8. Parthasarathy, K.R.: Introduction to Probability and Measure, Springer-Verlag, New York, NY
(1978)
9. Yosida, K.: Functional Analysis. Springer, Berlin (1965)
Double-Barrier Option Pricing Under
the Hyper-Exponential Jump Diffusion
Model
S. M. Grudsky and O. A. Mendez-Lara
Abstract This work is devoted to the study of the valuation problem for a
two-barrier option where the stock price is modeled by a Lévy process whose
characteristic function is rational (hyper-exponential jump diffusion (HEJD) case).
In a standard way, this problem is reduced to a convolution equation on a finite
interval, which is solved explicitly using the modified Wiener–Hopf method. On the
basis of the obtained explicit formulas, an algorithm for calculating the price of the
option under consideration is developed and numerically implemented. On the other
hand, with the help of the theory of generalized Toeplitz operators, theorems are
proved about the existence and uniqueness of the solution of the original convolution
equation and whether it belongs to a certain natural class of functions associated
with the Sobolev spaces.
Keywords Lévy processes · Barrier options · Toeplitz operators · Wiener–Hopf
method
Mathematical Subject Classification (2010) Primary 47G20; Secondary 60G40,
60G51
S. M. Grudsky ()
Department of Mathematics, CINVESTAV–IPN, Mexico City, Mexico
Regional Mathematical Center of the Southern Federal University, Rostov-on-Don, Russia
e-mail: grudsky@math.cinvestav.mx
O. A. Mendez-Lara
Department of Mathematics, CINVESTAV–IPN, Mexico City, Mexico
e-mail: omendez@math.cinvestav.edu.mx
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_10
197
198
S. M. Grudsky and O. A. Mendez-Lara
1 Introduction
The pricing problem of a double-barrier option when the log stock price is modeled
by the Brownian motion (classical hypothesis) is considered in [1–7]. The articles
[4–7] are devoted to an approach connected with a solution of the Black–Scholes
(partial) differential equation on a strip of finite width. But it should be noted that
for many cases, the Brownian motion is not an adequate model for the log stock
price. Therefore, in recent years, many authors have used Lévy processes as models
for the log stock price [8–24].
In the papers [15–24], the problems of pricing one-barrier and two-barrier
options in the case of Lévy processes are considered. The most natural way
for solving the one-barrier option problem is the Wiener–Hopf method, various
modifications of which are proposed in [15–21]. This method allows one to obtain
explicit formulas for the price of options in the form of multiple integrals which
are numerically implemented using different approximations. The situation changes
significantly in the case of a two-barrier option; the matrix Wiener–Hopf equation
that arises, generally speaking, does not have an explicit solution. Therefore, special
cases when the explicit solution can be obtained are of great importance.
The hyper-exponential jump diffusion (HEJD) model, which has a rational
symbol (see [19]), is attractive. On one hand, it is well known in the theory of
options [19–24], and on the other hand, it makes possible to solve explicitly the
associated matrix Wiener–Hopf equation, also known as the modified Wiener–Hopf
equation. Note also that the HJED can also be used as an approximation for other
stochastic processes used in the modern option theory (see [24]).
In the present article, we consider the problem of pricing a double-barrier option
in the case where the stock price is modeled by the HEJD process.
Following the monograph [8], we use the generalized Black–Scholes equation
approach. That is, we reduce the original option problem to a partial pseudodifferential equation. This equation is then reduced to the Wiener–Hopf equation,
which is solved explicitly taking into account the zeros and poles of the characteristic function. Based on the obtained explicit formulas, an algorithm for calculating
the option price is developed and implemented numerically.
In the second part of the work, a theoretical substantiation of the obtained
solution is carried out. This justification is based on the method developed in
[21]. Using the method of sectorial operators in the theory of generalized Toeplitz
operators, the existence and uniqueness of the solution of the partial pseudodifferential equation in Sobolev spaces are proved.
In Sect. 2, the necessary definitions are introduced, the partial pseudo-differential
equation is formulated, and it is reduced to the convolution equation on a finite
interval and then to the generalized Wiener–Hopf equation. In Sect. 3, these
problems are solved for the case of the HEJD process, and the formula for the price
of a double-barrier option is constructed in an explicit form. Section 4 is devoted
to the numerical implementation of the obtained formulas. We analyze here the
accuracy and speed of the obtained algorithm. One reduces in Sect. 5 the problem
under consideration to the so-called generalized Toeplitz operator, and using the
Double-Barrier Option Pricing under the HEJD Model
199
theory of sectorial operators, one proves that this Toeplitz operator is invertible in
the space L2 (R, 2s ). On this basis, in Sect. 6, the existence and uniqueness of a
solution of the generalized Wiener–Hopf equation are proved, and the formula for
the option price is written in terms of this solution.
2 Preliminaries
The Fourier transform of a function f is defined by
eixξ f (x) dx,
(F f ) (ξ ) =
∀ ξ ∈ R;
R
and the Laplace transform of a function f as
∞
e−τ ω f (τ ) dτ,
f4(ω) = (Lf ) (ω) =
∀ ω ∈ iR.
0
Let X be a Lévy process under the probability P; according to the LévyKhintchine representation
#
% P
eiξ x dμXt (x) = e−t ψ (ξ ) , ξ ∈ R, t ≥ 0,
E P eiξ Xt =
R
the function ψ P is known as the characteristic exponent of X and has the form
ψ P (ξ ) =
σ2 2
ξ − iμξ + ϕ (ξ ) ,
2
with
ϕ (ξ ) = −
∞#
−∞
%
eiuξ − 1 − iuξ χ(−1,1) (u) (du) ,
$
where is a measure on R satisfying R max x 2 , 1 (dx) < ∞. ϕ characterizes
the pure non-Gaussian component of the process. Taking the positive sign in the
exponent in the Fourier transform as defined above matches the Fourier-Stieltjes
transform in the definition of the characteristic exponent.
Definition 1 (See [8, p. 59]) Let X be a Lévy process and q > 0; the resolvent
operator U q is defined by
U q f (x) = E
∞
e−qt f (x + Xt ) dt
0
for every measurable nonnegative function f .
The Lévy process X satisfies the ACP condition if the resolvent operators have
the strong Feller property, that is, for every q > 0 and f ∈ L∞ (R), U q f is
continuous.
200
S. M. Grudsky and O. A. Mendez-Lara
Consider r > 0 the risk-free interest and an arbitrage-free market and it is true
that the existence of the solution of the valuation problem on such market depends
on the existence of a locally equivalent martingale measure (LEMM), let’s see how
the representations of a Lévy process between the historic measure P and the LEMM
Q are related.
Definition 2 Let P and Q be measures on the space (Ω, F , {Ft }) with right
continuous filtration {Ft }. Denote the restriction measures Pt = P|Ft , Qt = Q|Ft ,
dPt
and Zt = dQ
as the density of Qt with respect to Pt . If 0 < Zt < ∞ a.e, then the
t
measures P and Q are called locally equivalent.
Definition 3 A measure Q that is locally equivalent to the measure P such that the
discounted process St∗ = e−rt St is a Q-martingale is called a locally equivalent
martingale measure.
Since for a non-Gaussian process it is typical for infinitely many LEMMs to exist,
we assume that such measure has been chosen, so we will work the Lévy process X
under the LEMM Q, with characteristic exponent ψ Q .
The formula of the pseudo-differential operator given the characteristic exponent
ψ of the Lévy process is
(Lf ) (x) = F −1 [−ψ (−·) (F f )] (x) .
Lemma 1 (See [9]) Let X be a Lévy process under the probability P and ψ P its
characteristic exponent; then, /ψ P (ξ ) ≥ 0 for arbitrary ξ ∈ R.
2.1 Boundary Value Problem
Let us consider a general-type barrier option with deterministic expiration date T >
0, if X is a Lévy process that satisfies the ACP-condition, the stock prices {St }t ≥0
are given by the relation St = S0 eXt where S0 is the spot price of the stock at the
time of the valuation. Consider the sets
D ⊂ R × (−∞, T ],
D T = {x ∈ R | (x, T ) ∈
/ D} ,
c
B = {(x, t) ∈
/ D | t ∈ (−∞, T )} .
and the functions
g D ∈ L∞ (D) ,
g T ∈ L∞ D T ,
g c ∈ L∞ B c .
(1)
D stands as the barrier region where the option gets terminated by reaching it before
the expiration date; g D serves as a rebate paid out as a fraction of the premium.
D T are the values at the expiration time where the payoff g T is paid. B c is the
Double-Barrier Option Pricing under the HEJD Model
201
region where the option remains valid before the expiration date, and g c is an early
exercise compensation. Let’s use the hitting time TD of D by the process Yt % :=
(x + Xt % , t + t % ), and for (x, t) ∈ R × [0, T ], define the stochastic integral under a
chosen measure
%
#$ T ∧T
V (x, t) := E 0 D e−rs g c (Ys ) ds | Y0 = (x, t)
#
%
(2)
+E e−rTD g YTD χ{T <T } | Y0 = (x, t)
D
#
%
−rT
T
+E e
g (x + XT ) χ{T >T }
D
as the valuation of the option with conditions defined as above.
For the set of functions with support on D, define C0 (D) as the space of continuous functions vanishing at infinity, S (D) as the space of infinitely differentiable
functions vanishing at infinity faster than any negative power of |x| together with
all the derivatives, and S (D) as the set of all continuous linear functionals, or
distributions, on S (D).
Theorem 1 ([8, p. 65]) Let X be a Lévy process satisfying the ACP condition. Then,
the stochastic integral V is a bounded solution to the following boundary value
problem
⎧
⎨ [r − ∂t − L] V (x, t) = g c (x, t) , ∀ (x, t) ∈ B c ;
V (x, T ) = g T (x) , ∀ x ∈ D T ;
⎩
V (x, t) = g D (x, t) , ∀ (x, t) ∈ D.
(3)
The first equation is understood in the sense of generalized functions
$V , [r − ∂t − L] u% = 0,
for all u ∈ S (R × R) such that supp u ⊆ B c .
In the remaining of this work, we will specifically study a double-barrier
knockout option with expiration date T (see [19]).
Perform the change of variable τ = T −t; thus, with x = ln Sτ , the observed state
(x, τ ) becomes the initial spot of the stochastic process. Let’s set two predefined
values H1 < H2 as the barriers of the option; let h1 := ln H1 , h2 := ln H2 , and h =
h2 −h1 . Then, for this particular option D = {0, h}×(0, ∞), B c = (0, h) ×(0, ∞),
D T = (0, h), g c = g D = 0, and g T = g is a defined payoff. So, we can rewrite (3)
as the pseudo-differential problem
⎧
[r + ∂τ − L] V (x, τ ) = 0, (x, τ ) ∈ (0, h) × (0, ∞) ;
⎪
⎪
⎨
V (x, 0) = g (x) , x ∈ (0, h) ;
⎪
V (0, τ ) = 0,
⎪
⎩
V (h, τ ) = 0.
(4)
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S. M. Grudsky and O. A. Mendez-Lara
2.2 The Convolution Equation
Applying the Laplace transform, the problem becomes
⎧
4 (x, ω) = g (x) ,
⎨ [r + ω − L] V
4 (0, ω) = 0,
V
⎩
4
V (h, ω) = 0.
Let’s define, for s ∈ R, H s (D), the subset of distributions on S (D) with norm
$
·H s
s
f 2H S = D |(F f ) (ξ )|2 1 + |ξ |2 dξ < ∞. Since C∞
= H s (D), the
0 (D)
problem can be written as the convolution equation
4
P(0,h) F −1 [r + ω + ψ (−·)] F V
(x, ω) = g (x) ,
4 (·, ω) ∈ H s1 (0, h),
where P(0,h) is the restriction operator to the interval (0, h), V
s
2
and g ∈ H (0, h), for some s1,2 ∈ R. Let’s notice that the convolution equation
acts on the finite interval (0, h) with the symbol
a (ξ, ω) := r + ω + ψ (−ξ ) .
It is desirable to find functions with supports (−∞, 0] and [h, ∞), such that
4 (x, ω) will be continuous on the whole real line and their values on {0, h} match.
LV
4 (0, ω) = V
4 (h, ω) = 0. Consider
This will include the boundary conditions V
4[h,∞) such functions and denote V
4[0,h] := V
4 and 4
4(−∞,0] and V
g := χ[0,h] g. Then,
V
the problem can be stated (∀ x ∈ R) (∀ ω ∈ C) as the convolution equation
4(−∞,0] (x, ω) + [r + ω − L] V
4[0,h] (x, ω) + V
4[h,∞) (x, ω) = 4
V
g (x) .
2.3 The Modified Wiener–Hopf Equation
s
Definition 4 For s ∈ R, consider the weight function 2 (ξ ) := ξ 2 + 1 (see [25]).
Let 1 < p < ∞ and define Lp (R, s) as the 2-weighted Banach space with norm
φp,s :=
∞
−∞
1/p
|φ (ξ )|p 2 (ξ ) (ξ ) dξ
< ∞.
Applying the Fourier transform, the convolution equation can be stated as the
modified Wiener–Hopf equation (mWH equation)
φ− (ξ, ω) + a (ξ, ω) φh (ξ, ω) + eihξ φ+ (ξ, ω) = G (ξ ) ,
(5)
Double-Barrier Option Pricing under the HEJD Model
203
where
φh (·, ω) ∈ L2 (R, s1 ) ,
φ+ (·, ω) , φ− (·, ω) , G ∈ L2 (R, s2 ) .
(6)
The real numbers s1 ands2 will be chosen later. This problem, known as the modified
Wiener–Hopf equation with symbol a, has for solution a triple (φ− , φh , φ+ ) of
unknown functions such that the supports before the Fourier transform is applied
are (−∞, 0], [0, h] and [h, ∞), respectively.
3 HEJD Symbols
In terms of the model for the stock price, let’s focus on a specific set of the rational
symbols, those of HEJD type. Let’s first introduce a wider variety of processes.
Definition 5 Let − < 0 < + and ν ∈ (0, 2]. A Lévy process is called a regular
Lévy process of exponential type [− , + ] and order ν > 0 if the following two
conditions are satisfied:
(i) The characteristic exponent admits a representation ψ(ξ ) = −iμξ + φ(ξ ),
where φ is holomorphic in the strip 0ξ ∈ (Λ− , Λ+ ), is continuous up to the
boundary of the strip, and admits a representation φ(ξ ) = c|ξ |ν + O(|ξ |ν1 )
O(|ξ |ν1 ) as ξ → ∞ in the strip
(ii) There exist ν2 < ν and C such that the derivative of ψ in (i) admits a bound
|ψ (ξ )| ≤ C(1 + |ξ |)ν2 , 0ξ ∈ [Λ− , Λ+ ].
It is well known that an RLPE satisfies the ACP condition (see [8, p. 84–85]),
so the problem in the previous section is well defined for RLPEs. Grudsky [21]
is devoted to RLPEs with σ = 0 and ν ∈ (0, 2). The aim of this work is the
HJED symbols; this processes are RLPEs of order 2, if σ > 0; they have nontrivial
Gaussian component ϕ with asymptotic behavior ϕ (ξ ) ∼ ξ .
The characteristic exponent of a HEJD process is given by (see [19])
,
+ M
2
λ̌m
σ
,
ξ 2 − i μ̌ξ + ξ
ψ P (ξ ) =
2
ξ − i η̌m
m=1
where σ > 0, μ ∈ R, λ̌m , and η̌m represent the intensity and the tail-decay behavior
of the m-th jump, all λ̌m are positives, and η̌m is positive or negative if it is an
positive or negative jump, respectively. P is the historic measure, but the stochastic
integral in (2) will be under the measure Q, the LEMM associated to the Esscher
transform (see [8, p. 98]), so the new characteristic exponent is given by
ψ Q (ξ ) = ψ P (ξ − iθ ) − ψ P (−iθ ) ,
204
S. M. Grudsky and O. A. Mendez-Lara
where θ is the solution of ψ P (−i [1 + θ ]) − ψ P (−iθ ) + r = 0 (see [8, p. 100]).
With the adjusted parameters
μ = μ̌ + σ 2 θ,
λm = λ̌m
η̌m
,
θ + η̌m
4
ηm = η̌m + θ,
ψ Q also has the form
,
+ M
λm
σ2 2
ψ (ξ ) =
.
ξ − iμξ + ξ
2
ξ − i4
ηm
Q
(7)
m=1
The sign of the 4
ηm remains for positive and negative jumps.
Let’s use the Wiener–Hopf factorization a (ξ, ω) = a + (ξ, ω) a − (ξ, ω), where
a ± and a1± are analytic on the half-planes ± . Given that the symbol is expressed
ηm . Now, we can order the parameters
in terms of ψ Q (−ξ ), let’s denote ηm := −4
M
{ηm }M
1 (coupled with {λm }1 ) in a manner such that from 1 to J ≤ M, the parameters
are positive, and thus, corresponding to the poles on the upper half-plane and the
remaining K = M − J , negative parameters correspond to the poles on the lower
half-plane. Notice that a has for numerator a polynomial with two degrees more than
the denominator, and by Lemma 1, /a (ξ, ω) ≥ 0; hence, both a + and a − have one
−
−
K
more zero than poles, that is, a + has poles {iηn− }K
1 ⊂ and zeros {ξk (ω)}0 ⊂
+ J
+
−
+
+
−
K
and a has poles {iηl }1 ⊂ and zeros {ξj (ω)}0 ⊂ . So, we have the
following representations for a ± and also the partial fraction decompositions
−
2 K
k=0 ξ−ξk (ω)
−
#K
n=1 ξ −iη%n
Jj=0 ξ −ξj+ (ω)
Jl=1 ξ −iηl+
a + (ξ, ω) = σ2
a − (ξ, ω) =
,
,
bk (ω)
1
= σ22 K
k=0 ξ −ξ − (ω) ,
a + (ξ,ω)
k
c (ω)
1
= Jj=0 j + ,
a − (ξ,ω)
ξ −ξj (ω)
#
%
2
4
bn (ω)
a + (ξ, ω) = σ2 ρ (ξ, ω) + K
n=1 ξ −iη− ,
n
with ρ (ξ, ω) = ξ − d (ω), where d (ω) is determined in the operation of partial
fraction decomoposition of a + and {cj (ω)}, {bk (ω)}, {4
bn (ω)} ⊂ C.
Remark Without losing perspective of the dependence on ω ∈ C, we will drop its
notation when possible.
Divide the mWH equation by a − , the problem can now be separated according
to the support of the inverse Fourier transform of the summands of the equation,
and the
φ− , φ+ , and φh can be expressed in terms of the unknown values
functions
+
and φh iηn− . In particular, the terms corresponding to the segment
φ+ ξj
Double-Barrier Option Pricing under the HEJD Model
205
[0, h] yield the formula
J
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎫
cj
⎪
⎪
⎪
j =0 ξ −ξ + G (ξ )
⎪
j
⎪
⎡
⎤
⎪
−h
⎪
K 4
1
σ2
−
η
⎬
n
iη
e
φ
b
−
1
n
n=1 n h
2
ξ −iηn ⎦ ihξ
⎣
.
φh (ξ ) = +
+
e
J
+
1
⎪
a (ξ ) ⎪
−
c
φ
ξ
⎪
⎪
j
+
+
j
=0
j
⎪
⎪
ξ −ξj
⎪
⎪
⎪
⎪
# ⎪
⎪
+%
⎪
⎪
1
⎭
⎩ − J cj G ξ + − φ+ ξ + eihξj
+
j =0
j
j
ξ −ξj
(8)
Once the unknown values are solved, the functions φ− , φ+ , and φh can be explicitly
determined. To do so, the domains where φh and φ + are analytic can be used to get
the linear system
⎡
⎤⎡
⎤ ⎡
⎤
A00 A01 · · · A0J B01 B02 · · · B0K
φ+ ξ0+
E0
+ ⎥
⎢ A A ··· A
⎥⎢
⎢E ⎥
1J B11 B12 · · · B1K ⎥ ⎢ φ+ ξ1 ⎥
⎢ 10 11
⎢ 1⎥
⎢ .
⎥
⎥
⎢
⎢ . ⎥
.. . .
..
..
.. . .
.. ⎥ ⎢
..
⎢ ..
⎥
⎢ . ⎥
. .
. . ⎥⎢
.
.
.
.
⎢
⎥ ⎢ . ⎥
⎢
⎥⎢
⎥
⎢
+ ⎥
⎢ AK0 AK1 · · · AKJ BK1 BK2 · · · BKK ⎥ ⎢ φ+ ξJ ⎥ ⎢ EK ⎥
⎢
⎥⎢
⎥,
− ⎥=⎢
⎢ A10 A11 · · · A1J B11 B12 · · · B1K ⎥ ⎢ φh iη1 ⎥ ⎢ 0 ⎥
⎢
⎥⎢
⎥
⎥
⎢
⎢ A20 A21 · · · A2J B21 B22 · · · B2K ⎥ ⎢ φh iη2− ⎥ ⎢ 0 ⎥
⎢
⎥⎢
⎥ ⎢
⎥
. ⎥⎢
.
..
.. . .
.. . .
..
⎢ ..
⎥ ⎢ .. ⎥
⎣ .
⎦ ⎣ . ⎦
. .. ⎦ ⎣
. ..
.
.
.
.
−
AJ 0 AJ 1 · · · AJ J BJ 1 BJ 2 · · · BJ K
φh iηK
0
where
Ek =
Ak,j =
+
J
,
G ξk− −G ξj+
j =0 cj
ξk− −ξj+
ihξ +
ihξ −
cj e k −e j
,
ξk− −ξj+
− ihξ −
b n e ηn h e k
σ2 4
−
2
ξk −iηn−
cj
iηl+ −ξj+
−
b n e ηn h σ 2 4
2 i η+ −ηn−
l
Bk,n = −
Al,j =
Bl,n = −
,
,
,
, k = 0, . . . , K;
k = 0, . . . , K, j = 0, . . . , J ;
k = 0, . . . , K, n = 1, . . . , K;
l = 1, . . . , J,
j = 0, . . . , J ;
l = 1, . . . , J.
n = 1, . . . , K.
Solving the system (for each ω), φh is known, and the valuation can be obtained by
applying the inverse Fourier transform followed by the Inverse Laplace Transform.
Once a payoff is chosen, numerical results can be obtained by a direct computation
of the line integral. Let’s consider as an example the super-share option.
206
S. M. Grudsky and O. A. Mendez-Lara
3.1 Super-Share Option
Let K1 < S < K2 be fixed parameters; the payoff is given by
S/K1 S ∈ [K1 , K2 ]
0
c.c.
0
g (S) =
Let S = ex and k1,2 := ln K1,2 and then g (x) = e−k1 ex χ[k1 ,k2 ] (x).
Remark k1 and k2 are interpreted as barriers of the payoff and are not necessary
equals to the ones in the option contract (h1 , h2 ). It is then useful to set b2 :=
g (x) = e−k1 ex χ[b1 ,b2 ] (x) .
min {h2 , k2 } > 0 and b1 := max {h1 , k1 } < 0, so 4
To make the notation easier and making use of our assumptions for the option
barriers,
hletihξh1 = k1 and h2 = k2 . Applying the Fourier transform, G (ξ ) =
i
e e − 1 . For this particular option, the exact valuation is given by
− ξ −i
V[0,h] (x, τ ) =
K J 1 bk (ω) cj (ω) fk,j (x, ω) eτ ω dω,
iπσ 2
iR
(9)
k=0 j =0
⎡
⎡
⎢
⎢
⎢
⎢
fk,j (x, ω) = ⎢
⎢
⎢
⎢
⎣
⎢
⎢
⎢
1
⎢
ξk− (ω)−ξj+ (ω) ⎢
⎣
⎡
e
⎣
−
+ e−ixξk (ω)
e#x
%
ξk (ω)−i ξj+ (ω)−i
+ −
ihξj+ (ω) ⎢
⎤ ⎤ ⎤
⎥⎥⎥
⎦⎥⎥
⎥⎥
⎥⎥
⎥⎥,
⎦⎥
⎥
⎥
⎦
eh
− iφ+ ξj+ (ω)
ξj+ (ω)−i
#
%
−ixξj+ (ω)
−ixξ − (ω)
× e
k
−e
1
1
− ξ + (ω)−i
ξk− (ω)−i
j
where K and J represent the number of downward and upward jumps in the model
as stated in Sect. 3.
4 Numerical Example
There are several problems with formula (9). There are no closed formulas for
ξj+ (ω) and ξk− (ω), and numerical values are obtained for each ω by a root-finding
algorithm for polynomials. The values bk (ω) and cj (ω) are obtained numerically
from linear systems
along
the discretized contour. The same situation is true for
the values φ+ ξj+ (ω) ; they are solutions of the linear system in Sect. 3, which
is dependent on ω. A discretization of a finite segment is used as the integration
contour, and discretized integrals are valuated. The result is an approximation,
but it will serve us as a benchmark for accuracy and computation time of other
approximation methods.
Double-Barrier Option Pricing under the HEJD Model
207
As the value of the roots is given numerically as a modulus-sorted array, it is
necessary to classify them to form continuous paths. The asymptotic behavior of
the roots is helpful. Two roots, one in each half-plane, behave as the roots in the
Wiener–Hopf factorization of the Black–Scholes model’s symbol; the remaining
roots tend to the poles iηm as |0ω| → ∞. Thus, the finite segment needs to contain
at least [−iW, iW ] where W = max {|ηm |}. This segment is not enough, as the roots
that behave as in the Black–Scholes model do not change as rapidly as the ones that
tend to the mentioned poles; this led us to use the segment [−20W i, 20W i] over the
imaginary axis as the contour of integration to perform the numerical valuations.
Given the oscillatory nature of the integrand, we decided to use a nonuniform
grid on the finite contour. As all the factors decay exponentially, the discretization
consists of 2X points on [−2i, 2i], Y points in [−10W i, −2i) and (2i, 10W i], and
Z points on [−20W i, −10W i) and (10W i, 20W i]. Each subinterval was discretized
with equidistant points, and the values of X, Y, andZ were determined for a
particular numerical example.
We consider the following parameters: H1 = K1 = 80, H2 = K2 = 140,
τ = 2.0, σ = 0.16, μ̌ = 0.12, r = 0.05, {λ̌m } = {0.03, 0.04, 0.02, 0.025, 0.02},
{η̌m } = {−10, 15, −15, 20, −20}, and λ = M
1 λ̌m . A grid of initial stock values is
formed for S ∈ [90, 130] with increments of 0.25, i.e., 160 values.
As the number of points X, Y, and Z in the discretization affects the accuracy
and the calculation time of the valuation, we calculated them for several values
presented in Table 1, Figs. 1 and 2. Notice that all the linear systems are independent
of the pairs (S, τ ) and the time that took to solve the systems to get the values of
bk , cj , ξj+ , andξk− is on the 2nd column. The next two columns reflect the time to
calculate the valuation once the previous values were computed. The last column
shows the percent error against the finest discretization (FD) at S = 110.
The present algorithm of valuation takes a considerable amount of time, but it
will serve as a benchmark to measure precision for other approximation methods. If
needed, a coarser discretization can be used to save time; for example, if X = 2000,
Y = 1000, and Z = 500, the valuations are approximately within 5% of the finest
discretization for S ∈ [90, 130], and the computation time is 4–5 times faster. A
Table 1 Computation times for several discretizations of the finite segment
Discretization
X
Y
10,000
5000
8000
4000
5000
2500
2000
1000
1000
400
750
300
500
200
Z
1000
800
500
500
200
150
100
Solving
Lin-Sys (s)
46.4897
37.7685
23.2303
10.0892
6.4721
3.5796
2.3412
Integration (s)
S = 110
S ∈ [90, 130]
6.0143
968.3023
4.7670
767.4870
2.7245
438.6445
1.2691
204.3251
0.8309
133.7749
0.4194
67.5234
0.2872
46.2392
Percent error vs
FD at S = 110
–
0.1070
0.4175
1.4484
3.1461
5.3019
6.7165
208
S. M. Grudsky and O. A. Mendez-Lara
Fig. 1 HEJD valuations for several discretizations
Fig. 2 HEJD: Comparison of valuation errors (%) for several discretizations against the finest
coarser discretization can also be viewed as benchmark for computation time against
other methods once a percent error is fixed.
When the jump intensities are multiplied by 1, 0.5, 0.25, and 0.1, so, as the
global intensity of jumps λ decreases, the HEJD valuations using the algorithm tend
to the Black–Scholes valuation; at S = 110, the difference with the Black–Scholes
valuation is of 22.68%, 7.94%, 4.10%, and 1.78%, respectively.
Double-Barrier Option Pricing under the HEJD Model
209
Note The computations were performed on an Arch Linux® machine with an
Intel® Core™ i5-4570T Processor (4M Cache, up to 3.60 GHz). The algorithm was
developed with Python 3.8.
5 General Toeplitz Operators: Sectoriality Approach
In the previous sections, we presented an explicit solution and implemented it
numerically with an algorithm for solving (4) in the case of a HEJD process. In the
subsequent sections, we justify this solution in the sense of proving the existence
and uniqueness of the solution of (4) in the Sobolev spaces that arise naturally. The
theory of generalized Toeplitz operators and the concept of sectoriality are presented
in this section.
Let H be a Hilbert space and L (H ) the algebra of bounded linear operators in
H.
Definition 6 A ∈ L (H ) is called normally solvable if im A = im A.
Lemma 2 If A is normally solvable, H = im A ⊕ ker A∗ .
Definition 7 A is called left-invertible (right-invertible) if A−L A−R ∈ L (X)
exists such that A−L A = I AA−R = I . If there exists A−1 ∈ L (H ) such that
A−1 A = AA−1 = I , A is called an invertible operator.
Remark Since the kernel of a left-invertible operator is {0} and for a right-invertible
operator its image is H , a one-side invertible operator is also normally solvable.
Definition 8 The analytic projectors are given by P ± = 12 [I ± SR ], where SR is
the Cauchy-Lebesgue singular integral operator given by
1
(SR f ) (ξ ) =
iπ
f (x)
dx,
x
R −ξ
ξ ∈ R,
and such integral is understood in the sense of the principal value.
The operators SR andP ± are well defined on Lp (R) for 1 < p < ∞. The
operator SR is bounded in the space Lp (R, s) for 1 < p < ∞ and −1/2 < s <
(p − 1)/2 (see [26]).
Given (6), in the remainder of the work, let’s work with p = 2, so the
boundedness is given for |s| < 1/2.
2
On L2 (R, s) for |s| < 1/2, the analytic projectors are bounded, satisfy P ± =
P ± , P + P − = P − P + = 0, P + + P − = I , and generate the spaces L±
2 (R, s) :=
P ± (L2 (R, s)). Since χ(0,∞) is the Heaviside step function, then it is easy to see
that
P + := F χ(0,∞) F −1 ,
P − := F χ(−∞,0) F −1 .
210
S. M. Grudsky and O. A. Mendez-Lara
If |s| ≥ 1/2, the analytic projectors are not bounded; nevertheless, one can define
s
L±
2 (R, s) := F (H (R± )), where R+ = (0, ∞) , R− = (−∞, 0).
Definition 9 For a ∈ L∞ (R) , introduce the operator Ma given by Ma f = af for
all f ∈ L2 (R, s). The operator
+
Ta := T (a) := P + Ma : L+
2 (R, s) → L2 (R, s) ,
is called a Toeplitz operator with symbol a.
+
∗
Its adjoint operator Ta∗ : L+
2 (R, s) → L2 (R, s) is given by Ta = Ta .
Note that Ta , Ta∗ = Ta are bounded operators on L+
2 (R, s) for |s| < 1/2.
Proposition 1 (See [27]) If a ∈ L∞ (R) satisfies a ≡ 0, for Ta to be normally
solvable in L+
2 (R, s) for |s| < 1/2, it is necessary that ess inf ξ ∈R |a (ξ )| > 0.
5.1 The Douglas Algebra
Let’s define the set of all continuous functions f such that limx→−∞ f (x) = z =
limx→∞ f (x) as C• (R) and H ∞ (R) the class of functions of bounded norm with
f H ∞ = supz∈R |f (z)|. Consider the set H ∞ (R) + C• (R) ⊂ L∞ (R) of all
functions f such that f = h + c where h ∈ H ∞ (R) and c ∈ C• (R); this set is
called the Douglas algebra.
Introduce the functions γ − (ξ ) := ξ − i, γ + (ξ ) := ξ + i, u := γ + /γ − , and
eh (ξ ) := eihξ , so uh := eh u is element of the Douglas algebra. The following result
assures the normal solvability of Toeplitz operators with symbols on the Douglas
algebra, in particular for uh .
Lemma
3 (See, e.g.,
[27]) If the symbol a is in H ∞ (R) + C• (R)
H ∞ (R) + C• (R) and satisfies ess inf ξ ∈R |a (ξ )| > 0 and a1 is not in
H ∞ (R) + C• (R) H ∞ (R) + C• (R) , thenTa is left-invertible (right-invertible)
in L+
2 (R, s) for |s| < 1/2.
5.2 Sectoriality
Definition 10 Let A ∈ L (H ). A is called a ε-sectorial operator for ε > 0 if
infxH =1 $Ax, x%H = ε.
Clearly, if a ∈ L∞ (R), the multiplication operator Ma is ε-sectorial if and only
if ess inf ξ ∈R /a (ξ ) = ε > 0.
Double-Barrier Option Pricing under the HEJD Model
211
Remark Lemma 1 and the last fact enable us to use the sectoriality approach on the
valuation problem.
Definition 11 A function a ∈ L∞ (R) is called ε-sectorial if there is θ ∈ (−π, π)
such that Maθ is a ε-sectorial operator where aθ (ξ ) := eiθ a (ξ ).
Theorem
22 (See [21]) Let A be a ε-sectorial operator on H ; then, A is invertible,
2
and 2A−1 2L(H ) < 2ε−1 .
Let |s| < 1/2, given G is a subspace of the Hilbert space L+
2 (R, s) and its
as
the
orthoprojectors
onto each
orthogonal complement G⊥ ; consider PG and P⊥
G
subspace, that is, for f ∈ L+
∈
G,
f2 ∈ G⊥
s),
there
are
unique
functions
f
(R,
1
2
+
⊥
such that f = f1 + f2 . Let’s note that P = PG + PG .
+
For a ∈ L∞ (R), introduce the operator D : L+
2 (R, s) → L2 (R, s) given by
+
D := P⊥
G + P aPG .
Theorem 3 (See [21], Theorem 4.2) If a is a ε-sectorial function, D is invertible,
and the solution g of
f ∈ L+
2 (R, s) , |s| < 1/2,
Dg = f,
satisfies the estimate
g1 L+ (R,s) ≤ 2ε−1 f1 L+ (R,s) ,
2
2
where g1 := PG g and f1 := PG f .
⊥
Moreover, if g2 := P⊥
G g and f2 := PG f and we consider the operator D1 :=
PG aPG : G → G, D1 is also invertible, and the unique solution of the equation
has the form
%
#
−1
⊥
f.
g = D1−1 PG + P⊥
−
P
aD
P
G
G
G
1
6 Unique Solvability of Boundary Value Problems for HEJD
Symbols
In this section, we use properties of the symbol to construct an equivalent problem
to our pricing problem and whose solvability is given by Theorem 3; then, some
analytic properties of the solution are explored.
Introduce the function
2
σ 2
φ (ξ )
2 ξ + ϕ (−ξ )
4
c (ξ ) :=
=
.
1 + ξ2
1 + ξ2
It is clear that 4
c ∈ L∞ (R) and ∃ M > 0, ε1 > 0 | inf|ξ |≥M /4
c (ξ ) = ε1 .
212
S. M. Grudsky and O. A. Mendez-Lara
The assumption
∀ ω ∈ C ∃ ε2 (ω) > 0 | r + /ω ≥ ε2 (ω) .
(10)
is trivially checked at the contour for the inverse Laplace transform: Γ = iR. Let’s
add the assumption that ε2 is independent of ω.
Define the function
c (ξ, ω) :=
a (ξ, ω)
r + ω + ψ (−ξ )
=
.
1 + ξ2
1 + ξ2
(11)
Condition (10) and definition (11) seem arbitrary and of technical nature but follow
directly from Lemma 1, which is a standard tool in the theory of Lévy process, and
make it possible to implement the concept sectoriality. The following statements are
proved in the same way as Lemma 5.1 and Lemma 5.2 in [21].
Lemma 4 If ψ has the form (7) and ω ∈ iR, the function (11) is ε-sectorial for
some ε > 0 nondependent of ω.
Let’s use an alternative to condition (10).
Lemma 5 If ψ has the form (7) and exists, θ0 > 0 such that, for ω = 0,
Kθ0 := ω ∈ C | |arg ω| ≤
π
+ θ0 .
2
There exists ε > 0 such that c (·, ω) is ε-sectorial for all ω ∈ Kθ0 .
6.1 Solvability of Equivalent Problems
Since we introduced the function γ (ξ ) := γ + (ξ ) γ − (ξ ) = ξ 2 + 1 with c(·, ω),
let’s consider s1 = s + 1/2 and s2 = s − 1/2, where |s| < 1/2; then, (6) turns into
φh (·, ω) ∈ L2 (R, s + 1/2) ,
φ+ (·, ω) , φ− (·, ω) , G ∈ L2 (R, s − 1/2) .
(12)
Equation (5) can be rewritten as
%
#
φ− (ξ, ω) + 1 + ξ 2 c (ξ, ω) φh (ξ, ω) + eihξ φ+ (ξ, ω) = G (ξ ) .
(13)
Notice that γ ±1(ξ ) are analytic on ± , so dividing Eq. (12) by γ − (ξ ) and denoting
∈ L±
Ψ ± (ξ, ω) := φγ±±(ξ,ω)
2 (R, s) ,
(ξ )
+
+
Ψh (ξ, ω) := γ (ξ ) φh (ξ, ω) ∈ L+
2 (R, s) ;
(14)
Double-Barrier Option Pricing under the HEJD Model
)
Ψ − (ξ, ω) + c (ξ, ω) Ψh+ (ξ, ω) + eihξ u (ξ ) Ψ + (ξ, ω) = γG(ξ
− (ξ ) .
213
(15)
The following result will help us with the spaces where the solutions must belong.
Lemma 6 (See [25])
1. The set of functions where the unknown function φh (ξ, ω) must coincide with the
set of "+"-components of solutions of the boundary value problem
e−ihξ φh+ (ξ, ω) = φh− (ξ, ω) ,
φh± (·, ω) ∈ L±
2 (R, s + 1/2)
(16)
2. The set of functions where the unknown function Ψh+ (ξ, ω) must coincide with
the set of "+"-components of solutions of the boundary value problem
e−ihξ u (ξ ) Ψh+ (ξ, ω) = Ψh− (ξ, ω) ,
Ψh± (·, ω) ∈ L±
2 (R, s)
(17)
That is, the set where the function Ψh+ lies coincides with the subspace
ker Tuh L+ (R,s) .
2
In particular, Problems (16) and (17) are called Riemann boundary value
problems with coefficients e−h and uh = e−h u, respectively. Now, the classes of
functions where the unknown functions can be found have been narrowed by the
images of the analytic projectors, for Eq. (15),
⎧
⎪
⎨
⎪
⎩
G(ξ )
= Ψ − (ξ, ω) + c (ξ, ω) Ψh (ξ, ω) + eihξ u (ξ ) Ψ + (ξ, ω) ,
γ − (ξ )
Ψh+ (·, ω) ∈ ker Tuh L+ (R,s) ,
2
Ψ ± (·, ω) ∈ L±
2 (R, s) .
)
If we apply P + to Eq. (15) and define g + (ξ ) := P + γG(ξ
− (ξ ) ,
P + (c (ξ, ω)) Ψh+ (ξ, ω) + P + uh (ξ ) Ψ + (ξ, ω) = g + (ξ ) .
Noting that P + uh (ξ ) = Tuh , the problem becomes
P + (c (ξ, ω)) Ψh+ (ξ, ω) + Tuh Ψ + (ξ, ω) = g + (ξ ) .
(18)
Given the construction of Problem (18), Problem (13) has a solution if and only
if Problem (18) has it; furthermore, one can obtain the solutions of one problem
according to the other by using the definitions of Ψ ± and Ψh+ .
By Lemma 3, Tuh is left-invertible in L+
2 (R, s). Hence, Tuh is normally solvable,
and according to Lemma 2, L+
s)
=
im
Tuh ⊕ ker Tuh .
(R,
2
214
S. M. Grudsky and O. A. Mendez-Lara
Taking the corresponding orthogonal projectors
Puh : L+
2 (R, s) → ker Tuh ,
+
P⊥
uh : L2 (R, s) → im Tuh ,
and the operator
+
Duh : L+
2 (R, s) → L2 (R, s)
+
Duh := P c (·, ω) Puh + P⊥
uh ,
(19)
we can formulate the equation
Duh Y + (ξ ) = g + (ξ ) ,
(20)
and state the following equivalence of problems.
Lemma 7 (See [21], Lemma 5.4) Equation (18) has solution if and only if Problem
a left-inverse of Tuh and Y + a solution
(20) has a solution. Furthermore, given Tu−L
h
of Problem (20), the pair
Ψh+ (ξ ) = Puh Y + (ξ ) ,
⊥ +
P
Y
Ψ + (ξ ) = Tu−L
(ξ ) ,
uh
h
is a solution of Problem (18).
6.2 Uniqueness of the Solution
The existence of the solutions and the equivalency of problems are granted; the
uniqueness and the relationship between the solutions are given by the next theorem.
Theorem 4 If the function c has the form (11) with ψ as in (7) and ω ∈ Kθ0 :
1. The operator Duh on L+
2 (R, s) defined as above is invertible, and the solution
Y + of Problem (20) satisfies the estimate
2
2
2 Pu Y + 2 +
h
L2 (R,s)
≤
2
22
2Pu g + 2 +
,
h
L2 (R,s)
ε
where ε does not depend on ω.
2. Problem (18) has for unique solution the pair
+
Ψh+ (ξ, ω) = Puh Du−1
g
(ξ ) ,
h
⊥ −1 +
P
D
g
Ψ + (ξ, ω) = Tu−R
(ξ ) .
uh uh
h
Double-Barrier Option Pricing under the HEJD Model
215
3. Problem (13) has for unique solution the triple
φh+ (ξ, ω) = γ +1(ξ ) Ψh+ (ξ, ω) ,
φ + (ξ, ω) = γ + (ξ ) Ψ + (ξ, ω) ,
φ − (ξ, ω) = γ − (ξ ) Ψ − (ξ, ω) ,
where Ψ − (ξ, ω) can be obtained from Eq. (15).
Proof Statement 1 is a consequence of Theorem 3 and Lemma 5. Statement 2
follows from the previous one, and Lemma 7 gives Y + = Du−1
g + , which is
h
the unique solution of Problem (20). Finally, for Statement 3, the equivalency of
Problems (13)–(18) is straightforward, and the relation between its solutions is given
by (14). In particular, Ψ − , Ψh+ , Ψ + satisfies Eq. (15) if and only if Ψh+ , Ψ +
satisfies the equation in Problem (18) and
−
Ψ (ξ, ω) = P
−
G (ξ )
γ − (ξ )
− P − uh (ξ ) Ψ + (ξ, ω) − P − c (ξ, ω) Ψh+ (ξ, ω) .
Once φh+ (ξ, ω) is obtained, the inverse Fourier transform should be applied, and
given that ω must be in Kθ0 for small enough θ0 in order to use the Theorem 4, the
contour in which the inverse Laplace transform is performed is Γθ0 = ∂Kθ0 . So, the
solution of the Problem (4) has the form
1
eτ ω e−ixξ φh+ (ξ, ω) dξ dω.
V (x, τ ) =
(21)
(2π)2 i Γθ0 R
Let’s take a moment to study the functional space where V as obtained with the
sectoriality approach lies.
For the first variable, fix ω in Kθ0 , since φh+ (ξ, ω) is the solution of Problem
(16), φh+ (·, ω) ∈ L±
2 (R, s + 1/2), and its norm is bounded uniformly; therefore,
4 (·, ω) ∈ H s1 (0, h) = H s+1/2 (0, h). This remains true for
F −1 φh+ (·, ω) = V
4 (·, τ )
V (·, τ ) = L−1 V
For the second variable, fix x, V H s1 (0,h) is continuous on [0, ∞) as a function
of τ ; furthermore, it vanishes as τ → ∞; hence, V (x, ·) ∈ C0 [0, ∞).
We will resume the behavior of V on both variables by using the notation
V ∈ H s+1/2 (0, h) , C0 [0, ∞) ,
|s| < 1/2,
which is consistent with our initial definition of the problem. Once the payoff
function is assumed in the proper space, the existence, uniqueness, form, and space
where the solution is can be stated in the following result.
216
S. M. Grudsky and O. A. Mendez-Lara
Theorem 5 Let g (x) ∈ H s−1/2 (0, h) for |s| < 1/2, and let ψ Q be the
characteristic exponent (under a chosen LEMM) of a HEJD process, i.e., it has
the form (7). Problem (4) has a unique solution in C 0 [0, ∞), H s+1/2 (0, h) , and
it has the form (21).
This theorem is a direct consequence of Theorem 4 and given that
/ω → −∞ ⇒ eτ ω → 0, /ω < 0 ∀ ω ∈ Γθ0 , ω ∈ Γθ0 , |ω| → ∞ ⇒ /ω → −∞.
We have obtained and exact formula for the pricing problem specified, the computational implementation of (21) is easier when expressed as (9), and the numerical
result is in reality an approximation; nevertheless, this will serve as a benchmark
for accuracy against other approximation methods. As shown in Sect. 4, the
discretization and the method to find the roots of the symbol can be tweaked to
consume less time at the cost of accuracy, but this approximations will also provide
a reference to compare accuracy to time with alternative methods.1
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Single Jump Filtrations: Preservation
of the Local Martingale Property with
Respect to the Filtration Generated
by the Local Martingale
Alexander A. Gushchin and Assylliya K. Zhunussova
Abstract Let M be a local martingale with respect to a so-called single jump
filtration F = F(γ , F ) generated by a random time γ on a probability space
(Ω, F , P). It was recently mentioned by Herdegen and Herrmann (2016) that M
is also a local martingale with respect to the filtration H = FM that it generates
if F is the smallest σ -field with respect to which γ is measurable. We provide
an example of a local martingale with respect to a general single jump filtration
which is not a local martingale with respect to H. Then, we find necessary and
sufficient condition for preserving the local martingale property with respect to H.
The main idea of our constructions and the proofs is that H is also a single jump
filtration generated, in general, by other random time and σ -field. Finally, we prove
that every σ -martingale in considered models is still a σ -martingale with respect to
the filtration that it generates.
Keywords Filtration · Local martingale · σ -martingale · Single jump filtration
1 Introduction
Let γ be a random variable with values in [0, +∞] on a probability space
(Ω, F , P). We tacitly assume that P(γ > 0) > 0. According to [3], we call a
filtration F = (Ft )t ∈R+ a single jump filtration generated by γ and F if A ∈ Ft ,
t ∈ R+ , if and only if A ∈ F and A ∩ {t < γ } is either ∅ or coincides with
{t < γ }. This means that all randomness occurs at time γ . A special case where
A. A. Gushchin ()
Steklov Mathematical Institute, Moscow, Russia
Lomonosov Moscow State University, Moscow, Russia
e-mail: gushchin@mi-ras.ru
A. K. Zhunussova
Faculty of Computational Mathematics and Cybernetics of Lomonosov Moscow State University,
Moscow, Russia
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_11
219
220
A. A. Gushchin and A. K. Zhunussova
F = σ {γ }, i.e., F is the smallest σ -field with respect to which γ is measurable,
will be referred to as the Dellacherie model. In fact, Dellacherie [2] assumes that
Ω = R+ and γ (ω) = ω, but this specification is not essential. In the Dellacherie
model, the single jump filtration is the smallest one with respect to which γ is a
stopping time.
Additional randomness admitted in a general single jump filtration makes the
considered model more flexible. As it is shown in [3], every σ -martingale is a local
martingale in the Dellacherie model, while there are σ -martingales which are not
local martingales with respect to a general single jump filtration. In this paper, we are
interested in a similar phenomenon. It is well known that, if M is a local martingale
on an arbitrary stochastic basis (Ω, F , F, P), it is not necessarily a local martingale
relative to the filtration FM that it generates. However, Herdegen and Herrmann [4]
show that every local martingale in the Dellacherie model is also a local martingale
with respect to FM . We aim to show that it is not the case for a general model with a
single jump filtration F, and our main goal is to characterize all F-local martingales
which are not local martingales relative to FM . This may happen if a local martingale
with a positive probability equals a constant on a time interval starting before γ
and including γ . Then, the filtration FM has no information about γ at all, and
it may happen that there are not enough FM -stopping times to provide a localizing
sequence for M. Our example resembles Ruf’s counterexample [5] where a concrete
single jump filtration is considered. It has a property that, for every finite stopping
time σ , σ ∧ γ is bounded. In our example, the filtration FM is also a single jump
filtration where there are not enough stopping times.
A σ -martingale on an arbitrary stochastic basis also needs not be a σ -martingale
with respect to the filtration that it generates; see [1]. Our final result states that a
σ -martingale with respect to a general single jump filtration preserves this property
relative to the filtration that it generates.
We use two facts that simplify proofs and constructions. The first is that, for a
local martingale M relative to a single jump filtration F, FM is also a single jump
filtration. The second fact is a simple characterization of all local martingales with
respect to single jump filtrations.
2 Main Results
We assume unless otherwise stated that there is given a probability space (Ω, F , P)
with a random variable γ with values in [0, +∞]. Let G(t) = P(γ ≤ t) be a
distribution function of γ , G(t) = 1 − G(t), tG = sup {t ∈ R+ : G(t) < 1}, and
T = {t ∈ R+ : P(γ ≥ t) > 0}.
By default, (Ω, F , P) is equipped with the single jump filtration F = (Ft )t ∈R+
generated by γ and F . We will also write F(γ , F ) to distinguish it among other
filtrations. Recall that A ∈ Ft if and only if A ∈ F and A ∩ {t < γ } is
either ∅ or {t < γ }. A convenient way to characterize an Ft -measurable random
variable is that it is a constant on the set {γ > t} (and F -measurable). No
Single Jump Filtrations: Preservation of the Local Martingale Property
221
completeness assumption is made. Every σ -martingale is assumed to have all paths
right continuous everywhere.
For ease of references, we reformulate some statements from [3] in a convenient
form.
Theorem 1
(i) Every F-σ -martingale M has a representation
Mt = F (t)1{t <γ } + L1{t ≥γ } ,
t ∈ R+ ,
(1)
where F : R+ → R is a deterministic right-continuous function and L is a
random variable.
(ii) In order a process M given by (1) to be a σ -martingale, it is necessary and
sufficient that
E |L|1{0<γ <∞} γ ) < ∞,
dF
E L γ = t = F (t) − G(t−)
(t),
dG
0 < t < tG ,
(2)
dG(t)−a.s.,
(3)
and, additionally, if P(γ = tG < ∞) > 0, that
E L γ = tG = lim F (t).
t tG
(iii) In order for a process M given by (1) to be a local martingale, it is necessary
and sufficient that
E|Mt | < ∞,
t ∈ T,
and
E(Mt ) = E(M0 ),
t ∈ T.
(4)
(iv) Let a process M given by (1) be a local martingale. Then, it is not a martingale
if and only if tG < ∞, P(γ = tG ) = 0, and one of the two conditions is
satisfied:
E|L| = ∞
(5)
or
E|L| < ∞
and
lim F (t)G(t) = 0.
t tG
(6)
222
A. A. Gushchin and A. K. Zhunussova
Proof
(i) See [3, Proposition 2 (v)].
(ii) See [3, Theorem 3 and Proposition 3].
(iii) See [3, Theorem 1].
(iv) It follows from [3, Theorem 1] that M is a martingale if tG = ∞ or P(γ =
tG < ∞) > 0. So let us consider the case where tG < ∞ and P(γ = tG ) = 0.
If E|L| = ∞, then MtG = L is not integrable; hence, M is not a martingale.
Note that E|L| < ∞ if and only if E|L−E[L|γ ]| < ∞ and E(|E[L|γ ]|) < ∞.
Thus, this case precisely corresponds to the assumptions of statement (iii) in
[3, Theorem 4]. Moreover, the process is not a martingale only in the part (iii.i)
of this statement, and the assumption for this part is exactly the second relation
in (6).
2.1 Filtrations Generated by a Process
The aim of this subsection is to prove the following result.
Theorem 2 Let X be a process on a probability space (Ω, F , P) of the form
Xt = F (t)1{t <γ } + L1{t ≥γ } ,
t ∈ R+ ,
(7)
where F : R+ → R is a deterministic right-continuous function and γ and L are
random variables with values in [0, +∞] and R, respectively. Put
F X = σ {Xs , s ∈ R+ },
FtX =
A
σ {Xs , s ≤ t + ε}
(t ∈ R+ ),
(8)
ε>0
and
γ̃ = inf{t ∈ R+ : Xt = F (t)}
(inf ∅ = +∞).
(9)
Then, the filtration FX = (FtX )t ∈R+ coincides with the single jump filtration
H = (Ht )t ∈R+ generated by the random variable γ̃ and the σ -field F X , i.e.,
H = F(γ̃ , F X ).
Remark 1 The function F (t) in (7) is uniquely determined (and does not depend
on a version of X) for t < tG . On the contrary, since P(γ > t) = 0 for t ≥ tG ,
replacing F by another right-continuous function on [tG , ∞) changes X only on an
evanescent set. However, random variables γ̃ may differ significantly for different
versions of F on [tG , ∞). This is the price we pay for exact coincidence of filtrations
and not up to null sets.
Remark 2 Without loss of generality, we may assume that L = 0 on the set {γ =
∞}. Then, L is F X -measurable.
Single Jump Filtrations: Preservation of the Local Martingale Property
223
Proof Since
>
{γ̃ < t} =
{Xr = F (r)} ∈ FtX
rrational
r<t
and FX is right continuous, γ̃ is a stopping time relative to FX .
Next, we have γ ≤ γ̃ by the definition of γ̃ . Thus, on the set {γ̃ ≤ t} ∈ FtX ,
Xs = Xt = L for s > t. This means that the traces of the σ -fields F X and FtX on
{γ̃ ≤ t} are equal.
Now, let A ∈ F X . We have just proved that A ∩ {γ̃ ≤ t} ∈ FtX . If A belongs
to Ht , then either A ∩ {γ̃ > t} = {γ̃ > t} or A ∩ {γ̃ > t} = ∅. In both cases,
A ∩ {γ̃ > t} ∈ FtX and A ∈ FtX .
The converse inclusion FtX ⊆ Ht is obvious because we can rewrite (7) as
Xt = F (t)1{t <γ̃ } + L1{t ≥γ̃ } ,
t ∈ R+ .
2.2 A Counterexample
We now construct a probability space (Ω, F , P) with a random variable γ and a
local martingale M with respect to the single jump filtration generated by γ and F ,
which is not a local martingale with respect to the filtration that it generates.
Let Ω = [0, 1) × {−1, 0, 1} and F the product of the Borel σ -field on [0, 1) and
the σ -field of all subsets of {−1, 0, 1}. For ω = (ω1 , ω2 ) ∈ Ω, we put γ (ω) = ω1
and ε(ω) = ω2 . A probability measure P on (Ω, F ) is defined as the product of the
Lebesgue measure and the measure that attributes the probability 1/3 to every point
of the second space. In particular, γ and ε are independent under P.
Introduce also an increasing function S : [0, 1) → R+ such that S(0) = 0 and
$1
0 S(t) dt = ∞.
Define now
Mt = εS(γ )1{t ≥γ } ,
t ∈ R+ .
We see that M satisfies (1) with F (t) ≡ 0 and L = εS(γ ). Here, T = [0, 1) and
E|Mt | ≤
t
S(u) du < ∞,
E(Mt ) = E(ε)
0
t
S(u) du = 0,
0
Therefore, M is an F(γ , F )-local martingale by Theorem 1 (iii).
Define a stopping time γ̃ as in (9) with M instead of X. Then,
γ̃ (ω) =
ω1 , if ω2 = 0;
+∞, if ω2 = 0.
t ∈ [0, 1).
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A. A. Gushchin and A. K. Zhunussova
Therefore, P(γ̃ ≥ t) ≥ P(γ̃ = +∞) = 1/3, and the set {t : P(γ̃ ≥ t) > 0} is equal
to R+ . Since the filtration generated by M is the single jump filtration generated by
γ̃ and F M by Theorem 2, it is necessary for M to be a local martingale with respect
to its own filtration that E|Mt | < ∞ for all t ∈ R+ by Theorem 1 (iii). However,
for t ≥ 1,
E|Mt | = E |ε|S(γ ) =
2
E S(γ ) = ∞.
3
We conclude that M is not a local martingale with respect to the filtration FM that it
generates.
2.3 Examples of Appropriate Localizing Sequences
Let us assume that tG < ∞, P(γ = tG ) = 0, and M has form (1) and is a martingale
on [0, tG ). Take an arbitrary sequence tn tG . In [3], see the proof of Theorem 1;
the following sequence of F-stopping times is constructed:
Tn =
tn , if γ > tn ;
+∞, otherwise.
It is proved that Tn → ∞ a.s. and M Tn = M tn , which implies that M is an Flocal martingale with a localizing sequence {Tn }. We will show how this idea can be
modified to prove that M is an FM -local martingale; see also [4, Proof of Theorem
3.5 (a)]. These constructions aim to explain preliminarily how some conditions
appear in our main result in the next subsection. They are not used in its proof.
Assume that tG < ∞, P(γ = tG ) = 0, and F is not constant on any interval
[t ∗ , tG ). Then, there is an increasing sequence {tn } converging to tG and such that
F (tn ) = F (tn−1 ) for all n. Put
Sn =
tn , if Mtn = Mtn−1 ;
+∞, otherwise,
where M is as above. Then, obviously, Sn is an FM -stopping time and Sn ≤ Tn
for all n. Therefore, M Sn is an F(γ , F )-martingale and hence an FM -martingale.
Moreover, since P(γ < tG ) = 1, we have P(tn > γ ) → 1 as n → ∞. We conclude
that Sn ↑ ∞ a.s. and M is an FM -local martingale.
Now, assume that F (t) ≡ c on [t ∗ , tG ) and P(L = c) = 0. Take an increasing
sequence t ∗ < t1 < · · · < tn < · · · < tG , tn → tG , and define Rn by
Rn =
tn , if Mtn = c;
+∞, otherwise.
Single Jump Filtrations: Preservation of the Local Martingale Property
225
Then, Rn = Tn a.s. and Rn is an FM -stopping time for all n. We arrive at the
conclusion that again, M is an FM -local martingale.
2.4 Necessary and Sufficient Conditions
In this subsection, we give necessary and sufficient conditions for a local martingale
with respect to a single jump filtration to be a local martingale with respect to the
filtration that it generates.
Theorem 3 Let M be an F(γ , F )-local martingale of the form (1). Denote by FM
the filtration generated by M; see (8). Then, M is not an FM -local martingale if and
∗
only if tG < ∞, P(γ = tG ) = 0, there is t ∗ < tG such that F (t)
≡ c on [t , tG ) for
∗
some constant c, P(L = c, γ ≥ t ) > 0, and E L − E(L|γ ) = ∞.
Corollary 1 Let M be an F(γ , F )-local martingale and G = (Gt )t ≥0 be any
filtration such that FtM ⊆ Gt ⊆ Ft for every t ≥ 0. Suppose that the assumptions
of Theorem 3 are not satisfied. Then, M is a G-local martingale.
Corollary 2 (Herdegen and Herrmann [4]) In the Dellacherie model, every
F(γ , F )-local martingale is an FM -local martingale.
Proof of Theorem 3 According to Theorem 1 (i), M has a representation (1).
Obviously, every F(γ , F )-martingale is an FM -martingale. Thus, we may and will
assume that M is not an F(γ , F )-martingale. By Theorem 1 (iv), this means that
tG < ∞, P(γ = tG ) = 0, and either (5) or (6) holds.
Put
γ̃ = inf{t ∈ R+ : Mt = F (t)}
(inf ∅ = +∞).
By Theorem 2, the filtration FM coincides with the single jump filtration F(γ̃ , F M ).
Rewriting (1) in the form
Mt = F (t)1{t <γ̃ } + L1{t ≥γ̃ } ,
t ∈ R+ ,
we deduce from Theorem 1 that M is an F(γ , F )-local martingale if
E|Mt | < ∞ and E(Mt ) = E(M0 ),
t ∈ T,
while M is an FM -local martingale if and only if
E|Mt | < ∞
and E(Mt ) = E(M0 ),
where T M = {t ∈ R+ : P(γ̃ ≥ t) > 0},
t ∈ T M.
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A. A. Gushchin and A. K. Zhunussova
Obviously, γ̃ ≥ γ ; hence, T ⊆ T M . Therefore, if M is not an FM -local
martingale, it is necessary that T T M , i.e., tG ∈ T M . In other words, P(γ̃ ≥
tG ) > 0. But
{γ̃ ≥ tG } ⊆ {ω : L(ω) = F (t) for all t ∈ [γ (ω), tG )} =: B.
Put B ∗ := B ∩ {γ < tG }. Since P(γ < tG ) = 1, we have P(B ∗ ) > 0. We obtain
that F (t) is constant on any interval [γ (ω), tG ) if ω ∈ B ∗ . Thus, putting
t ∗ = inf∗ γ (ω),
ω∈B
we obtain that F (t) ≡ c := F (t ∗ ) on [t ∗ , tG ), and the set {L = c, γ ≥ t ∗ } contains
B ∗ and, hence, has a positive probability.
To complete the proof of necessity, it remains to show that E L − E(L|γ ) =
∞. Note that F is bounded in the left neighborhood of tG and G(t) → 0 as t tG .
Hence, the second condition in (6) is not satisfied. Thus, we have (5). On the other
hand, denote H (t) = E[L|γ = t]; see details in [3]. Then,
E E(L|γ ) = E H (γ ) =
[0,tG )
|H (s)| dG(s)
and
E(Mt ) = F (t)G(t) +
[0,t ]
H (s) dG(s),
t < tG ,
(10)
$
in particular, [0,t ] |H (s)| dG(s) < ∞. Differentiating (10) in t and taking
∗
(4)
$ into account, we obtain that H (t) = c dG(t)-a.s. on (t , tG ). Thus,
[t ∗ ,tG ) |H (s)| dG(s) < ∞. The claim follows.
Conversely, let us show that the conditions of the theorem are sufficient. If
∗
F (t) ≡ c on [t ∗ , tG ), then γ̃ ≥
tG on the set {L
= c, t ≤ γ ≤ tG }. Finally,
it follows from E L − E(L|γ ) = ∞ that E L = ∞. Hence, E MtG = ∞,
and M is not an F(γ̃ , F M )-local martingale by Theorem 1 (iii).
2.5 Preservation of the σ -Martingale Property
In this subsection, we prove that every F-σ -martingale is also a σ -martingale with
respect to the filtration that it generates.
Theorem 4 Let M be a σ -martingale with respect to a single jump filtration F.
Then, it is also a σ -martingale with respect to the filtration FM that it generates.
Single Jump Filtrations: Preservation of the Local Martingale Property
227
Corollary 3 Let M be an F(γ , F )-σ -martingale and G = (Gt )t ≥0 be any filtration
such that FtM ⊆ Gt ⊆ Ft for every t ≥ 0. Then, M is a G-σ -martingale.
Corollary 4 Every F(γ , F )-local martingale is an FM -σ -martingale.
Proof of Theorem 4 Every F(γ , F )-σ -martingale has a representation (1). Without
loss of generality, we may assume that F (t) ≡ y, t ≥ tG , where y is a number such
that P(L = y) = 0.
Put
γ̃ = inf{t ∈ R+ : Mt = F (t)}
(inf ∅ = +∞).
The above assumption guarantees that P(γ̃ > tG ) = 0. By Theorem 2, the filtration
FM coincides with the single jump filtration F(γ̃ , F M ). Rewrite (1) in the form
Mt = F (t)1{t <γ̃ } + L1{t ≥γ̃ } ,
t ∈ R+ .
(11)
Let us consider all half-open intervals J = [a, b), 0 ≤ a < b ≤ tG , on which
F (t) is constant and which are maximal among all intervals with these properties,
i.e., (i) F (t) = c for some c for all t ∈ [a, b); (ii) if a > 0, then for any ε ∈ (0, a)
there is s ∈ (a −ε, a) with F (s) = c; (iii) if b < tG then for any ε ∈ (0, tG −b) there
is s ∈ (b, b + ε) with F (s) = c. Obviously, there are at most countable numbers
of such intervals, and they do not intersect. Enumerate those of them that satisfy
P(γ ∈ J, L = F (a)) > 0 as J1 , . . . , Jk , . . . . Let ak < bk be the end points of Jk
and ck = F (ak ).
If γ̃ (ω) = γ (ω), then γ̃ (ω) > γ (ω) and
F (t) = L(ω),
for all t such that γ (ω) ≤ t < γ̃ (ω).
Excluding P-null sets such as {γ > tG } and {γ = tG , L = F (tG )}, we obtain
{γ < γ̃ } =
>
Dk
a.s.,
k
where
Dk = {ω : γ (ω) ∈ Jk , L(ω) = ck }
= {ω : γ (ω) ∈ Jk , γ̃ (ω) = bk , L(ω) = ck }.
Our purpose is to check that the conditions of Theorem 1 (ii) imply the
corresponding conditions for representation (11) and γ̃ .
228
A. A. Gushchin and A. K. Zhunussova
Assume that there exists a strictly positive function ψ : (0, +∞) → (0, +∞)
such that
|L|ψ(γ ) dP < ∞,
{0<γ <∞}
which is equivalent to (2). Take δk > 0 such that
|ck |δk P(Dk ) < ∞
k
and put
ψ(t),
t = bk for all k;
ψ(t) ∧ δk , t = bk for some k.
ψ̃(t) =
Then,
{0<γ̃ <∞}
|L|ψ̃(γ̃ ) dP =
{0<γ̃ <∞, γ =γ̃ }
|L|ψ̃(γ̃ ) dP
+
=
{0<γ̃ <∞, γ =γ̃ }
{0<γ <∞, γ =γ̃ }
+
≤
|L|ψ̃(γ̃ ) dP
|L|ψ̃(γ ) dP
k
{0<γ̃ <∞}∩Dk
{0<γ <∞}
|L|ψ̃(bk ) dP
|L|ψ(γ ) dP +
|ck |δk P(Dk ) < ∞,
k
which proves (2) for γ̃ .
The last step is to prove that
dF
(t),
E L γ̃ = t = F (t) − P(γ̃ ≥ t)
d G̃
0 < t < tG ,
d G̃(t)−a.s.,
(12)
where G̃(t) = P(γ̃ ≤ t) and, additionally, if P(γ̃ = tG < ∞) > 0, that
E L γ̃ = tG = lim F (t).
t tG
(13)
We consider separately the following cases: (i) t ∈
/ ∪k [ak , bk ]; (ii) t ∈ (ak , bk ) for
some k; (iii) t = ak for some k, and t ∈
/ ∪l {bl }; (iv) t = bk for some k, and t = tG ;
(v) t = bk < tG for some k, and F (bk ) = F (ak ); (vi) t = bk < tG for some k,
Single Jump Filtrations: Preservation of the Local Martingale Property
229
F (bk ) = F (ak ), and t ∈
/ ∪l {al }; and (vii) t = bk < tG for some k, F (bk ) = F (ak ),
and t ∈ ∪l {al }:
(i) Let E be a Borel subset of (0, ∞) \ ∪k [ak , bk ]. Then, for every t ∈ E, {γ ≥
t} = {γ̃ ≥ t} a.s., in particular, the distribution functions of γ and γ̃ coincide
on E. On the other hand, {γ ∈ E} = {γ̃ ∈ E}, which implies that E(L|γ ) =
E(L|γ̃ ) on E a.s. The claim follows.
(ii) $Let E be a Borel subset of (ak , bk ) such that P(γ ∈ E, L = ck ) > 0 and
{γ ∈E} |L| dP < ∞. It follows from (3) that E(L|γ = t) = ck for t ∈ (ak , bk )
dG(t)-a.s. Hence,
L1{γ̃ ∈E} dP = L1{γ ∈E, L=ck } dP = L1{γ ∈E} dP
−
L1{γ ∈E, L=ck } dP
= ck P(γ ∈ E, L = ck ) = ck P(γ̃ ∈ E),
which implies E(L|γ̃ ) = ck on (ak , bk ) dG(t)-a.s. Obviously, the expression
on the right in (12) is equal to ck as well.
(iii) Let ak ∈
/ ∪l {bl }. If P(γ̃ = ak ) = 0, then the value E(L|γ̃ = ak ) at a single
point ak does not influence on (12). So let us suppose that P(γ̃ = ak ) > 0.
Then, according to (3),
E L1{γ =ak } = P(γ = ak )E(L|γ = ak ) = ck P(γ = ak )−(ck −ck )P(γ ≥ ak ),
where ck is the left limit of F at ak . We have
E(L|γ̃ = ak ) =
=
E L1{γ̃ =ak }
P(γ̃ = ak )
=
E L1{γ =ak } − E L1{γ =ak , L=ck }
P(γ̃ = ak )
ck P(γ = ak ) − (ck − ck )P(γ ≥ ak ) − ck (P(γ = ak ) − P(γ̃ = ak ))
P(γ̃ = ak )
= ck − P(γ ≥ ak )
(ck − ck )
P(γ̃ = ak )
= F (ak ) − P(γ̃ ≥ ak )
F (ak ) − F (ak −)
.
P(γ̃ = ak )
(iv) Here, condition (13) has to be verified. But {γ̃ = tG } = {γ ∈ [ak , bk ), L =
ck }, and the requested equality is trivial.
(v) Since F is assumed to be continuous at bk in this case, E(L|γ = bk ) = ck if
P(γ = bk ) > 0, as it follows from (3). Thus,
E[L|γ̃ = bk ] =
E L1{γ ∈[ak ,bk ), L=ck } + E L1{γ =bk }
= ck .
P(γ ∈ [ak , bk ), L = ck ) + P(γ = bk )
230
A. A. Gushchin and A. K. Zhunussova
(vi) Note that F is not continuous at bk in (vi) and (vii), so P(γ = bk ) > 0 in these
cases.
Since E(L|γ = bk ) =
E L1{γ =bk }
P(γ =bk )
, we obtain from (3) that
E L1{γ =bk } = F (bk )P(γ = bk ) − (F (bk ) − ck )P(γ ≥ bk ).
(14)
Substituting it in the formula below, we get
E(L|γ̃ = bk ) =
E L1{γ =ak , L=ck } + E L1{γ =bk }
E L1{γ̃ =bk }
=
P(γ̃ = bk )
P(γ̃ = bk )
=
ck P(γ = ak , L = ck ) + F (bk )P(γ = bk ) − (F (bk ) − ck )P(γ ≥ bk )
P(γ̃ = bk )
=
ck P(γ̃ ≥ bk ) − F (bk )P(γ > bk )
P(γ̃ = bk )
= F (bk ) − P(γ̃ ≥ bk )
F (bk ) − ck
,
P(γ̃ = bk )
which coincides with (12).
(vii) To simplify the notation, put pk = P(γ ∈ [ak , bk ), L = ck ) and qk = P(γ =
bk , L = F (bk )).
In this case, we again obtain (14) from (3). Then, substituting (14) in the
formula below, we get
E(L|γ̃ = bk ) =
E L1{γ̃ =bk }
P(γ̃ = bk )
=
E L1{γ ∈[ak ,bk ), L=ck } + E L1{γ =bk , L=F (bk )}
P(γ̃ = bk )
=
ck pk + E L1{γ =bk } − E L1{γ =bk , L=F (bk )}
P(γ̃ = bk )
=
ck pk + F (bk )P(γ = bk ) − (F (bk ) − ck )P(γ ≥ bk ) − F (bk )qk
P(γ̃ = bk )
c P(γ̃ ≥ bk ) − F (bk )P(γ̃ > bk )
= k
P(γ̃ = bk )
= F (bk ) − P(γ̃ ≥ bk )
F (bk ) − ck
.
P(γ̃ = bk )
Acknowledgment A.G. was supported by Russian Foundation for Basic Research grant 20-0100646.
Single Jump Filtrations: Preservation of the Local Martingale Property
231
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135–156 (2020)
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Process. Appl. 125(10), 3657–3662 (2015)
Local Time and Local Reflection
of the Wiener Process
I. A. Ibragimov, N. V. Smorodina, and M. M. Faddeev
Abstract In this paper we introduce a concept of a Brownian trajectory local
reflection. Ideologically, this concept is close to the concept of the Brownian local
time, which can be considered as the integrated (over time) Dirac delta function of
a Brownian trajectory. In the concept of the local reflection, we replace the Dirac
delta function by its first derivative.
Keywords Wiener process · Local time · Limit theorem
Let w(s), s ⩾ 0, be a standard Wiener process and A be an arbitrary measurable
subset of R. Denote by λ(U ), U ⊂ R, the Lebesgue measure of a set U . The sojourn
time of the process w(s) in the set A up to the time t > 0 is defined by
μ(t, A) = λ{s : w(s) ∈ A, s ∈ [0, t]}.
For a fixed t, the set function μ(t, ·) is a random measure; moreover, it is known
(see [1], ch. 1, §4) that the measure μ(t, ·) is absolutely continuous with respect to
the Lebesgue measure λ with probability 1. The Radon–Nikodym derivative
l(t, x) =
dμ
(t, x)
dλ
is called the local time of the process w(s) at the point x up to the time t.
I. A. Ibragimov · N. V. Smorodina ()
St. Petersburg Department of V.A. Steklov Institute of Mathematics of the Russian Academy of
Sciences, St. Petersburg, Russia
e-mail: ibr32@pdmi.ras.ru; smorodina@pdmi.ras.ru
M. M. Faddeev
St. Petersburg State University, Physical Faculty, St. Petersburg, Russia
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_12
233
234
I. A. Ibragimov et al.
It is also known that, with probability 1, there exists a continuous two-parameter
process l(t, x) , (t, x) ∈ (0, ∞) × R, satisfying the relation
t
f (w(τ )) dτ =
R
0
f (y)l(t, y) dy
(1)
for any t > 0 and any function f integrable on an arbitrary finite interval.
With probability 1, the local time l(t, x) also satisfies the relation (see [2], ch.9,
§4)
1
l(t, x) = lim
ε↓0 2ε
t
0
1[−ε,ε] (x − w(τ ))dτ.
(2)
On the other hand, in the distribution sense, we have
1
1[−ε,ε] = δ,
ε↓0 2ε
lim
(3)
where δ is the Dirac delta function. Taking into account (2) and (3), we can rewrite
(2) in the following form:
l(t, x) =
t
δ(x − w(τ ))dτ.
(4)
0
Using Fourier analysis we can give a precise meaning to (4). The corresponding
approach belongs to S.M. Berman (see [3]). In the distribution sense, we have
δ(x − y) =
1
2π
δ(x − w(τ )) =
1
2π
R
eip(y−x) dp,
(5)
eip(w(τ )−x) dp.
(6)
so that
R
Note that for fixed τ the rhs of (6) is not defined in the usual sense (as a random
variable), but it will be defined correctly after integrating with respect to τ , namely:
t
0
1
2π
M t
1
lim
eip(w(τ )−x) dp dτ =
eipw(τ ) dτ e−ipx dp.
2π M→+∞ −M
0
R
(7)
The last limit exists in the metric of the space L2 (Ω × R, P × λ) (see [1], ch.1, §4).
Note that it follows from (3) that the local time l(t, x) in some sense can be
considered as a number of crossings of the level x by the process w(·) up to the time
t (for more details, see, e.g., [4]).
Local Time and Local Reflection of the Wiener Process
235
The next theorem shows that the Brownian local time can be used to construct
probabilistic representations of some second-order differential equation’s solutions.
Namely, for a fixed function f ∈ L2 (R), consider a second-order ordinary
differential equation
u(2) = f,
(8)
where u is an unknown function and we are looking for u ∈ L2 (R). Passing to the
Fourier transform, we get
− p20
u(p) = f0(p).
(9)
Suppose in addition that the function f satisfies the condition
R
(1 + x 2 )|f (x)|dx < ∞.
(10)
It can be shown, in the usual way, that in this case Eq. (8) has a solution u ∈ L2 (R)
if and only if the function f satisfies a condition
(1) (0) = 0,
f0(0) = fB
(11)
or, which is equivalent, the condition
R
f (x) dx =
R
xf (x) dx = 0.
(12)
Denote by f ∗ l the convolution of the function f with the function l(t, ·).
The following statement holds:
Theorem 1 Suppose that a function f ∈ L2 (R)∩C(R) satisfies the conditions (10)
and (12). Define a function u(t, x), t > 0, x ∈ R, by
u(t, x) = −
1
Ef ∗ l(t, x).
2
Then, uniformly in x
lim u(t, x) = u(x),
t →∞
where u is the unique weak solution of Eq. (8), such that u ∈ L2 (R).
Proof Using (1) we get
f ∗ l(t, x) =
R
f (x − y)l(t, y)dy =
t
0
f (x − w(τ ))dτ.
(13)
236
I. A. Ibragimov et al.
Further, let function u ∈ L2 (R) be a solution of the equation u(2) = f . Using Itô
formula we get
t
1
Eu(x − w(t)) − u(x) = E
f (x − w(τ ))dτ.
2
0
For completing the proof, it is sufficient to note that uniformly in x ∈ R
Eu(x − w(t)) → 0 as t → ∞.
Now, we define another object. For this we substitute in (6) instead of the delta
function δ its first derivative δ (1).
In the distribution sense (see [5])
(1) (p) = ip,
δB
so we get
1
2π
δ (1) (x − w(τ )) =
R
ip eip(w(τ )−x) dp.
(14)
As above (14) is not defined for fixed τ as a random variable. To define a correct
object, we integrate (14) with respect to τ twice. Namely, we consider
t
t
dτ1
0
dτ2 δ
(1)
(x − w(τ2 )) =
τ δ (1) (x − w(τ )) dτ
0
τ1
=
t
i
2π
t
τ
0
R
p eip(w(τ )−x) dp dτ.
(15)
Note that in the distribution sense
1
(1[−ε,0] − 1[0,ε] ) = δ (1),
ε↓0 ε 2
lim
(16)
$t
so 0 δ (1) (x − w(τ )) dτ can be considered in some sense as a number of reflections
for w(·) from a level x up to time t. More precisely, it is equal to the number of
reflections from a level x from below minus the number of reflection from above.
Unfortunately the rhs. of (15) is not defined in the L2 -sense similar to (7).
This means that we need some regularization. One of the possible ways of this
regularization is the following: for (t, x) ∈ (0, ∞) × R, let us define a random
Local Time and Local Reflection of the Wiener Process
237
variable l1 (t, x) by
l1 (t, x) =
i
2π
i
=
2π
=
t R
0
R
i
lim
2π M→+∞
pe
p τ eip(w(τ )−x)1[−t /τ, t /τ ](p) dp dτ
t ·r(p)
−ipx
τ eipw(τ ) dτ dp
0
M
−M
p e−ipx
t ·r(p)
τ eipw(τ ) dτ dp,
(17)
0
where
r(p) = min(1,
1
).
|p|
(18)
Now, let us show that the limit (17) almost surely exists in the metric of the space
L2 (R).
It follows from (17) that
0
l1 (t, p) = ip
t ·r(p)
τ eipw(τ ) dτ,
(19)
0
so it is sufficient to prove that almost surely 0
l1 (t, ·) ∈ L2 (R).
We have
t
2
0
l1 (t, ·)2 ⩽ 2
p2 dp
τ dτ
|p|<1
+2
|p|⩾1
p2
t /|p|
2
τ dτ
dp ⩽
0
0
t4
t4
+
3
2
dp
< ∞.
2
|p|⩾1 p
Thus, we have proved that the limit in (17) a.s. exists. By analogy with the local
time, we call the process l1 (t, x), (t, x) ∈ (0, ∞) × R, a local reflection of the
Wiener process w(·).
Next, we prove an analog of the relation (1) for the local reflection. Suppose that
f ∈ W21 (R). Then, we have
−i
f (y)l1 (t, y) dy =
2π
R
=
−i
2π
t
τ dτ
R
0
=
t
0
R
f0(p)p
t ·r(p)
τ e−ipw(τ ) dτ dp
0
e−ipw(τ ) f0(p)p 1[−t /τ, t /τ ](p) dp
τf (1) ∗ Dt /τ (w(τ )) dτ,
238
I. A. Ibragimov et al.
where for every M > 0 the function DM is the Dirichlet kernel defined by
DM (x) =
sin(Mx)
B
or D
M (p) = 1[−M,M] (p).
πx
The last formula means that, for the local reflection l1 (t, x), we have
t
τf (1) ∗ Dt /τ (w(τ )) dτ =
R
0
f (y)l1 (t, y) dy
(20)
instead of (1).
The following theorem shows that the Brownian local reflection can be used
for constructing probabilistic representations of solutions of some third-order
differential equations. Namely, for a fixed function f ∈ L2 (R), consider the
equation
u(3) = f,
(21)
where u is an unknown function and we are looking for a solution u ∈ L2 (R).
Passing to the Fourier transform, we get
− ip30
u(p) = f0(p).
(22)
Suppose in addition that a function f satisfies the condition
R
(1 + |x|3)|f (x)|dx < ∞.
(23)
It can be shown that in this case Eq. (21) has a solution u ∈ L2 (R) if and only if the
function f satisfies the condition
B
(1) (0) = f
(2) (0) = 0,
f0(0) = fB
(24)
or, which is equivalent, the condition
R
f (x) dx =
R
xf (x) dx =
R
x 2 f (x) dx = 0.
(25)
Denote by f ∗ l1 the convolution of the function f with the function l1 (t, ·).
The following statement holds:
Theorem 2 Suppose that a function f ∈ L2 (R) satisfies the conditions (23) and
(25). Define a function u(t, x), t > 0, x ∈ R, by
u(t, x) = −
1
Ef ∗ l1 (t, x).
4
Local Time and Local Reflection of the Wiener Process
239
Then, uniformly in x
lim u(t, x) = u(x),
t →∞
where u is a unique weak solution of Eq. (21), such that u ∈ L2 (R).
Proof It follows from (24) that
f0(p) = O(p3 ) as p → 0.
(26)
Further, we have
i
Ef ∗ l1 (t, x) =
2π
=
i
2π
R
R
e
t ·r(p)
−ipx 0
f (p) p
p2 τ
τ e− 2 dτ dp
0
e−ipx f0(p) p A(t, p) dp,
(27)
where
A(t, p) =
t ·r(p)
p2 τ
τ e− 2 dτ
0
t ·r(p)
p2 τ
2 − p2 τ τ =t ·r(p)
2
2
= − 2 τe
+ 2
e− 2 dτ
p
p 0
τ =0
=−
2
4
4 − p2 t·r(p)
2
− p t·r(p)
2
2
r(p)e
t
+
−
e
p2
p4
p4
= R1 (t, p) + R2 (t, p) + R3 (t, p).
So we get
Ef ∗ l1 (t, x) =
+
i
2π
R
i
2π
R
e−ipx f0(p) pR1 (t, p)dp
e−ipx f0(p) pR2 (t, p)dp +
i
2π
R
e−ipx f0(p) pR3 (t, p)dp
= I1 (t, x) + I2 (t, x) + I3 (t, x).
(28)
Note that I2 does not depend on t. We have
u2 (x) =
−1
−i
I2 (t, x) =
4
2π
R
e−ipx f0(p)
1
dp.
p3
(29)
240
I. A. Ibragimov et al.
It is clear that this function u2 (x) is a solution of Eq. (21), so, to complete the proof,
it is sufficient to show that
lim sup |I1 (t, x)| = 0
(30)
lim sup |I3 (t, x)| = 0.
(31)
t →∞ x∈R
and
t →∞ x∈R
First, we prove (30). Since
1 e−ipx f0(p) pR1 (t, p)dp R
x∈R 2π
sup |I1 (t, x)| ⩽ sup
x∈R
⩽
=
t
π
|p|<1
|f0(p)|
p2 t·r(p)
1
r(p)e− 2 dp
|p|
p2 e− 2 dp +
t
f 2
π
p2
R
1 − p2 t
t
e 2 dp +
|p|
π
p2 t
|p|<1
C
⩽ √
t
R
|f0(p)|
⩽ Ct
t
π
p2 e− 2 dp + t 5/2 f 2
|p|⩾1
|f0(p)|
1 − |p|t
e 2 dp
p2
1 −|p|t 1/2
e
dp
4
|p|⩾1 p
1 −|p| 1/2
e
dp
4
|p|⩾t p
we obtain that sup |I1 (t, x)| tends to 0 as t → +∞.
x∈R
Next, we prove (31). Since
1 e−ipx f0(p) pR3 (t, p)dp 2π
R
x∈R
sup |I3 (t, x)| ⩽ sup
x∈R
2
⩽
π
2
=
π
R
|f0(p)|
1 − p2 t·r(p)
2
e
dp
|p|3
p2 t
1
2
|f0(p)| 3 e− 2 dp +
π
|p|
|p|<1
⩽C
p2 t
|p|<1
C
⩽ √
t
R
e− 2 dp +
p2
|p|⩾1
|f0(p)|
1 − |p|t
e 2 dp
|p|3
1 −|p|t 1/2
2
f 2
e
dp
6
π
|p|⩾1 p
e− 2 dp + t 3 f 2
1 −|p| 1/2
e
dp
6
|p|⩾t p
Local Time and Local Reflection of the Wiener Process
241
we obtain that sup|I3 (t, x)| also tends to 0 as t → +∞.
x∈R
Now, let us formulate a generalization of Theorem 2. The main idea is to use
the high-order derivatives of the Dirac delta function in (6). Namely, for k ∈ N and
t > 0, let us define a random process lk (t, x) by
ik
lk (t, x) =
2π
=
ik
2π
ik
lim
=
2π M→+∞
t R
0
R
pk τ eip(w(τ )−x)1[−t /τ, t /τ ](p) dp dτ
pk e−ipx
t ·r(p)
τ k eipw(τ ) dτ dp
0
M
−M
k −ipx
t ·r(p)
p e
τ k eipw(τ ) dτ dp,
(32)
0
where r(p) is defined by (18).
By the same reason as above, we call the process lk (t, x) the Brownian local
reflection of the k-th order.
The following statement holds:
Theorem 3 Suppose that the function f ∈ L2 (R) satisfies the conditions
R
(1 + |x|k+2)|f (x)|dx < ∞,
and
R
f (x) dx =
R
xf (x) dx = . . .
R
x k+1 f (x) dx = 0.
Define a function u(t, x), t > 0, x ∈ R, by
1
u(t, x) = − k+1 Ef ∗ lk (t, x).
2
Then, uniformly in x
lim u(t, x) = u(x),
t →∞
where u ∈ L2 (R) is the unique weak solution of the equation u(k+2) (x) =
f (x), x ∈ R.
This theorem can be proved by the same argument as Theorem 2.
Acknowledgments The research of the third author was partially supported by Russian Foundation for Basic Research, grant 19-01-00657a.
242
I. A. Ibragimov et al.
References
1. Borodin, A.N., Ibragimov, I.A.: Limit theorems for functional of random walks. Proc. Steklov
Inst. Math. 195, 1–259 (1995)
2. Chung, K.L., Zhao, Z.: From Brownian Motion to Schrodinger’s Equation. Springer-Verlag,
Berlin, Heidelberg (1995)
3. Berman, S.M.: Local times and sample function properties of stationary Gaussian process.
Trans. Am. Math. Soc. 137, 277–299 (1969)
4. Berzin-Joseph, C., León, J.R.: Weak convergence of the integrated number of level crossings to
the local time for Wiener processes. Theory Probab. Appl. 42(4), 568–579 (1998)
5. Gel’fand, I.M., Shilov, G.E.: Generalized Functions, Volume 1: Properties and Operations.
Academic, New York (1964)
Random Harmonic Processes
with New Properties
Elena Karachanskaya
Abstract In this paper, we introduce a new general model for random signals and
adjoining harmonic processes. This model is stochastic hierarchically correlated
series (SHCS). The sufficient conditions for both the wide-sense stationary property
and the mean ergodic property for random harmonic (trigonometric) processes
are set. Moreover, we describe a class of wide-sense stationary and mean ergodic
random harmonic processes function with nonuniformly random phase.
Keywords Stochastic hierarchically correlated series · Wide-sense stationarity ·
Mean ergodicity · Nonuniformly random phase
1 Introduction
Researchers have increasingly been studying models of signals with the random
behavior. In this paper [1], we introduced a new class of random harmonic processes
which differ from the processes that are ordinarily analyzed in [2]. In this article,
we research main properties of such processes – stationarity and ergodicity. Usually,
random phase for harmonic processes has uniform distribution. Note that some
actual technical processes do not have this property, for example, magnetic noise.
As well known, harmonic processes with random amplitudes are not ergodic. They
follow sub-optimal predictability for an investigated event. This paper is a survey of
some aspects of random harmonic processes.
E. Karachanskaya ()
Far Eastern State Transport University, Khabarovsk, Russia
Pacific National University, Khabarovsk, Russia
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_13
243
244
E. Karachanskaya
2 A Modern Model for Random Signals
Definition 1 Let An (t) and Bn (t) for all n ∈ N be random processes and {φs }∞
n=1
be a consequence of random variables. A harmonic series
Z(t) =
∞
n=1
+
An (t) cos ωn t +
n
φs
+ Bn (t) sin ωn t +
s=1
n
,
φs
,
s=1
(1)
∞
where ∞
n=1 |An (t)| +
n=1 |Bn (t)| < ∞, will be termed as stochastic hierarchically correlated series (SHCS).
These series differ from the ones studied in the theory of classical, stochastic
harmonic random series [1]. The series (1) may be contemplated as the sum of
strongly dependent terms. Moreover, in dependence on the conditions that put on
components, this series may be considered as doubly stochastic.
The representation (1) allows to describe the broad class of random processes
similar to harmonic ones.
Radar signals s(t) are represented as a form [3]
s(t) = g(t) cos [2π fc · t + ϕ(t)] ,
where g(t) is an envelope amplitude for the signal, ϕ(t) is a modulation phase, fc
is frequency, and t is a time variable. In cases where function g(t) is unknown, we
study an amplitude modulation model, and in cases where phase ϕ(t) is unknown,
we study a phase modulation one. For the reason for g(t) and ϕ(t) to be subjected
to random perturbations, we research s(t) as stochastic process.
An output signal of laser is described as the following expression [4]:
U (t) = (Uo + U (t)) · sin (2πυo t + φ(t)) ,
where U (t) is random changes or fluctuations of an amplitude with respect to Uo
and φ(t) is phase-modulation-determined random fluctuations.
According to [5], we can write a mathematical model for random harmonic of
magnetic noise as a form
unωo (t) = a(t) cos [nωo t − ψ(t)] ,
where ψ(t) is a random phase. This formula is in good agreement with the
experiment.
Random Harmonic Processes with New Properties
245
Random processes which are similar to
Z(t) =
n
Aj (t) · cos
j =1
j
ϕk (t) ,
(2)
k=1
are useful in radio communication and radio navigation [6]. There is an additive
mixture of narrowband random processes.
Suppose that
4 k (t),
)ϕk (t) = 2π(fk − fk−1 ) · t + ϕk (t) − ϕk−1 (t) = 2π)fk · t + )ϕ
Āj (t) = A(t), fj = f, ϕj (t) = ϕ(t),
then, we obtain one version for phase modulation model:
Z1 (t) = A(t) cos (2πf · t + ϕ(t)) .
Let fn = −
(3)
ñωo
be ñth harmonic wave; then, we set a model
2π
Z1 (t) = A(t) cos (2πfñ · t + ψ(t))
for magnetic noise.
3 Stationarity
Some of harmonic processes are passing by homogeneously in time. They look like
continuous random oscillations near nonrandom values.
On average, amplitude and nature of fluctuations do not change with time. Thus,
the process is stationary in the wide sense. An ergodic property is more global for
all random processes.
The random harmonic process is stationary since the phase of this processes
is uniform variable on [0, 2π]. Randomness of its amplitudes violates ergodic
property.
We represent some random harmonic processes which are both wide-sense
stationary and ergodic with respect to average.
Let X(t) be a centered random function, i.e., E[X(t)] = 0. A canonical
representation for the wide-sense stationary function is
X(t) =
∞
Uk cos ωk t + Vk sin ωk t,
k=0
Uk2 + Vk2 ≤ const,
246
E. Karachanskaya
where Uk and Vk (k ∈ {1, 2, . . .}) are centered independent random variables and
D[Uk ] = D[Vk ] = Dk for every k.
N
Uk cos ωk t +Vk sin ωk t, N = ∞, it is a spectral decomposition
When X(t) =
k=0
of stationary random function with a discrete spectrum.
An autocorrelation function for this process is
KX (τ ) =
N
Dk cos ωk τ.
k=0
Let )ϕs (t) be random phase increments defined as
)ϕs (t) = ψs (t) − ψs−1 (t), ψs (t) = ωs · t + φs .
Assume Al (t) = Al and Bj (t) = Bj are independent identically distributed (IID)
non-centered random variables.
Then, the stationary function is the SHCS:
Z(t) =
∞
An cos ωn t + φn + Bn sin ωn t + φn
(4)
n=1
where all the phases φn are independent random variables uniformly distributed
on the segment [0, 2π] and the amplitudes Al for every l and Bj for every j are
mutually independent, and they are both independent on the phases φn for every n
[7]).
Let us consider the Gaussian Fourier series [2]:
Z(t) =
∞
an [Xn cos nt + Yn sin nt] .
(5)
n=1
Note that the expression (5) is a particular case of SHCS: An (t) = an Xn and
Bn (t) = an Yn , where X0 , Y0 , X1 , Y1 , . . . are Gaussian sequence and phases
ωn · t + φn = nt are not random.
Now, we introduce the new class of random harmonic process with the widesense stationary property.
Definition 2 A function f (x) defined on an open interval [a, b) is called (b − a)/k
periodic function (k ∈ N), if it fulfills the following:
⎧ #
b−a
(k − 1)(b − a) ⎪
⎪
,
,
if x ∈ a, a +
⎨f x +
k
k
f (x) =
#
(k − 1)(b − a)
(k − 1)(b − a)
⎪
⎪
, if x ∈ a +
,b .
⎩f x −
k
k
Random Harmonic Processes with New Properties
247
Theorem 1 Let Z(t) be a random process and have a form
Z(t) =
∞ #
%
ηn cos ωn t + φn + νn sin ωn t + φn ,
(6)
n=1
where the amplitudes ηn and νn are independent of each other and of the phases
∞
∞
E[ηn ]2 < +∞,
E[νn ]2 < +∞, E[ηn ] = E[νn ] = 0, and the
φn for every n,
n=1
n=1
phases φn and φn ∈ [0, 2π) (n = 1, 2, . . .) are IID. The process (6) is stationarity
π
periodic
in the wide sense, if the probability density ρ(φ) for the phase φ is
2
function, i.e.,
⎧ # 3π π
⎪
ρ
φ
+
,
if
φ
∈
0,
,
⎪
⎪
⎨ 2 2 #
ρ(φ) = ρ φ − 3π , if φ ∈ 3π , 2π ,
⎪
⎪
2
2
⎪
⎩
0,
if φ ∈
/ [0, 2π).
(7)
Proof By definition, a random function Z(t) is wide-sense stationary one, if it holds
the following:
#
%
E Z(t) = const,
#
%
cov Z(t), Z(t + τ ) = K(τ ).
(8)
π
π
π
, we hold ρ (φ + π) = ρ φ + +
=
Taking into account ρ(φ) = ρ φ +
2
2
2
π
ρ φ+
= ρ(φ), and in such a way, we check the conditions (8).
2
Since sin(φ + π) = − sin φ, cos(φ + π) = − cos φ, and the random phases and
random amplitudes are mutually independent, we set
∞ #
#
%
#
% %
E Z(t) = E
ηn cos ωn t + φn + νn sin ωn t + φn
=
n=1
∞
∞
% %
# % #
# % #
E ηn E cos ωn t + φn +
E νn E sin ωn t + φn .
=
n=1
n=1
Applying the series expansions for sine and cosine for the π/2− periodic function
ρ(φ) yields a result:
π
2π
#
% 2π
E cos φn =
cos φn ρ(φn )dφn =
cos φn ρ(φn )dφn +
cos φ ρ(φn )dφn =
π
0 π
0π
π
=
cos φn ρ(φn )dφn +
cos(φ + π) ρ(φn + · 2)dφn =
2
0
0
π
π
cos φn ρ(φn )dφn −
cos φn ρ(φn )dφn = 0.
=
0
0
248
E. Karachanskaya
In such a way, we have E [sin φn ] = 0 and, consequently,
E [Z(t)] = const = 0.
Check the fulfillment of the second condition. Using the result above and mutual
independence for random phases, it holds the following:
∞
# % #
%
#
% E ηn2 E cos φn + ωn (t + τ ) cos φn + ωn t +
cov Z(t), Z(t + τ ) =
n=1
∞
# % #
%
+
E νn2 E sin φn + ωn (t + τ ) sin φn + ωn t .
n=1
(9)
It is necessary to calculate E[sin 2φn ] and E[cos 2φn ]. Since ρ(φ) is the π/2−
periodic function, we set the following:
E[sin 2φn ] = 0, E[cos 2φn ] = 0.
Substituting the results above into (9) yields the following:
∞
∞
# % 1 # %
#
% 1 cos(ωn τ )E ηn2 +
sin(ωn τ )E νn2 = K(τ ).
cov Z(t), Z(t + τ ) =
2
2
n=1
n=1
All conditions (8) are fulfilled. The process Z(t) under conditions (6) and (7) is
stationary in the wide sense.
4 Construction of the Phase Density Function that Provides a
Wide-Sense Stationarity of a Harmonic Process
We apply the correlated sums idea for construction of the phase density function for
the wide-sense stationary harmonic processes.
∞
be a random variables sequence with the value from
Theorem 2 Let φ̃β
β=1
the interval [0, 2π) and ρβ (φ̃; k) be the distribution density function. Assume that
ρβ (φ̃; k) is the π/2− periodic function. Then:
1. A random variables sequence {φn }∞
n=1 exists, and every distribution density
function ρ(n) is (π/2)− periodic one, and it is defined by formula
ρ(φn ) =
n
k=1
⎤
βk
1
⎣ak ·
ρβ (φ̃; k)⎦ , βk ∈ N,
βk
⎡
β=1
(10)
Random Harmonic Processes with New Properties
249
where ak is fixed real numbers independent on φ̃ or random
variables from the
segment [0; 1] with its own distribution law under condition nk=1 ak = 1, ak ∈
[0, 1].
$ 2π
π
2. ρβ (φ̃; k) = ρβ φ̃ +
; k ; 0 ρβ (φ̃; k)d φ̃ = 1.
2βk
3. Random process (1) is stationary in the wide sense.
Proof Firstly, prove that function (10) is π/2-periodic. Indeed, every function
ρβ (φ̃; k) is π/2-periodic, since
π
π
π ; k = ρβ φ̃ +
· βk ; k = ρβ φ̃ + ; k .
ρβ (φ̃; k) = ρβ φ̃ +
2βk
2βk
2
Hence, the period for the function ρβ (φ̃; k) equals π/2, and we set the following:
2π
ρ(φn )dφ =
0
=
n
k=1
ak ·
1
βk
2π n
0
k=1
β
k 2π
⎤
βk
1
⎣ak ·
ρβ (φ̃; k)⎦ d φ̃ =
βk
⎡
β=1
ρβ (φ̃; k)d φ̃ =
β=1 0
n
ak = 1.
k=1
The application of Theorem 1 completes this theorem.
Note that the 2π-periodic function g(x) =
k
f
s=1
2π
· s is (2π/k)x+
k
periodic one.
φ
and φ ∈ [0, 2π)
4π 2
is 2π-periodic. Then, we construct π/2-periodic function g(φ) (here, k = 4):
Let us consider an example. Suppose the function f (φ) =
g(φ) = f (φ +
π
2π
3π
4π
) + f (φ +
) + f (φ +
) + f (φ +
),
2
2
2
2
φ ∈ [0, 2π) .
Therefore, the function g(φ) is (see Fig. 1)
g(φ) =
φ
3
,
+
2π 2
8π
g(φ +
π
) = g(φ),
2
if
and
g(φ) = 0
if
φ∈
/ [0, 2π) ,
φ ∈ [0, 2π) ,
250
E. Karachanskaya
Fig. 1 The graph of the
(π/2)-periodic function g(φ)
has property
$which
+∞
−∞ g(φ) dφ =
$ 2π
0 g(φ) dφ = 1
and we can define the distribution density function, because
+∞
−∞
g(φ)dφ =
2π
g(φ)dφ = 1.
0
5 Ergodicity
It’s well known that ergodicity is an important property for practice.
We will investigate mean ergodic property for a random harmonic function Z(t).
It means
1 T
Z(t)dt = E[Z(t)].
lim
T →∞ T 0
Here, we use Slutsky’s ergodic theorem [8]; it is a sufficient condition for the
mean ergodic function.
Theorem 3 (Slutsky) For a wide-sense stationary random function Z(t) it holds
the relations
1
T →∞ T
T
lim
K(τ )dτ = 0, K(τ ) = cov [Z(t), Z(t + τ )] .
0
Theorem 4 Let (4) be a wide-sense stationary random harmonic process. Suppose
that:
1. An and Bm are non-centered and mutually independent for any n, m random
variables, and E[An ]2 < ∞ and E[Bn ]2 < ∞ for every n.
2. φn are random phases defined on the interval [0; 2π) and independent on An
and Bm , and its distribution
density function
to (10).
∞ is similar
2
2
3. the random series ∞
n=1 E[An ] and
n=1 E[Bn ] converge.
4. {(ωn )−1 }∞
n=1 is a monotone nonincreasing and boundary sequence.
Then, the process (4) is mean ergodic.
Random Harmonic Processes with New Properties
251
Proof According to Theorem 1, we set E [Z(t)] = 0.
Then, we obtain
1
T →∞ T
T
lim
1
= lim
T →∞ T
1
= lim
T →∞ 2T
T
0
+ ∞
n=1
K(τ )dτ =
0
∞
∞
1
1
2
2
E[An ] cos ωn τ +
E[Bn ] sin ωn τ dτ =
2
2
n=1
n=1
sin ωn T
E[A2n ]
ωn
−
∞
n=1
cos ωn T
E[Bn2 ]
ωn
+
∞
n=1
1
E[Bn2 ]
ωn
,
Using Abel-Dirichlet’s test for series in this expression and taking into account
assumption of this theorem, it follows that
1
T →∞ T
T
lim
K(τ )dτ = 0.
0
The proof is complete.
6 Conclusion
We have presented the stochastic hierarchically correlated series (SHCS) that serve
as generalization model for many harmonic processes, like different radio signals.
The new class of random harmonic processes with the property of stationarity in the
wide sense has been constructed. In addition, these random harmonic processes can
be mean ergodic. The concept of hierarchically correlated series makes it possible
to study a radio signal containing multi-noise mixture.
References
1. Doobko, V.A., Karachanskaya, E.V.: Classification and simulation of the Random Harmonic
Processes on the base of SHCS - series. Yakutian Math. J. 18, 36–54 (2011) (In Russian)
2. Kahane, J.-P.: Some random series of functions. In: Cambridge Studies in Advanced Mathematics, vol. 5. Cambridge University Press, Cambridge (1985)
3. Cook, Ch.E., Bernfield, M.: Radar Signals. An Introduction to Theory and Application.
Advanced Radar Studies Department Sperry Gyroscope Company, Great Neck, Long Island,
New York (1967)
4. Machehin, Yu.: Estimation of result of measurements of instability of frequency outcomes of
lasers, on a basis fractal properties diffusion processes. Sistemi obrobki informatsii 4(71), 139–
142 (2008) (In Russian)
252
E. Karachanskaya
5. Kolachevskiy, N.N.: Magnetic Noise. Nauka, Moscow (1971) (in Russian)
6. Antipenskiy, R.: Working out of the random signals models. Komponents Technol. 11, 146–151
(2007) (In Russian)
7. Yaglom, A.M.: An Introduction to the Theory of Stationary Random Functions. Dover Publications Inc., Mineola, NY (2004)
8. Yaglom, A.M.: Correlation Theory of Stationary and Related Random Functions. Vol. 1: Basic
Results. Springer, New York (1987)
On Solvability of One Nonlinear Integral
Equation Arising in Modelling
of Geographical Spread of Epidemics
A. Kh. Khachatryan
Abstract In the present work, one nonlinear convolution integral equation on the
whole line is considered. This equation arises in theory of temporal-spatial spread
of epidemics. For the above equation, the existence and uniqueness theorems are
proved. Based on theoretical convergence, results are applied to epidemic model
with diffusion term, describing geographical spread of infection diseases. At the end
of work for influenza epidemic, several numerical simulations are implemented.
Keywords Nonlinearity · Monotonicity · Existence · Uniqueness · Bounded
solution · Diffusion · Epidemics
Mathematical Subject Classification (MSC 2010) 45Gxx, 92Bxx
1 Introduction
We consider the following nonlinear convolution integral equation on the whole axis
∞
(z) = α
H (z − s)G((s), ϕ(s))ds, z ∈ R
(1)
−∞
with respect to the unknown function (z), satisfying the boundary conditions
lim (z) = λ1 ,
z→−∞
lim (z) = λ2 .
z→+∞
(2)
A. Kh. Khachatryan ()
Department of Higher Mathematics and Physics ANAU, Yerevan, Armenia
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_14
253
254
A. Kh. Khachatryan
Here,
G(, ϕ) = (1 − ϕ) − 2 ,
(3)
ϕ(z) is a continuous bounded monotonically nondecreasing given function with
properties:
0 ≤ ϕ(z) ≤ ε <
1
, z ∈ R,
3
(4)
lim ϕ(z) = 0,
lim ϕ(z) = ε.
(5)
z→−∞
z→+∞
We also assume that parameter α including in Eq. (1) belongs to the interval
α∈
2
1
,
.
1−ε 1+ε
(6)
The kernel H (z) is defined on R and satisfies conservativity condition
∞
H (z) ≥ 0, z ∈ R,
H (s)ds = 1.
(7)
−∞
Equation (1) arises in the theory of geographical spread of epidemics (Sect. 3). As
known, at present the problem of the spread of infectious diseases is one of the
important and rapidly developing branches of mathematical biology ([1–15] and
references therein).
Below, we prove the existence and uniqueness results for boundary value problem (1)–(2) (Sect. 2), and results are applied to the population problem modelled
by spatial-temporal spread of infection diseases with diffusion term (Sect. 3). At
the end of the work, several numerical results, based on experimented data, are
presented (Sect. 4).
2 Existence and Uniqueness of the Solution of Boundary
Value Problem (1)–(2)
Our main goal in this paragraph is to prove the existence and uniqueness of boundary
value problem (1)–(2).
Nonlinear Integral Equation Arising in Modelling of Geographical Spread of Epidemics
255
2.1 Existence of the Solution
Theorem 1 Let ϕ be the given continuous nondecreasing bounded function on R
with properties (4) and (5). Let conditions (6) and (7) be fulfilled. Then, Eq. (1) has
a continuous nonincreasing bounded solution on R. Moreover, the following limit
relations hold
lim (z) =
z→−∞
α−1
,
α
lim (z) =
z→+∞
α − 1 − αε
.
α
Proof We consider the following iteration for Eq. (1):
∞
n+1 (z) = α
#
%
H (z − t) (1 − ϕ(t))n (t) − 2n (t) dt, z ∈ R,
(8)
−∞
0 (z) =
α−1
,
α
n = 0, 1, 2, . . . .
(9)
Applying the method of mathematical induction, below we prove that
n (z) ↓ in n,
n (z) ≥
(10)
α − 1 − αε
, z ∈ R, n = 0, 1, 2, . . . .
α
(11)
α − 1 − αε
, z ∈ R.
α
(12)
First, we verify that
1 (z) ≤ 0 (z) and 1 (z) ≥
Indeed, due to (4), (7), and (9) from (8), it follows
∞
1 (z) ≤ α
#
%
H (z − t) 0 (t) − 20 (t) dt
−∞
α − 1 (α − 1)2
=α
−
α
α2
Observe that inequality 0 (z) ≥
approximation.
∞
H (z − t)dt =
−∞
α−1
= 0 (z)
α
(13)
α − 1 − αε
immediately follows from zero
α
256
A. Kh. Khachatryan
Now, we show that
1 (z) ≥
α − 1 − αε
, z ∈ R.
α
(14)
From (8), due to nonnegativity of function 0 (z), we get
∞
1 (z) ≥ α
#
%
H (z − t) (1 − ε)0 (t) − 20 (t) dt, z ∈ R.
(15)
−∞
We consider the following function:
1−ε
χ(u) = (1 − ε)u − u , u ∈ 0,
,
2
2
%
#
,
and taking into account the monotonicity of it with respect to u on interval 0, 1−ε
2
from (15) we obtain
(α − 1 − αε) (α − 1 − αε)2
α − 1 − αε
1 (z) ≥ α (1 − ε)
−
,
=
α
α2
α
(16)
since 0 ∈ 0, 1−ε
.
2
for some n ∈ N. Due to
Assume that n (z) ≤ n−1 (z) and n (z) ≥ α−1−αε
α
%
#
1
2
we obtain
α ∈ 1−ε , 1+ε , n (z) ≤ 0 (z), and χ(u) ↑ on interval 0, 1−ε
2
∞
n+1 (z) ≤ α
#
%
H (z − t) (1 − ϕ(t))n−1 (t) − 2n−1 (t) dt = n (z)
(17)
−∞
and
∞
n+1 (z) ≥ α
#
%
H (z − t) (1 − ε)n (t) − 2n (t) dt
−∞
≥ (α(1 − ε) − α + 1 + αε)
α − 1 − αε
α − 1 − αε
=
, z ∈ R.
α
α
(18)
By induction, it is easy to check that
n ∈ C(R), n = 0, 1, 2, . . . .
(19)
Nonlinear Integral Equation Arising in Modelling of Geographical Spread of Epidemics
257
Last inclusion can be shown using the fact that convolution operator for integrable
and bounded function is a continuous function [16]. Now, we prove that
n (z) ↓ in z on R, n = 0, 1, 2, . . . .
(20)
In monotonicity of zero approximation, it follows immediately from definition.
Assume that n (z) ↓ in z on R for some n ∈ N. Representing the iteration (8)
in the form
∞
n+1 (z) = α
#
%
H (t) (1 − ϕ(z − t))n (z − t) − 2n (z − t) dt,
−∞
(21)
α−1
0 (z) =
, n = 0, 1, 2, . . . , z ∈ R,
α
taking into account monotonicity of the function ϕ and χ and induction assumption,
from (21) we conclude that n+1 (z) is decreasing in z on R.
Thus, the sequence of continuous functions {n (z)}∞
n=0 has a pointwise limit as
n → +∞
lim n (z) = (z).
n→∞
(22)
Observe that from (19), (20), and (11), it follows
•
(z) ↓ in z on R.
(23)
•
α−1
α − 1 − εα
≤ (z) ≤
, z ∈ R.
α
α
(24)
In accordance with B. Levi’s theorem [17], limit function (z) satisfies Eq. (1).
Using the continuity of convolution operation for summable and bounded function
from (19), we have
∈ C(R).
(25)
Next, we consider the asymptotic behavior of solution at ±∞. In view of (23)–(25),
it follows that there exist
α−1
,
α
(26)
α − 1 − αε
.
α
(27)
lim (z) = λ1 ≤
z→−∞
lim (z) = λ2 ≥
z→+∞
258
A. Kh. Khachatryan
Using the following well-known limit relation [18]
∞
∞
H (z − t)F (t)dt = F (±∞)
lim
z→±∞
−∞
H (t)dt
(28)
−∞
and also condition (5), we obtain
λ1 = lim (z) = α(λ1 − λ21 ),
(29)
λ2 = lim (z) = α (1 − ε)λ2 − λ22 .
(30)
z→+∞
z→+∞
So,
λ1 =
α−1
,
α
λ2 =
α − 1 − αε
.
α
(31)
Thus, the theorem is proved.
Remark 1 If we replace property (5) of the given continuous bounded monotonically nondecreasing function ϕ(z) with
lim ϕ(z) = ε,
z→−∞
lim ϕ(z) = 0,
z→+∞
(32)
we may similarly state that
•
•
•
α−1
α − 1 − αε
≤ (z) ≤
.
α
α
(z) ↑ in z on R.
lim (z) =
z→+∞
α−1
α − 1 − αε
, lim (z) =
.
z→−∞
α
α
(33)
(34)
(35)
2.2 Uniqueness of the Solution
In this section, we prove that the solution of Eq. (1) is unique and bounded and
belongs to the following class of functions:
M = f ∈ C(R) :
α−1
α − 1 − αε
≤ f (z) ≤
, z∈R .
α
α
(36)
Nonlinear Integral Equation Arising in Modelling of Geographical Spread of Epidemics
259
%
1
Theorem 2 Let conditions ε ∈ [0, 16 ), α ∈ 1−2ε
, 32 , and (7) be satisfied. Then,
Eq. (1) has a unique solution in the class of functions M.
Proof Suppose the contrary. Let Eq. (1) have two solutions 1 (z) and 2 (z), j ∈
M, j = 1, 2. In view of (1) and (3), we have
|1 (z) − 2 (z)|
∞
≤α
H (z − t) |1 (t) − 2 (t)| |1 − ϕ(t) − 1 (t) − 2 (t)| dt.
(37)
−∞
Observe
1 (t) + 2 (t) ≤ 1 − ϕ(t), t ∈ R.
(38)
Indeed, Inequality (38) immediately follows from the following simple estimates:
2
, t ∈ R,
3
2
2(α − 1)
≤ , t ∈ R,
1 (t) + 2 (t) ≤
α
3
%
1
since ε ∈ [0, 16 ), α ∈ 1−2ε
, 32 .
Denote
1 − ϕ(t) ≥ 1 − ε >
m := sup |1 (z) − 2 (z)| ≥ 0.
(39)
z∈R
From (37) and (39), we get
∞
m ≤ αm
−∞
2(α − 1 − αε)
dt = m(2 − α + 2αε).
H (z − t) 1 −
α
Last, inequality leads to
m [α(1 − 2ε) − 1] ≤ 0.
1
Since α > 1−2ε
, ε < 16 , and m ≥ 0, m = 0 and 1 (z) = 2 (z), z ∈ R. The
last means that Eq. (1) has a unique solution in M.
260
A. Kh. Khachatryan
3 On Mathematical Modelling of Geographical Spread of
Epidemics
In this section, we consider one mathematical epidemic model of spatial-temporal
spread of infection diseases. The results, obtained in the previous paragraph, we
apply to the above model.
3.1 Mathematical Model: Notations
Presently, there are numerous mathematical models describing the spread of
epidemic in population [1–5, 8, 9, 11, 13, 14]. Among them, an important role
plays model with diffusion term. Epidemic diseases are transmitted through contacts
between a susceptible and an infected individual, and the virus is propagated
diffusely.
The population is divided into three groups: Let S(t, x) be the number at time
t and location x of susceptible, those who can catch the disease; I (t, x) be the
number at time t and location x, those who are infected and can transmit the disease;
and, finally, R(t, x) be the number at position x and time t of removed individuals,
including isolated and dead people.
We also introduce the following notations. Let β be the transmission rate of
infection, representing the ratio of infected people to the susceptibles, and D be
the diffusion coefficient. By æ, we denote the removed rate from class R to the
class of susceptibles S, and by γ removal rate, representing the ratio of removed
individuals R to the infected class I . It is assumed that β and γ are constants.
Further, we assume that population size N is large and constant. Hereinafter, for
convenience, we will take N = 1, D = 1.
Under the abovementioned assumptions and notations, the corresponding system
of nonlinear differential equations with partial derivatives can be written as follows:
∂I (t, x)
∂ 2 I (t, x)
= βS(t, x)I (t, x) +
− γ I (t, x),
∂t
∂x 2
(40)
∂R(t, x)
∂ 2 R(t, x)
= −æR(t, x) +
+ γ I (t, x),
∂t
∂x 2
(41)
S(t, x) = 1 − R(t, x) − I (t, x).
(42)
In literature, this model is called SIRS model.
Nonlinear Integral Equation Arising in Modelling of Geographical Spread of Epidemics
261
3.2 Basic System of Nonlinear Integral Equations
The solution of the system of Eqs. (40) and (41) is sought in the form of travelling
wave:
I (t, x) = (x + ct) = (z),
(43)
R(t, x) = R(x + ct) = ϕ(z),
(44)
where c is the wave speed, z = x + ct.
Since and ϕ depend on a linear combination of the independent variables x
and t, then Eqs. (40) and (41) can be reduced to the following system of nonlinear
ordinary differential equations:
(z) − c (z) − γ (z) = −β (z) − 2 (z) − (z)ϕ(z) , z ∈ R,
ϕ (z) − cϕ (z) − æϕ(z) = −γ (z), z ∈ R.
(45)
(46)
We assume that the unknown functions and ϕ belong to the class of twice
differentiable continuous functions on R, i.e., , ϕ ∈ C 2 (R).
We complete Eqs. (45) and (46), giving the appropriate boundary conditions.
(−∞) = λ1 ,
(+∞) = λ2 ,
(47)
ϕ(−∞) = 0,
ϕ(+∞) = ε.
(48)
The boundary value problems (45)–(48) in turn can be reduced to the following
system of convolution-type nonlinear integral equations on whole line
∞
(z) = α
H (z − s)G((s), ϕ(s))ds,
(49)
−∞
γ
ϕ(z) =
æ
∞
H0 (z − s)ϕ(s)ds,
(50)
−∞
where parameter α = γβ > 1 is the so-called contact number and G is given by
Formula (3),
H (s) =
⎧
γ
⎪
eν1 s , if s ≤ 0,
⎪
⎪
2
⎪
⎨ c + 4γ
⎪
⎪
⎪
⎪
⎩
(51)
γ
c2 + 4γ
e ν2 s ,
if s > 0,
262
A. Kh. Khachatryan
H0 (s) =
⎧
æ
⎪
eμ1 s , if s ≤ 0,
√
⎪
⎪
2
⎨ c + 4æ
⎪
⎪
æ
⎪
⎩√
eμ2 s , if s > 0.
c2 + 4æ
(52)
ν2 < 0 < ν1 and μ2 < 0 < μ1 are the roots of the following characteristic
equations, respectively,
ν 2 − cν − γ = 0,
(53)
μ2 − cμ − æ = 0.
(54)
It is easy to check that
∞
H (s)ds = 1,
(55)
H0 (s)ds = 1.
(56)
−∞
∞
−∞
Note that Eq. (49) for a given ϕ coincides with Eq. (1).
Our main goal is to study and solve the system of nonlinear Eqs. (49) and (50)
with boundary conditions (47) and (48). Observe that the corresponding nonlinear
integral operator of the systems (49) and (50) does not pass the monotonicity
property. The latter makes it very difficult to construct a fixed point for the above
system. Below, we suggest the approximate (iterative) way to solve it.
3.3 Auxiliary Nonlinear Integral Equation
In this section, we will consider the so-called SIS model, which is simple comparing
with SIRS model discussed in the previous paragraph. In the framework of the SIS
model of epidemics, the corresponding equations can be written as follows:
∂ 2 I (t, x)
∂I (t, x)
= −βS(t, x)I (t, x) +
− γ I (t, x),
∂t
∂x 2
(57)
S(t, x) = 1 − I (t, x).
(58)
In fact, we divide the population into two classes, S and I , and for simplicity, we
assume (inexorable generality) that the removed rate from class I to class S is the
Nonlinear Integral Equation Arising in Modelling of Geographical Spread of Epidemics
263
same as the removed rate from class I to class R, considered in the previous SIRS
model.
From the solution of Eq. (57), we search in the form of travelling wave:
I (t, x) = I (x + ct) = y(z).
(59)
Equation (57) is easily reduced to the following boundary value problem:
y − cy − γ y = −β y − y 2 ,
(60)
y(−∞) = 0, y(+∞) = η,
(61)
which is in turn can be transformed into following integral equation:
∞
H (z − s)g(y(s))ds
(62)
β
(u − u2 ) = α(u − u2 ).
γ
(63)
y(z) =
−∞
with nonlinearity
g(u) =
Observe that Eq. (62) coincides with (49) if we take ϕ = 0 (first approximation).
Let η be the fixed point of the function g, i.e.,
g(η) = η.
(64)
From (63), it follows that
η=
Since g(u) =
α−1
β −γ
=
, (β > γ ).
β
α
(65)
β
(u − u2 ) monotonically increasing on u ∈ (0, 12 ), we have
γ
β ∈ (γ , 2γ ).
(66)
Notice that y = 0 and y = η satisfy Eq. (62) (trivial solutions). Our main goal is
to construct a nontrivial solution for Eq. (62), satisfying boundary conditions (61).
It should be noted that Eq. (62) with symmetric kernel in the class of bounded
functions has only trivial solutions y = 0 and y = η. Observe that kernel
H is asymmetrical (c > 0). The existence of a nontrivial solution for Eq. (62)
immediately follows from Diekman’s Theorem 6.1 (see [5]). Hereinafter, we’ll use
this theorem for our consideration.
264
A. Kh. Khachatryan
We introduce the Diekman’s function [5]:
∞
L(σ ) = g (0)
e−σ s H (s)ds, σ ≥ 0
(67)
−∞
and consider the characteristic equation
L(σ ) = 1.
(68)
Using representations (51) and (63), integrating (67), from (68) we get
σ1,2 =
c∓
c2 − 4(β − γ )
.
2
(69)
Observe that ν1 > max(σ1 , σ2 ) = σ2 , which provides convergence of integral (67).
The relation (69) contains the so-called threshold condition. If c > cm = 2(β − γ )
(minimum speed), then there is an epidemic and an epidemic wave passes through
the population; otherwise, (0 < c < cm ) the epidemic does not occur.
Let us consider the following iteration for Eq. (62):
∞
yn+1 (z) =
H (z − s)g(yn−1 (s))ds, z ∈ R,
(70)
−∞
as a zero approximation was chosen
y0 (z) =
η,
for z ≥ 0,
ηeσ1 z ,
for z < 0,
(71)
where σ1 ∈ (0, σ0 ) is the root of Eq. (68) and σ0 is the positive number for which
L(σ0 ) < 1.
(72)
The sequence of functions {yn (z)}∞
n=0 is monotonically decreasing. The sequence
of these functions is bounded from below by function (z):
yn (z) ≥ (z) =
η
ηeσ1 z − Me(σ1 +δ)z , for z ≤ 1δ ln M
,
0,
η
,
for z > 1δ ln M
(73)
Nonlinear Integral Equation Arising in Modelling of Geographical Spread of Epidemics
265
where
0 < δ < min{σ1 , σ0 − σ1 },
(74)
σ1 η
η2
;
.
σ1 + δ 1 − L(σ1 + δ)
M ≥ max
(75)
Since yn (z) is monotonically decreasing and bounded from below, there exists a
pointwise convergence:
lim yn (z) = y(z),
n→∞
where y(z) is the monotonically increasing solution of Eq. (62) and satisfies boundary condition (61). Several numerical simulations concerning with construction of
function y(z) are reported in Sect. 4.
3.4 Scheme of Solving a System of Eqs. (49) and (50)
Now we are ready to describe the approach for solving of system of nonlinear
integral Eqs. (49) and (50), satisfying the boundary conditions (47) and (48).
In Eq. (50), we replace the function (z) with y(z) (first approximation for ϕ(z)).
We have
∞
ϕ(z) =
H0 (z − s)y(s)ds, z ∈ R,
(76)
−∞
where y(z) is the solution of Eq. (62). From (76), it is obvious that function ϕ(z)
will “inherit” all the properties of function y(z).
1. Since y(z) ↑ in z on R, the function will also be monotonically increasing on R.
2. Taking into consideration (4), (28), (56), and (61) from (76), we get
∞
ϕ(+∞) = y(+∞)
H0 (s)ds = η
−∞
γ
= ε,
æ
(77)
∞
ϕ(−∞) = y(−∞)
−∞
Note that ϕ(z) satisfies conditions (4) and (5).
H0 (s)ds = 0.
(78)
266
A. Kh. Khachatryan
Now we substitute the already constructed solution ϕ(z) in (49). All conditions
of Theorem 1 are satisfied. So Eq. (49) has bounded nonincreasing solution and
satisfying double estimate
η − ε ≤ (z) ≤ η, z ∈ R
(79)
(−∞) = η, (+∞) = η − ε.
(80)
and limit relations
4 Numerical Results
In this section, we present several numerical results, which are based on real data.
For influenza epidemic, a blood test was performed among students during 1 year
at Yale University. It leads to a contact number estimate of α = γβ = 1, 5 (see
[19, 20]).
4.1 SIS Model
We start with Eq. (70). It should be noted that all approximations can be calculated
analytically. For the first iteration, we get
⎧
#
%
η
1−η
1
⎪
y1+ (z) = η + √ βη
,
for z ≥ 0,
eν2 z σ1 −ν
−
+
⎪
⎪
2σ
−ν
ν
2
2
1
2
2
⎪
c +4γ
⎪
⎪
⎨ −
βη
η
1
ν1 z 1−η +
y1 (z) = y1 (z) = √c2 +4γ e
ν1
σ1 −ν1 − 2σ1 −ν1 +
⎪
⎪
⎪
⎪
⎪
1
1
1
1
σ
z
2σ
z
1
1
⎪
, for z < 0.
⎩ +e
σ1 −ν2 − σ1 −ν1 + ηe
2σ1 −ν1 − 2σ1 −ν2
As for the initial parameters, we took β = 9, γ = 6, and c = 4. Then, from
relations (53), (63), (65), (69), and (51), we obtain
ν1,2 = 2 ±
√
1
3
10, g (0) = , η = , σ1 = 1, σ2 = 3,
2
3
⎧
⎨ √3 eν1 s , if s ≤ 0,
10
H (s) =
⎩ √3 eν2 s , if s > 0.
10
9
.
Minimization of function L(σ ) leads to σ0 = 2. So δ = 1, L(δ + σ1 ) = 10
Nonlinear Integral Equation Arising in Modelling of Geographical Spread of Epidemics
η2
η
,
M = max
2 1 − L(2)
(z) =
zmax =
10
,
9
=
z0 =
267
η
1
ln
= −1, 204,
δ M
1 z
10 2z
3e − 9 e ,
for z ≤ −1, 204,
0,
for z > −1, 204,
ησ1
1
ln
= −1, 897, max = 0, 025,
δ M(σ1 + δ)
y0 (z) =
1
3,
1 z
3e ,
for z ≥ 0,
(81)
for z < 0.
Recently in work [10], it was proved the following uniform estimate for iterations
(70):
n = e−(δ+σ1 )z |yn+1 (z) − yn (z)| ≤ α0 Ln (σ1 + δ),
(82)
where
α := sup e−(δ+σ1 )z |y1 (z) − y0 (z)| .
(83)
From (70) and (71), one can easily calculate the first iteration:
y1 (z) =
√
√
1
10+3 (2− 10)z
,
3 −
60 e √
√
1 z
e2z
10−3 (2+ 10)z
,
3 e − 10 −
60 e
if z ≥ 0,
if z < 0.
(84)
Taking into account (81), (84), and (82), we get
α0 = 0.102, 4 ≤
9
10
4
· α0 ≈ 0.066.
Observe that after the fourth iteration, the error is less than 6.6%.
In Fig. 1 is plotted the solution of Eq. (62) y(z) for the given parameters.
It can be easily proved that this solution is unique for the given cone segment
(z) ≤ y(z) ≤ y0 (z).
Fig. 1 Graph of the solution of Eq. (62) for the parameters β = 9 and γ = 6
268
A. Kh. Khachatryan
Nonlinear Integral Equation Arising in Modelling of Geographical Spread of Epidemics
269
4.2 SIRS Model
Now, we are ready to construct the function ϕ(z) using the solution y(z). We rewrite
Eq. (76) in the form of
ϕ (x) = √
0
γ
+
c2 + 4æ
∞
+
e
x
μ2 (x−t ) −
y (t)dt +
−∞
eμ2 (x−t )y + (t)dt
0
eμ1 (x−t )y + (t)dt ,
(85)
x
ϕ (x) = √
x
γ
−
c2 + 4æ
∞
+
e
0
μ2 (x−t ) −
y (t)dt +
−∞
eμ1 (x−t )y − (t)dt
x
eμ1 (x−t )y + (t)dt .
(86)
0
From (4), (77) follows that æ > 3γ η. As for æ, we choose æ = 7. Therefore, from
(54) and (52) we have
√
ηγ
11, ε =
= 0, 29,
æ
⎧
⎨ √3 eμ2 s , if s ≥ 0,
μ1,2 = 2 ±
H0 (s) =
⎩
11
√3 e μ1 s ,
11
if s < 0.
Notice that the intersection of sets (6) and (66) will be
β∈
2γ
γ
,
.
1−ε 1+ε
(87)
So one can be seen that all parameters are self-consistent.
Figure 2 shows the graphs of functions 0 ≤ ϕ(z) < ε and η − ε ≤ (z) ≤ η
which are determined via relations (86) and (49), respectively.
270
A. Kh. Khachatryan
Fig. 2 Graphs of the functions 0 ≤ ϕ(z) < ε and η − ε ≤ (z) ≤ η for the parameters β =
9, γ = 6, and æ = 7
Fig. 3 Graphs of the functions 0 ≤ ϕ(z) < ε and η − ε ≤ (z) ≤ η for the parameters β =
9, γ = 6, and æ = 14
In Fig. 3 are plotted graphs of the functions ϕ(z) and (z) for the parameters
β = 9, γ = 6, and æ = 14. Observe that solution (z) in this cone segment
[η − ε, η] is unique (see Theorem 2).
Nonlinear Integral Equation Arising in Modelling of Geographical Spread of Epidemics
271
Acknowledgments The author is grateful to Prof. Kh.A. Khachatryan for the useful discussions
and comments in preparing this paper.
The author thanks the GKN MON RA for the support in the framework of the project
№SCS 18T-1A004.
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1003061
A Simple Wiener-Hopf Factorization
Approach for Pricing Double-Barrier
Options
Oleg Kudryavtsev
Abstract This paper suggests a new approach to pricing double-barrier options
under pure non-Gaussian Lévy processes with jumps of finite variation. The key idea
behind the method is to represent the process under consideration as a difference
between two subordinators. We use such a splitting rule to the Lévy process at
exponentially distributed randomized time points. Then, we obtain the doublebarrier option price by solving recurrent simple Wiener-Hopf equations on the
interval defined by the barriers.
Keywords Wiener-Hopf factorization · Barrier options · Levy processes ·
Numerical methods · Laplace transform
1 Introduction
A standard option pricing problem in computational finance deals with the numerical methods to compute an expectation of a specific-type function G, which
depends on a stochastic process St modeling a stock price dynamic. If the payoff
function G depends not only on the stock price at the terminal date but also on its
observed trajectory, we obtain the typical setting for pricing path-dependent options
in finance.
The most popular path-dependent options are barrier options, which include
double-barrier options. Recall that a double-barrier option is a contract which pays
the specified amount G(ST ) at the terminal date T , provided during the lifetime of
the contract; the price of the stock does not cross specified constant barriers D from
O. Kudryavtsev ()
Russian Customs Academy, Rostov Branch, Rostov-on-Don, Russia
Southern Federal University, Rostov-on-Don, Russia
e-mail: koe@donrta.ru
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_15
273
274
O. Kudryavtsev
above and U from below. When at least one of the barriers is crossed, the option
expires worthless, or the option owner is entitled to some rebate.
In recent years, researchers pay much attention to stochastic models of financial
markets that depart from the classical Black-Scholes model. At this moment, a broad
range of models is available. We concentrate on the class of one-factor non-Gaussian
exponential Lévy processes that admit jumps in asset prices and can reproduce
the well-known volatility smile phenomena. For an introduction to Lévy models
in application to finance, we refer to [1, 2].
From a probabilistic viewpoint, double-barrier option prices can be expressed in
terms of conditional expectation on a payoff function that depends on the underlying
stochastic process and its supremum and infimum. In analytical terms, a doublebarrier derivative’s value is the solution to the Kolmogorov backward equation of a
particular-type subject to appropriate initial and boundary conditions. In the case of
Lévy models, one needs to solve complex partial integrodifferential equations where
a spatial variable belongs to the interval specified by the lower and upper barriers.
By now, several large groups of relatively universal numerical methods exist for
pricing single-barrier options under Lévy processes. However, the known results on
pricing double-barrier options are somewhat limited.
Existing numerical methods in literature can be categorized into three groups:
Monte Carlo simulations (see, e.g., [3–6]), backward induction methods (see,
e.g., [7–13]), and numerical methods for solving integrodifferential equations (see,
e.g., [14–20]). Standard Monte Carlo methods for pricing path-dependent options
under Lévy models are rather slow. Incorporating the Wiener-Hopf factorization
technique into the Monte Carlo algorithms leads to efficient pricing single-barrier
and lookback options. In [6], the joint distribution of a Lévy process and its
supremum and infimum under the Laplace transform was computed for a particular
case of a Lévy measure whose restriction on a positive half line has a rational
Laplace transform. In the general case, such results are unavailable.
The backward induction methods are based on the fact that the risk-neutral
valuation formula for the European option can be seen as a convolution of the payoff
function with the transition density. The key idea is to set up a time lattice and view
the option as of the European type between two adjacent dates. In the case of doublebarrier options, we need to consider boundary conditions at each time step.
As a theoretical background behind the procedure, the third group uses the
method of horizontal lines [21], which includes a time discretization while a
space variable remains continuous. Carr suggested a meaningful probabilistic
interpretation of the method in [22], which we call a time randomization technique
or Carr’s randomization.
After the time discretization, a sequence of specific stationary boundary problems
for integrodifferential equations on an interval arises. In the case of continuously
monitored options, one can also reduce the initial Kolmogorov backward equation
to a convolution one on an interval applying the Laplace transform in time variable
(see, e.g., [15, 17, 19]). To solve them, one may apply either finite difference
methods like in [14] or Wiener-Hopf factorization method to solve two coupled
integrodifferential equations (see, e.g., [15–17]). In both cases, the numerical
A Simple Wiener-Hopf Factorization Approach for Pricing Double Barrier Options
275
methods are rather more involved than in the single-barrier case and require
approximate formulas for the Wiener-Hopf factors (see, e.g., [23, 24]). Analytical
solutions can be found in a few Lévy models only (see, e.g., [19]). The doublebarrier problem is related to a matrix Wiener-Hopf factorization (see details in
[18, 25]), which is not analytically available yet. Therefore, pricing double-barrier
options in exponential Lévy models remains a computational challenge.
The goal of the current paper is to suggest a new, easy, and effective method to
price double-barrier options under pure non-Gaussian Lévy processes with jumps
of finite variation. The main advantage of the approach is applying explicit WienerHopf factorization formulas similar to the ones suggested in [26] for options with
a single barrier. The key idea behind our method is to represent the process under
consideration as a sequence of positive and negative jumps. We apply such splitting
rule to the process at consecutive exponentially distributed randomized time points
and reduce the problem to iterative solving of simple Wiener-Hopf equations on the
interval.
2 Theoretical Background
2.1 Lévy Processes and Wiener-Hopf Factorization: Basic
Facts
A Lévy process is a stochastically continuous process with stationary independent
increments (for general definitions, see, e.g., [27]). A Lévy model may have a
Gaussian component and pure jump component. The latter is characterized by the
density of jumps, which is called the Lévy density. A Lévy process Xt can be
completely specified by its characteristic exponent, ψ, definable from the equality
E[eiξ X(t )] = e−t ψ(ξ ) (we confine ourselves to the one-dimensional case).
The Lévy-Khintchine formula gives the characteristic exponent
ψ(ξ ) =
σ2 2
ξ − iμξ +
2
+∞
−∞
(1 − eiξy + iξy1[−1;1](y))F (dy),
(1)
where σ 2 ≥ 0 is the variance of the Gaussian component, 1A is the indicator
function of the set A, and the Lévy measure F (dy) satisfies
min{1, y 2 }F (dy) < +∞.
R\{0}
If the jump component is a process of finite variation, equivalently, if
min{1, |y|}F (dy) < +∞,
R\{0}
(2)
276
O. Kudryavtsev
then (1) can be simplified
σ2 2
ξ − iμξ +
ψ(ξ ) =
2
+∞
−∞
1 − eiξy F (dy),
(3)
with a different μ, and the new μ is the drift of the Gaussian component.
Assume that the riskless rate r is constant and, under a risk-neutral measure
chosen by the market, the underlying process evolves as St = S0 eXt , where Xt
is a Lévy process. Then, we must have E[eXt ] < +∞, and therefore, ψ must admit
the analytical continuation into the strip 0ξ ∈ (−1, 0) and continuous continuation
into the closed strip 0ξ ∈ [−1, 0].
Further, if d ≥ 0 is the constant dividend yield on the underlying asset, then
the following condition (the EMM-requirement) must hold: E[eXt ] = e(r−d)t .
Equivalently,
r − d + ψ(−i) = 0,
(4)
which can be used to express the drift μ via the other parameters of the Lévy process:
σ2
+
μ=r −d −
2
+∞
−∞
(1 − ey + y1[−1;1](y))F (dy),
(5)
where 1A (x) = 1 if x ∈ A, and otherwise, it is equal to 0.
In the examples below, we list some popular classes of Lévy processes in
empirical studies of financial markets.
Example 1 (Tempered Stable Lévy Processes) The characteristic exponent of a pure
jump KoBoL process of order ν ∈ (0, 2), ν = 1 is given by
ψ(ξ ) = −iμξ + cΓ (−ν)[λν+ − (λ+ + iξ )ν ]
(6)
+cΓ (−ν)[(−λ− )ν − (−λ− − iξ )ν ],
where c > 0, μ ∈ R, and λ− < −1 < 0 < λ+ . The characteristic exponent (6) is
derived in [2] from the Lévy-Khintchine formula with the Lévy densities of negative
and positive jumps, F∓ (dy), given by
F∓ (dy) = ceλ± y |y|−ν−1 dy;
(7)
in the first two papers, the name extended Koponen family was used. Later, the same
class of processes was used in [28] under the name CGMY model. The following
relations between parameters of KoBoL model and C, G, M, Y parameters of
CGMY model are valid:
C = c, Y = ν, G = λ+ , M = −λ− .
A Simple Wiener-Hopf Factorization Approach for Pricing Double Barrier Options
277
More general version with c± instead of c and the different exponents ν± is
known as a tempered stable Lévy model [1]. In this case, we have for ν+ , ν− ∈
(0, 2), ν+ , ν− = 1
ν
ψ(ξ ) = −iμξ + c+ Γ (−ν+ ) λ++ − (λ+ + iξ )ν+
+c− Γ (−ν− )[(−λ− )
ν−
(8)
− (−λ− − iξ ) ],
ν−
where c+ , c− > 0, μ ∈ R, and λ− < −1 < 0 < λ+ .
Example 2 (Variance Gamma Processes) The Lévy density of a variance gamma
process is of the form (7) with ν = 0, and the characteristic exponent is given by
(see [29])
ψ(ξ ) = −iμξ + c[ln(λ+ + iξ ) − ln λ+ + ln(−λ− − iξ ) − ln(−λ− )],
(9)
where c > 0, μ ∈ R, and λ− < −1 < 0 < λ+ .
Example 3 (Kou Model) If F∓ (dy) are given by exponential functions on negative
and positive axis, respectively,
F∓ (dy) = c± (±λ± )eλ± y ,
where c± ≥ 0 and λ− < 0 < λ+ , then we obtain Kou model. The characteristic
exponent of the process is of the form
ψ(ξ ) =
ic− ξ
ic+ ξ
σ2 2
ξ − iμξ +
+
.
2
λ+ + iξ
λ− + iξ
(10)
The two-sided version was introduced in [30].
Example 4 (β-Class) The β-class of Lévy processes, introduced in [31], is a tenparameter Lévy process which has a characteristic exponent
1
ψ(ξ ) = aξ + σ 2 ξ 2
2
c1
iξ
+
B(α1 , 1 − λ1 ) − B α1 − , 1 − λ1
β1
β1
c2
iξ
+
B(α2 , 1 − λ2 ) − B α2 + , 1 − λ2
β2
β2
278
O. Kudryavtsev
with parameter range a, σ ∈ R, c1 , c2 , α1 , α2 , β1 , β2 > 0 and λ1 , λ2 ∈ (0, 3) \
{1, 2}. Here, B(x, y) = Γ (x)Γ (y)/Γ (x + y) is the beta function. The density of
the Lévy measure is given by
π(x) = c1
e−α1 β1 x
eα2 β2 x
1
+
c
1{x<0} .
{x>0}
2
(1 − e−β1 x )λ1
(1 − eβ2 x )λ2
2.2 Wiener-Hopf Factorization
There are several forms of the Wiener-Hopf factorization. Let X t = sup0≤s≤t Xs
and X t = inf0≤s≤t Xs be the supremum and infimum of Lévy process Xt . The
Wiener-Hopf factorization formula used in probability reads
iξ X E eiξ XTq = E eiξ XTq E e Tq ,
∀ ξ ∈ R,
(11)
where Tq ∼ Exp q is an exponentially distributed random variable and independent
of X. Notice that the following useful relations hold (see details in [27]):
XTq ∼ XTq − X Tq
(12)
XTq ∼ XTq − XTq .
(13)
One can easily show that the r.h.s. of (11) can be represented in terms of the
characteristic exponent ψ(ξ ) of Xt as follows:
E eiξ XTq = qE
∞
e
−qt iξ Xt
e
0
dt = q
∞
e−qt E[eiξ Xt ]dt =
0
q
.
q + ψ(ξ )
Introducing the notation
φq+ (ξ ) = qE
φq− (ξ ) = qE
∞
%
#
e−qt eiξ Xt dt = E eiξ XTq ,
0
#
%
iξ X
e−qt eiξ Xt dt = E e Tq ,
∞
0
we can rewrite (11) as
q
= φq+ (ξ )φq− (ξ ).
q + ψ(ξ )
(14)
A Simple Wiener-Hopf Factorization Approach for Pricing Double Barrier Options
279
If XTq = XTq , then X Tq = 0, and we obtain a trivial factorization identity:
φq− (ξ ) =
q
,
q + ψ(ξ )
φq+ (ξ ) = 1.
(15)
In another particular case, when X Tq = XTq , we have X Tq = 0, and the factorization
factors have a simple form:
φq+ (ξ ) =
q
,
q + ψ(ξ )
φq− (ξ ) = 1.
(16)
In order to write the factorization identity (11) in operator form, we define the
normalized resolvent of X for a function u(x) as
Eq u(x) = E[u(x + XTq )].
(17)
In [32], Eq is referred as the expected present value operator (EPV operator) due to
the observation that, for a stream u(Xt ),
Eq g(x) = E
+∞
qe
−qt
u(x + Xt )dt .
(18)
0
Replacing in (18) process X with the supremum X and infimum X of process X, we
obtain the EPV operators E±
q under X and X, respectively. Equivalently,
E+
q u(x) = E u(x + X Tq ) ,
E−
q u(x) = E u(x + X Tq ) .
(19)
(20)
It follows from the definition that
+
E+
q 1(−∞,h) (·) = 1(−∞,h) (·)Eq 1(−∞,h) (·);
(21)
−
E−
q 1(0,+∞) (·) = 1(0,+∞) (·)Eq 1(0,+∞) (·).
(22)
The operator form of the Wiener-Hopf factorization is written as follows (see
details in [32]):
−
− +
Eq = E+
q Eq = Eq Eq .
(23)
Note that (23) are understood as equalities for operators in appropriate function
spaces, for instance, in the space of semi-bounded Borel functions. Under appropriate conditions on the characteristic exponent, the EPV operators are defined as
operators in spaces of functions of exponential growth at infinity, and (23) holds in
these spaces.
280
O. Kudryavtsev
Let Z be a continuous random variable with probability density. Then, for a wide
class of functions u(x), the following useful relation holds
E[u(x + Z)] = (2π)−1
+∞
−∞
#
%
eixξ E eiξ Z û(ξ )dξ,
(24)
where û is the Fourier transform of a function u:
+∞
û(ξ ) =
e−ixξ u(x)dx.
−∞
−
Hence, we can interpret Eq , E+
q , andEq as pseudo-differential operators (PDO)
q
+
−
with the symbols q+ψ(ξ ) , φq (ξ ), andφq (ξ ), respectively. Denote by Dx the difd
. Recall that a PDO a(Dx ) with the symbol a(ξ ) acts as
ferential operator −i dx
follows:
+∞
−1
a(Dx )u(x) = (2π)
eixξ a(ξ )û(ξ )dξ,
(25)
−∞
−1 is the
Notice that Dx = Fξ−1
→x ξ Fx→ξ , where F is the Fourier transform and F
inverse Fourier transform:
+∞
−1
−1
eixξ û(ξ )dξ = u(x).
(F û)(x) = (2π)
−∞
Hence, one can represent (25) shortly as Fξ−1
→x a(ξ )Fx→ξ u(x).
±
Taking into account that Eq = q(q + ψ(Dx ))−1 and E±
q = φq (Dx ), we may
write (17) and (19)–(20) as
−1
Eq u(x) = Fξ−1
→x q(q + ψ(ξ )) Fx→ξ u(x);
(26)
−1
±
E±
q u(x) = Fξ →x φq (ξ )Fx→ξ u(x).
(27)
If φq± (ξ ) are known explicitly, then each operator on the rightmost part of (26) and
(27) can be numerically implemented by means of the fast Fourier transform (FFT).
3 Splitting Rule and Wiener-Hopf Factorization
Let T , K, D, and U be the maturity, strike, lower barrier, and upper barrier, and
the stock price St = DeXt be an exponential Lévy process under a chosen riskneutral measure (see (4)) which has no diffusion component (σ = 0) and only
jumps of finite variation (see (2)). Without loss of generality, we confine ourselves
A Simple Wiener-Hopf Factorization Approach for Pricing Double Barrier Options
281
to a double-barrier put option. Set the riskless rate and the dividend rate equal to
r and d, respectively. As a basic example to illustrate our method, we consider
an approach to pricing continuously monitored double-barrier put options without
rebate under a Lévy process with the characteristic exponent (3) that satisfies (2).
3.1 The Problem Setup and General Pricing Formulas
Let us introduce h = ln U/D. Then, the payoff at maturity is 1(0,h)(XT )G(XT ),
where G(x) = (K − Dex )+ , and the no-arbitrage price of the double-barrier put
option at the beginning of a period under consideration (t = 0) and Xt = x with
x ∈ (0, h) given by
#
%
V (T , x) = E x e−rT 1XT >0 1XT <h G(XT ) ,
(28)
where T is the final date and X t = inf0≤s≤t Xt and X t = inf0≤s≤t Xt are
the infimum and the supremum of the process Xt , respectively. The short-hand
notation E x [f (Xt , Xt , X t )]] means that we take the expectation of the function
f (x + Xt , x + X t , x + Xt ).
Denote by V̂ (q, x) the Laplace transform of V (T , x) w.r.t. T . Applying Fubini’s
theorem, we obtain that V̂ (q, x) is the discounted expected value of the payoff
function G(Xt )1Xt >0 1Xt <h at exponentially distributed time Tq+r :
V̂ (q, x) =
+∞
0
=E
x
+∞
e
= Ex
e−qt E x e−rt G(Xt )1Xt >0 1Xt <h dt
−(q+r)t
0
G(XTq+r )
q +r
G(Xt )1Xt >0 1Xt <h dt
(29)
1XTq+r >0 1XT
q+r <h
.
(30)
If we can compute the expectations similar to (30) efficiently, we can recover the
initial function V (T , x) by using various Laplace transform inversion techniques.
Let us recall the Post-Widder formula (see, e.g., [33]) for inverting the Laplace
transform. If f (t) is$ a nonnegative function of a real variable t and it’s Laplace
∞
transform f˜(q) = 0 e−qt f (t) dt is known, then the approximate Post-Widder
formula for f (t) reads as follows (see, e.g., [33])
f (t) = lim fN (t);
N→∞
(−1)N
fN (t) :=
N!
N
t
N+1
(31)
f˜(N)
N
t
,
(32)
282
O. Kudryavtsev
where f˜(N) (q)—Nth derivative of f˜ at the point q.
For applying the Post-Widder formula to the transformed function V̂ (q, x), we
need to find functions of the kind (32). Differentiating n − 1 times the expression
n−1 q n
(29) w.r.t q and multiplying it by (−1)
(n−1)! , we obtain
vn (q, x) :=
+∞
(−1)n−1 q n n−1
V̂ (q, x)
∂
(n − 1)! q
(33)
q n t n−1 −(q+r)t x e
E G(Xt )1Xt >0 1Xt <h dt
(n − 1)!
0
x G(XT(n,q+r) )
=E
1XT(n,q+r) >0 1XT(n,q+r) <h ,
(1 + r/q)n
=
(34)
where T(n, q) is a gamma random variable with the shape parameter n > 0 and the
rate parameter q > 0.
Taking into account that
T(n, q + r) ∼ T(n − 1, q + r) + Tq+r ,
we can represent XT(n,q+r) as a sum
n ,
XT 1 + . . . + XTq+r
q+r
1 ,. . . ,T n are consecutive time increments being independent exponenwhere Tq+r
q+r
tially distributed random variables with the parameter q + r. Using the relations
1XT(n,q+r) >0 = 1XT(n−1,q+r) >0 1XT(n−1,q+r) +XT n
>0 ,
1XT(n,q+r) <h = 1XT(n−1,q+r) <h 1XT(n−1,q+r) +X n
,
q+r
and
Tq+r <h
we obtain that for n = 1, 2, . . .,
vn (q, x) = E x
vn−1 (q, XTq+r )
(1 + r/q)
1XTq+r >0 1XT
q+r <h
,
(35)
where v0 (q, x) = G(x)1(0,h)(x) and the random time Tq+r ∼ Exp (q + r).
Let N be a sufficiently large natural number. Once we computed the sequence
(35) for q = T /N recurrently, we obtain vN (N/T , x). Further, we see that due to
the Post-Widder formula, for a fixed x, vN (N/T , x) converges to V (T , x) as N →
+∞. The procedure (35) with q = N/T is equivalent to Carr’s randomization,
A Simple Wiener-Hopf Factorization Approach for Pricing Double Barrier Options
283
introduced in [22] for the case of American call options. Later, it was generalized
for a wider class of stochastic control problems in paper [34].
Recall that the main idea of Carr’s randomization is in assuming the expiry of the
option to be random. If we suppose that the maturity of the double-barrier option is
gamma distributed with the shape parameter N and the rate parameter N/T , then
we obtain a randomized analog of (28):
V (T , x) ≈ E x e−rT(N,q) G(XT(N,q) )1XT(N,q) >0 1XT(N,q) >h ,
(36)
where q = N/T . Notice that the randomized time T(N, q) is chosen since
it converges in quadratic mean to T as N → +∞, E[T(N, q)] = T , and
V ar[T(N, q)] = T 2 /N.
It is easy to show that vN (q, x) defined by (34) is exactly the right-hand side of
(36). Hence, we see that the randomized option price V (T , x) converges to V (T , x)
as N → +∞ as well.
In [23], the technique was successfully used for pricing single-barrier options in
Lévy models. Proofs of Carr’s randomization convergence in the case of problems
with a single barrier for Lévy processes are presented in [35, 36]. Notice that in the
current paper, we applied a different technique for a double-barrier case.
3.2 Splitting Procedure
The state-of-art implementation of the Wiener-Hopf method for pricing singlebarrier options (see, e.g., [15, 17, 23]) leads to the factorization (14) of (q + r)/(q +
r + ψ(ξ )), where ψ(ξ ) is the characteristic exponent of the Lévy process Xt . Then,
using (12)–(13) and (17)–(20), one can calculate the sequence (35) with q = N/T
and h = +∞ as follows: for n = 1, . . . , N,
vn (q, x) =
1
·
(1 + r/q)
E x [vn−1 (q, (XTq+r − X Tq+r ) + X Tq+r )1XTq+r >0 ]
=
1
E− 1(0,+∞) E+
q+r vn−1 (q, x).
(1 + r/q) q+r
(37)
Unfortunately, in the case of double-barrier options (with h < +∞), the formulas
similar to (37) are not available for general Lévy models. Instead, one needs to
factorize the following matrix:
exp(iξ h)
0
(q + r + ψ(ξ ))/(q + r) exp(−iξ h),
284
O. Kudryavtsev
(see, e.g., [25])) or to find an iterative solution to the pair of certain coupled integral
equations (see details in [13, 15]).
The new approach to calculating (35) requires the following steps. First, we
represent Xt as a difference between two subordinators Xt+ and −Xt− :
Xt = Xt+ − (−Xt− ).
Recall that a subordinator is a Lévy process with sample paths being almost
surely nondecreasing. According to [1, Proposition 3.10], a subordinator has no
diffusion component, only positive jumps of finite variation and nonnegative drift.
Denote by ψ+ (ξ ) and ψ− (ξ ) the characteristic exponents of Xt+ and Xt− ,
respectively. If μ ≥ 0 in (3), we define ψ+ (ξ ) and ψ− (ξ ) as follows:
ψ+ (ξ ) = −iμξ +
ψ− (ξ ) =
0
−∞
+∞
(1 − eiξy )F (dy),
0
(1 − eiξy )F (dy),
otherwise,
ψ+ (ξ ) =
+∞
1 − eiξy F (dy),
0
ψ− (ξ ) = −iμξ +
0
−∞
1 − eiξy F (dy).
Notice that Xt− has almost surely nonincreasing sample paths. Hence, we have that
+
Xt = Xt+ ,
(38)
−
X−
t = Xt .
(39)
Let Xt+,1 and Xt+,2 be Lévy processes with the same characteristic exponent
ψ+ (ξ ), i.e., Xt+,1 ∼ Xt+ and Xt+,2 ∼ Xt+ . Due to the property of increments of a
−
Lévy process being stationary independent, we conclude that Xt and Xt+,1
/2 + Xt +
Xt+,2
/2 are identically distributed. It means that for a fixed t > 0, the current position
of Xt with starting point x has the same distribution as the final position of the
discrete-time process Yjt with the following dynamics:
•
•
•
•
Y0t = x.
Y1t = Y0t + Xt+,1
/2 —an upward movement.
−
t
t
Y2 = Y1 + Xt —a downward movement.
Y3t = Y2t + Xt+,2
/2 —an upward movement.
A Simple Wiener-Hopf Factorization Approach for Pricing Double Barrier Options
285
For a short time period [0, t], we may approximate the value of X and X at a
given time t by the correspondent maximum and minimum of Y t at the time j = 3.
Let us denote by Y t = min{Y0t , Y1t , Y2t , Y3t } and Y t = max{Y0t , Y1t , Y2t , Y3t }. Notice
that extrema of Y can be easily defined as follows:
−
Y t = min x, x + Xt+,1
/2 + Xt ,
+,1
+,2 −
Y t = max x, x + Xt+,1
/2 , x + Xt /2 + Xt + Xt /2 .
(40)
Let a natural number N be sufficiently large and q = N/T . As explained in
Sect. 3.1, vN (N/T , x) defined iteratively by (35) gives an approximate price of
the double-barrier option V (T , x) (see (28)). For computing (35), we approximate
XTq+r in (35) with Y Tq+r , since the randomized time Tq+r converges in quadratic
mean to 0 as N → +∞. Notice that in this case, the following relations hold:
Y3 q+r = x + XT+,1
+ XT−q+r + XT+,2
;
q+r /2
q+r /2
T
(41)
1Y Tq+r >0 = 1(0,+∞)(x)
× 1(0,+∞) (x + XT+,1
+ XT−q+r );
q+r /2
(42)
1Y Tq+r <h = 1(−∞,h) (x) × 1(−∞,h) (x + XT+,1
)
q+r /2
× 1(−∞,h) (x + XT+,1
+ XT−q+r + XT+,2
).
q+r /2
q+r /2
(43)
In some sense, we may consider such approximation as an analog to the operator
splitting method suggested in [7], where the backward jump-diffusion PIDE for
option prices is solved by splitting the related operator into the differential, positive
and negative jump parts. Recall that the fundamental idea behind the approach in
[7] involves representing a jump operator as a PDO with subsequent transforming
into the correspondent matrix exponential.
Introduce the following operators:
+
E+ u(x) = E u(x + X Tq+r /2 ) ,
E− u(x) = E u(x + X −
Tq+r ) .
(44)
(45)
Notice that Tq+r /2 is also an exponentially distributed random variable but with the
intensity parameter equal to 2(q + r).
Set
iξ X+
iξ X− φ+ (ξ ) = E e T2(q+r) , φ− (ξ ) = E e Tq+r .
286
O. Kudryavtsev
Taking into account (38)–(39), we obtain that XT+2(q+r) and XT−q+r admit trivial
factorizations described above in Sect. 2.2. Hence, we have the following (see (15)
and (16) for clarifications):
2(q + r)
,
2(q + r) + ψ+ (ξ )
q+r
φ− (ξ ) =
.
q + r + ψ− (ξ )
φ+ (ξ ) =
(46)
(47)
Now, we can rewrite the operators E+ and E− in (44)–(45) with symbols (46)–(47)
as follows:
E+ u(x) = E u(x + XT+2(q+r) )
+∞
eixξ 2(q + r)/(2(q + r) + ψ+ (ξ ))û(ξ )dξ,
= (2π)−1
(48)
= Fξ−1
→x φ+ (ξ )Fx→ξ u(x),
E− u(x) = E u(x + XT−q+r )
+∞
eixξ (q + r)/(q + r + ψ− (ξ ))û(ξ )dξ,
= (2π)−1
(49)
−∞
−∞
= Fξ−1
→x φ− (ξ )Fx→ξ u(x).
Due the relations (41)–(43), we may approximate vn (q, x) in (35) as follows:
vn (q, x) ≈
=
%
#
1
T
E vn−1 q, Y3 q+r 1Y Tq+r >0 1Y Tq+r <h
(1 + r/q)
1(0,h)(x)
·
(1 + r/q)
#
+,2
−
E x vn−1 q, XT+,1
+
X
+
X
Tq+r
Tq+r /2 1X +,1
q+r /2
·1X+,1
%
−
Tq+r /2 +XTq+r >0
=
Tq+r /2 <h
1X+,1
+,2
−
Tq+r /2 +XTq+r +XTq+r /2 <h
1(0,h)(x) +
E 1(−∞,h) E− 1(0,+∞) E+ 1(−∞,h) vn−1 (q, x).
(1 + r/q)
(50)
It follows from (21)–(22) that (50) can be rewritten in the following form:
vn (q, x) =
1(0,h)(x) +
E 1(0,h)E− 1(0,h) E+ vn−1 (q, x).
(1 + r/q)
(51)
A Simple Wiener-Hopf Factorization Approach for Pricing Double Barrier Options
287
Notice that in (51), we took into account that 1(−∞,h) vn−1 (q, x) = vn−1 (q, x).
Summing up, we conclude that the formula (51) leads to consecutive evaluations
of simple Fourier integrals similar to (48) or (49) with functions u(x) supported on
(0, 1).
The EPV operators E− and E+ can be efficiently implemented by using the fast
Fourier transform (FFT) for real-valued functions. Recall that the discrete Fourier
transform (DFT) is defined by
Fl = DF T [f ](l) =
M−1
fk e2πikl/M ,
l = 0, . . . , M − 1.
k=0
The inverse DFT recovers the set of fk s exactly from Gl s. The correspondent
formula reads:
1 fk = iDF T [F ](k) =
Fl e−2πikl/M ,
M
M−1
k = 0, . . . , M − 1.
l=0
If the data consist of a real-valued array {fk }M
k=0 , then the resulting transform
satisfies FM−l = F̄l . Since F0 and FM/2 are real, the transformed complex-valued
array has the same “ degrees of freedom” as the original real data set {fl }. In this
case, it is efficient to use FFT algorithm for real-valued functions (see [37] for
technical details). To distinguish the DFT of real functions, we will use notation
RDFT.
Fix the number of the space points M = 2m and the spatial step h > 0. Define the
Mh
Mh
partitions of the normalized log-price domain [− Mh
2 ; 2 ) by points xk = − 2 +kh
π π
and k = 0, . . . , M − 1 and the frequency domain [− h ; h ] by points ξl = 2πl
hM and
l = −M/2, . . . , M/2. Then, the Fourier transform of a function u on the real line
can be approximated as follows:
û(ξl ) ≈ h(−1)l RDF T [u](l),
l = 0, . . . , M/2.
Here and below, z denotes the complex conjugate of z. Using the notation (46)–(47),
we can approximate E± as follows:
(E± u)(xk ) = iRDF T [ φ± . ∗ RDF T [u]](k), k = 0, . . . , M − 1,
where .∗ is the element-wise multiplication of arrays that represent the functions.
The main parameters of the algorithm for evaluating the function V (T , x) are the
number of time steps N, the number of space points M, and the scaling factor L
for the localization interval (−L ln(2); L ln(2)) (for more details about the choice
of the corresponding parameters of the algorithm, see [23, 24]). For a given M and
L, one can define the step h of the mesh by the formula h = L ln(4)
M .
288
O. Kudryavtsev
Finally, we conclude that the complexity of the algorithm for implementing (51)
using FFT is O(NM ln M). We will refer to the method developed as “the simple
Wiener-Hopf factorization method” (SWHF method).
Numerical experiments in [26] for the case of single-barrier options show that
the approach similar to the SWHF method can be a competitor for the efficient
algorithm developed in [23] that uses more accurate construction of an approximate
Wiener-Hopf factorization. We expect that the SWHF method for double-barrier
options will demonstrate good computational efficiency as well.
4 Numerical Experiments
In this section, we compare the performance of the suggested SWHF method and
other numerical pricing methods. As a basic example, we consider the knock-andout put option with the strike K, the lower barrier D, the upper barrier U , and time to
expiry T . We take the KoBoL model of order ν ∈ (0, 1), with the parameters σ = 0,
ν = 0.5, λ+ = 9, λ− = −8, and c = 1. As in [16], we choose the instantaneous
interest rate r = 0.03, time to expiry T = 0.1 year, the strike price K = 3500,
and the barriers D = 2800 and U = 4200. In this case, the drift parameter μ is
approximately −0.0423.
In numerical examples, we implemented the algorithm of the SFWH method
described in Sect. 3. For comparison, we choose a Monte Carlo method (MC
method) and the approximate Wiener-Hopf factorization method (AWHF method)
developed in [16]. In Table 1, we present the option prices obtained by the SFWH
method at five spot levels (S/K = 0.81, 0.90, 1.00, 1.10, 1.19) and by those
methods as reported in Table 1 and [16].
We use the prices calculated in [16] by the MC method with 500,000 sample
paths simulations and 2000 time steps along each trajectory as the benchmark. In
Table 1, the sample mean values are compared with the prices computed by the
AWHF and SWHF methods. The results show a general agreement between the
Monte Carlo simulation results and those computed by the other methods, including
the new one. To study performance of the SWHF method, we include to Table 1 the
cases of smaller number of space points (M = 2048) and larger number of time
steps (N = 320), along with the parameters M = 4096 and N = 80 taken in the
paper [16]. Recall that the number of space points M and the number of time steps
N are the main parameters for both AWHF and SWHF methods. The scaling factor
L for the localization interval (−L ln(2); L ln(2)) in the SWHF method is set to
be 1.
The SFWH prices converge sufficiently fast, and the relative errors reported in
Table 2 show that increasing the number of time steps improves the accuracy of the
method as expected.
Notice that the same algorithm’s parameters for the AWHF and SWHF methods
lead to similar results. However, the SWHF method is rather simpler to implement
into program.
A Simple Wiener-Hopf Factorization Approach for Pricing Double Barrier Options
289
Table 1 Prices: knock-and-out put in KoBoL (CGMY) model, ν = 0.5, λ+ = 9, λ− = −8, and
c=1
M
N
S/K = 0.81
S/K = 0.90
S/K = 1.00
S/K = 1.10
S/K = 1.19
MCa
5 · 105
2000
222.545
303.337
78.672
14.578
3.541
AWHFb
1012
80
222.5256
302.2708
78.19000
14.5985
3.5222
SWHFc
1011
80
222.3233
302.3494
79.0976
14.8533
3.6042
SWHF
1012
80
220.3986
302.2064
79.1141
14.8482
3.5885
SWHF
1012
320
221.5624
302.8453
78.8553
14.6945
3.5624
Option parameters: K = 3500, D = 2800, U = 4200, r = 0.03, d = 0, T = 0.1
a The MC method parameters: M—the number of sample paths, N—the number of time steps
b The AWHF method parameters: M—the number of space points, N—the number of time steps
c The SWHF method parameters: M—the number of space points, N—the number of time steps
Table 2 Relative errors w.r.t.
MC: knock-and-out put in
KoBoL (CGMY) model,
ν = 0.5, λ+ = 9, λ− = −8,
and c = 1
M
N
S/K = 0.81
S/K = 0.90
S/K = 1.00
S/K = 1.10
S/K = 1.19
AWHFa
1012
80
−0.01%
−0.35%
−0.61%
0.14%
−0.52%
SWHFb
1011
80
−0.1%
−0.3%
0.5%
1.9%
1.8%
SWHF
1012
80
−1.0%
−0.4%
0.6%
1.9%
1.4%
SWHF
1012
320
−0.4%
−0.2%
0.2%
0.8%
0.6%
Option parameters: K = 3500, D = 2800, U = 4200,
r = 0.03, d = 0, T = 0.1
a The AWHF method parameters: M—the number of
space points, N—the number of time steps
b The SWHF method parameters: M—the number of
space points, N—the number of time steps
5 Conclusion
In the paper, we suggested a new approach for pricing path-dependent options with
a payoff depending on the infimum and supremum of Lévy processes at expiry. The
method suggested makes it easy to implement such a sophisticated tool as the matrix
Wiener-Hopf factorization for general Lévy models with jumps of finite variation. In
future research, we plan to extend our approach to general Lévy models with jumps
of infinite variation and compare computational efficiency of the SWHF method
with existing numerical methods for pricing double-barrier options.
Acknowledgement This work was supported by the Russian Foundation for Basic Research under
Grant No. 18-01-00910.
290
O. Kudryavtsev
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New Procedure for Applying the
Cramér–von Mises Test for Parametric
Families of Distributions
Gennady Martynov
Abstract Goodness-of-fit tests of the Cramér–von Mises type are considered for
testing hypotheses about the belonging of the observed random variable distribution
to a given parametric family of distributions. The limiting distribution of the
considered criteria statistic does not depend on the sample distribution in cases
where such a distribution is specified exactly. However, for parametric families,
the distribution of classical statistics can depend on the family under the hypothesis
and the specific distribution function within this family. The limiting distribution
of statistics using the Khmaladze transformation to an empirical process becomes
independent of both the hypothetical family of distributions and the values of the
unknown parameters of the sample distribution. The application of this criterion,
however, remains very difficult and requires special approaches to each family of
distributions. The paper recalls that for all the most commonly used families, the
distribution of classical statistics does not depend on distribution parameters. For
the remaining families, a new simple hypothesis testing procedure is proposed. This
procedure is applied in work for a family of gamma distributions. The question of
the possibility of using it for hyper-normal distribution families is also discussed.
The obtained results are partially applicable also to the Kolmogorov–Smirnov test.
Keywords Goodness-of-fit tests · Cramér–von Mises type tests · Parametric
families of distributions · Gamma distribution · Hyper normal distribution
G. Martynov ()
Institute for Information Transmission Problems of the Russian Academy of Sciences, Moscow,
Russia
e-mail: martynov@iitp.ru
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_16
293
294
G. Martynov
1 Introduction: The Cramér–von Mises Test—Simple and
Complex Hypotheses
The introduction sets out the known information necessary for use in the main part
of the work contained in Sects. 2 and 3.
1.1 The Classical Cramér–von Mises Test
Let Xn = {X1 , X2 , . . . , Xn } be the sample of observations of a random variable of
the r.v. with an unknown distribution function F (x), x ∈ R 1 . At first, we consider
the testing of simple hypothesis
H0 : F (x) = G(x),
where G(x) is the continuous distribution function. Let us transform the sample Xn
to be uniformly under H0 distributed sample T n = {T1 , T2 , . . . , Tn } where Ti =
G(Xi ), 0 ≤ Ti ≤ 1, and i = 1, 2, . . . , n. For testing the hypothesis H0 , it can be
used as the standard form of the Cramér–von Mises statistic
ωn2 = n
1
(Fn (t) − t) dt =
2
1
0
0
Bn2 (t) dt.
Here, Fn (t) is the empirical distribution function based on Ti , i = 1, 2, . . . , n, and
Bn (t) = n1/2 (Fn (t) − t)
is the empirical process weakly converging here to the Brownian bridge B(t).
The limit distribution of ωn2 under H0 is the distribution of the functional
ω2 =
1
B 2 (t)dt
0
represented as the quadratic form
ω2 =
∞
xk2
,
(πk)2
(1)
k=1
where {xi , i = 1, 2, . . . , } is the sequence of the independent standard normal
random variables. An important property of the statistic described here is that its
limiting distribution does not depend on the distribution G(x) of the observed
random variable under the simple hypothesis.
Cramér–von Mises Test for Parametric Families
295
For practical purposes, however, statistics designed to test complex hypotheses
are more interesting. Usually, it is tested whether the distribution of a sample
belongs to parametric distribution families. Let Xn = {X1 , X2 , . . . , Xn } be the
sample from the r.v. with the distribution function F (x), x ∈ R1 . We will test
the hypothesis
H0 : F (x) ∈ G = {G(x, θ ), θ = (θ1 , θ2 , . . . θk )5 ∈ ⊂ Rk },
where θ is an unknown vector of parameters. We will consider the Cramér–von
Mises statistic
∞
2
(Fn (x) − G(x, θ̂n ))2 dG(x, θ̂n ),
ωn = n
−∞
where θ̂n is the maximum likelihood estimator of θ and Fn (x) is the empirical
distribution function, based on Xn . The statistic ωn2 can be transformed to the
interval [0, 1]:
ωn2 = n
1
(F̂n (t) − t)2 dt,
0
F̂n (t) is the estimated empirical distribution function, based on the transformed
sample T n = (t1 , . . . , tn ) with ti = G(Xn , θ̂n ), i = 1, . . . , n.
Under the certain number of the regularity conditions [3, 7, 15, 18], the distribution of the statistic ωn2 converges in probab ility under H0 to the distribution of the
functional
ω2 =
1
ξ 2 (t, θ 0 )dt
0
of the Gauss process ξ(t, θ 0 ) with Eξ(t, θ 0 ) = 0 and with the covariance function
K(t, τ, θ 0 ) = E(ξ(t, θ 0 )ξ(τ, θ 0 ))
= (K0 (t, τ ) − q 5 (t, θ 0 )I −1 (θ 0 )q(τ, θ 0 )),
(2)
where K0 (t, τ ) = min(t, τ ) − tτ, t, τ ∈ (0, 1), θ 0 is an unknown value of the
parameter θ , at which the sample is obtained,
q 5 (t, θ 0 ) = (∂G(x, θ )/∂θ1, . . . , ∂G(x, θ )/∂θk )|t =G(x,θ 0) ,
I (θ 0 ) is the Fisher information matrix,
I (θ 0 ) = E((∂/∂θi ) log g(X, θ )(∂/∂θj ) log g(X, θ )) 1≤i,j ≤k ,
g(x, θ ) = (∂G(x, θ )/∂x).
296
G. Martynov
We note here only one condition, which is important for the Cramér–von Mises
test. This is the condition that the limit mathematical expectation of the statistic is
finite:
1
K(t, t, θ )dt < ∞ for all θ ∈ .
0
We notice that the expression (2) depends on the unknown parameter θ 0 , and
therefore, also, both the distribution of the Cramér–von Mises statistic itself and
its significance levels depend on it. It turned out that for the main families of
distributions, there is no dependence of the limiting distribution of the statistic on
the parameter θ 0 . However, in general, there is a difference in the distributions of
the statistic for different families.
Knowledge of the covariance function allows, in principle, an accurate calculation of the eigenvalues and eigenfunctions of the corresponding covariance operator.
Let λk , k = 1, 2, . . . , be such eigenvalues. Then, the limiting distribution of the
Cramér–von Mises statistic can be represented as the distribution of the quadratic
form
ω2 =
∞
x2
k
λ2
k=1 k
.
There are many ways to find the eigenvalues of covariance operators. One of them
is described in [13]. Calculation of the quadratic form distributions is described in
[11].
1.2 Two Special Families of Distributions
It is well known that the empirical process does not depend on unknown parameter
θ 0 for the family of the form (see [3–5, 7])
G = {G((x − θ1 )/θ2 ), −∞ < x < ∞, θ2 > 0}.
Most known example of such family is the normal and the logistic distribution
families.
Another class of the distribution family proposed in [12] is
R = {R((x/β)α ), α > 0, β > 0, x ∈ X ⊂ [0, ∞)}.
The limit distribution of the Cramér–von Mises statistic for this class does not
depend on the unknown parameters α and β. Most known examples of such family
are the Weibull and the lognormal distribution families. Both families are generated
Cramér–von Mises Test for Parametric Families
297
Table 1 Correspondence between the distribution families G and R
X = −ln Y (X = ln Y )
Y = e−X (Y = eX )
1
FX (x) = G( x−θ
θ2 ), θ2 > 0
G(x) = 1 − R(e−x )
FY (x) = R(( βx )α ), α, β > 0
R(x) = 1 − G(−ln x)
α = 1/θ2 , β = e−θ1
θ1 = −ln β, θ2 = 1/α
Normal distribution family
Negative exponential
distribution
F (x) = ex/θ2 , −∞ < x < 0
Extreme value
distribution
Lognormal distribution family
Pareto distribution
F (x) = 1 − x α , x > 1
Weibull distribution
α
F (x) = 1 − e−(x/β) ,
x−θ
− θ 1
F (x) = e−e 2 , x > θ1
Exponential distribution
F (x) = 1 − e−x/θ2 , x > 0
Logistic distribution
1
, −∞ ≤ x ≤ ∞
F (x) =
π x−θ1
1+exp − √
3
θ2
x>0
Power distribution
F (x) = x α , x ∈ [0, 1]
Heavy-tailed distribution
1 √
F (x) = 1 −
, x≥0
(π/ 3)α
1+(x/β)
by the root distribution functions G(x) and R(x). They are related by the forms
G(x) = 1 − R(e−x ) and vice versa R(x) = 1 − G(−ln x). The corresponding
random variables X and Y are transformed by the formulas X = −ln Y and Y =
e−X or X = ln Y and Y = eX . The parameters of the two families are converted by
the formulas θ1 = −ln βand θ2 = 1/α and vice versa α = 1/θ2 and β = e−θ1 . The
results are shown in Table 1. When testing a hypothesis, a sample can be converted
from one family to another. Moreover, the distributions of the statistics are the same
in both cases.
It can be seen that although the Table 1 presents many important distributions, it
does not contain the gamma distribution. The distribution of the Cramér–von Mises
statistic for gamma distribution is studied in Sect. 2. It turns out that the limiting
distribution in this case depends on unknown parameters. What to do in this case?
Section 2 proposes some method to overcome this difficulty. Section 1.4 contains the
Khmaladze method for the same purpose. Both methods have their own advantages
and disadvantages.
1.3 Formula for Calculation of the Cramér–von Mises Statistic
The simple formulas for computing the value of the Cramér–von Mises statistic exist
(see, e.g., [2, 3]): For traditional Cramér–von Mises type tests, there is a very simple
formula for calculating the values of their statistic. This is the formula:
ωn2 =
n 1
i − 1/2 2
+
t(i) −
12n
n
i=1
(3)
298
G. Martynov
where
t(i) = G(X(i) ), X(1) ≤ X(2) ≤ · · · ≤ X(n) ,
designed for the simple hypothesis, and
t(i) = G(X(i) , θ̂n ), X(1) ≤ X(2) ≤ · · · ≤ X(n) ,
designed for the parametric hypothesis. Here, X(1) ≤ X(2) ≤ · · · ≤ X(n) is an
ordered sample X.
1.4 Khmaladze Transformation of the Parametric Empirical
Process
Khmaladze has proposed in [8]
√ (see also [9]) the following transformation of the
parametric empirical process n(Fn (x) − G(x, θ̂n )) to the form
wn (x) =
√
n{Fn (x) − Kn (x, θ̂n )},
where compensator Kn is
Kn (x, θ̂n ) =
x#
q 5 (y, θ̂n )
∞
×
∞
,
−1 ∞
5
q(s, θ̂n )q (s, θ̂n )dG(s, θ̂n )
q(z, θ̂n )dFn (z)
y
y
× dG(y, θ̂n ) .
The analog of the Cramér–von Mises statistic is
ωn2 =
1
0
2
wn (x)|t =G(x,θ̂)
dt.
Here, the empirical process weakly converges to the Wiener process, that is,
in the limit, it does not depend on the unknown parameters of sufficiently regular
families and from the families themselves. The advantage is that in practice, only a
few percentage point values need to be remembered. The disadvantage is that each
parametric family requires a separate algorithm to be developed, which can be very
complex.
Cramér–von Mises Test for Parametric Families
299
1.4.1 Application to the Exponentiality Test
We will now consider the problem of testing hypothesis
H0 : F (x) ∈ {G(x, λ) = 1 − e−λx }, x > 0, λ > 0.
Then, the compensator Kn (x, θn ) can be represented as (see [6])
λ̂ Xi (2 − (λ̂/2)Xi ))
n
Kn (x, θn ) =
i:Xi≤t
+λ̂x(2 + (λ̂/2)x)(1 − Fn (x)) − λ̂2 x
1 Xi
n
i:Xi≤t
where λ̂ is the sample mean.
2 Cramér–von Mises Test for the Gamma Distribution
Family
In this section, a short derivation of the formula for the traditional empirical process
covariance function for the parametric family of the gamma distribution will be
given. Exact distribution tables of Cramer–Mises statistics for the family of gamma
distributions will be presented here. The Tables 2, 3 and 4 are obtained by exact
calculations.
Table 2 Exact critical values of the significance levels 0.1 for the Cramér-von Mises statistic for
gamma distribution with κ >= 1.0
κ
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
t0.1,κ
0.1112
0.1105
0.1099
0.1093
0.1089
0.1085
0.1082
0.1079
0.1077
0.1074
κ
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
t0.1,κ
0.1072
0.1069
0.1066
0.1063
0.1061
0.1059
0.1058
0.1056
0.1055
0.1054
κ
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
t0.1,κ
0.1053
0.1051
0.1049
0.1048
0.1047
0.1046
0.1045
0.1045
0.1044
0.1043
κ
9.0
10
12
14
18
20
25
40
50
90
t0.1,κ
0.1043
0.1042
0.1041
0.1040
0.1039
0.1039
0.1038
0.1037
0.1037
0.1036
300
G. Martynov
Table 3 Exact critical values of the significance levels 0.05 for the Cramér-von Mises statistic for
gamma distribution with κ >= 1.0
κ
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
t0.05,κ
0.1362
0.1353
0.1344
0.1338
0.1332
0.1327
0.1322
0.1318
0.1315
0.1312
κ
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
t0.05,κ
0.1309
0.1304
0.1300
0.1297
0.1294
0.1292
0.1290
0.1288
0.1286
0.1285
Table 4 Exact critical values
of the significance levels 0.1
for the Cramér-von Mises
statistic for gamma
distribution with κ < 1.0
κ
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
κ
0.01
0.02
0.03
0.04
0.05
0.06
t0.05,κ
0.1284
0.1281
0.1279
0.1277
0.1275
0.1274
0.1273
0.1272
0.1271
0.1271
t0.1,κ
0.1711
0.1679
0.1649
0.1621
0.1595
0.1571
κ
0.07
0.08
0.09
0.10
0.20
0.30
κ
9.0
10
12
14
18
20
25
40
50
90
t0.1,κ
0.1548
0.1526
0.1506
0.1488
0.1351
0.1271
t0.05,κ
0.1270
0.1269
0.1268
0.1266
0.1265
0.1264
0.1263
0.1262
0.1262
0.1261
κ
0.40
0.50
0.60
0.70
0.80
0.90
t0.1,κ
0.1221
0.1187
0.1163
0.1145
0.1131
0.1120
Let’s define the density function and the gamma distribution function as follows:
g(x; θ, κ) =
1
(κ, x/θ )
.
x κ−1 e−(x/θ), G(x; θ, κ) =
κ
(κ)θ
(κ)
We will derive the covariance function in accordance with the formula (2). First,
we find, using formulas from [1], the derivatives of the distribution function with
respect to κ and θ :
Gκ (x; θ, κ) = −
1
W (x, θ, κ),
(κ)
where
W (z, θ, κ) =
z 1 z κ
2 F2 {κ, κ}, {1 + κ, 1 + κ}; −
2
κ θ
θ
z
z
− (κ) ln
− ψ(κ) ,
+ κ,
θ
θ
2 F2 (·) is a generalized hypergeometric function,
(κ, z) is the incomplete gamma function,
(4)
Cramér–von Mises Test for Parametric Families
301
ψ(z) is the digamma function:
Gθ (x; θ, κ) = −e− θ
x
x κ
θ
/(θ (κ)).
The vector q(x; θ, κ) as a result will be
q(x; θ, κ) = (Gκ (x; θ, κ), Gθ (x; θ, κ)),
The information matrix will be as follows:
I (θ, κ) =
ψ (κ)
1
θ
1
θ
κ
θ2
.
Covariance function of the standard empirical process is
K(t, τ ; θ, κ) = min(t, τ ) − t τ
− q(x; θ, κ)T I (θ, κ)q(y; θ, κ)
t =G(x;θ,κ), τ =G(y;θ,κ)
.
Theorem 1 The covariance function of the considered limiting empirical process
can be represented as follows:
K(t, τ, θ, κ) = min(t, τ ) − tτ
⎛
xy κ − x + y
k − x
1
2 x
⎝
θ
κ
− 4 2
e
e θ W (y, θ, κ)
+θ
θ (κ)
θ2
θ
⎞
y κ − y
e θ W (x, θ, κ) + θ 4 ψ (κ)W (x, θ, κ)W (y, θ, κ)⎠
+θ 2
θ
x = G−1 (t; θ, κ)
y = G−1 (τ ; θ, κ)
where W (x, θ, κ) is determined by the formula (4).
From the formulas obtained, it is easy to conclude that the variation function
depends only on the κ parameter. This fact was noted for the first time in [17]
and then by many authors. In work [17] and more recent works, tables obtained by
modeling are given. Here for the first time, four-digit tables are obtained by precise
calculations. The change in the critical value when changing the value θ at a given
significance level is relatively small. Since the distribution of the statistic depends
on an unknown parameter, the hypothesis will be tested according to the following
general procedure.
Procedure 1 If there is a table of critical value corresponding to the exact unknown
values of the vector of parameters under the null hypothesis, then we select the level
302
G. Martynov
in it corresponding to the estimate of this vector. If there is no such table, then we
carry out a sufficiently exact modeling, from which we find an approximate critical
value.
Statistics are calculated using the formula (3). The advantage of this criterion
is the ease of calculating the statistic after finding the parameter estimates. The
disadvantage is that it is necessary to find the critical value in one way or another
each time. Both the proposed procedure and Khmaladze’s method [8] face the
problem of finding parameter estimates, which, with a large dimension of the
parameter vector, can be a difficult task. The disadvantage of Khmaladze’s method,
on the contrary, lies in the difficulty of calculating the statistics itself, expressed in
the form of a multiple integral.
3 Hypernormal Distribution Family
In this section, we will consider a three-parameter hypernormal distribution. An
increase in the number of parameters in the family of distributions leads to the
fact that, using goodness-of-fit tests, we approximate the unknown distribution of
the sample by some distribution from this family. The probability of rejecting the
hypothesis with such a family of distributions will be negligible. The family under
consideration here includes several simpler families.
3.1 Test with the Maximum Likelihood Estimation
There, we will consider the three parametric distribution functions with the density
h(x) =
1
2θ1(1 + θ12 )
e
x−θ θ2
− θ 0 1
(see [10, 14, 16]). In Fig. 1, there are several representatives of the family of
hypernormal distributions.
Let X = (X1 , X2 , . . . , Xn ) be a sample from a random variable having
a hypernormal distribution. To estimate the parameters, we will maximize the
logarithm of the likelihood function Ln ,
i=n Xi − θ0 θ2
1
,
−
log Ln = −n log 2 + log θ1 + log (1 +
θ
θ2
1
i=1
Cramér–von Mises Test for Parametric Families
303
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Fig. 1 Three parametric hypernormal distributions
Derivatives with respect to parameters are
i=n
Xi − θ0 θ2 −1
∂
θ2 log Ln =
sign(Xi − θ0 ) ,
∂θ0
θ1
θ1 i=1
i=n ∂
n
θ2 Xi − θ0 θ2 −1
log Ln = − +
,
θ
∂θ1
θ1 θ1
1
i=1
i=n ψ(1 + θ12 ) Xi − θ0 θ2
∂
log Xi − θ0 .
log Ln = n
−
2
∂θ2
θ1
θ1 θ2
i=1
We obtain equations for estimates
⎧
⎪
⎪
⎪
⎪
⎨
Xi −θ0 θ2 −1
sign(X
−
θ
)
= 0,
i
0
i=1
θ
1
θ 2
Xi −θ0
−1 + θn2 i=n
i=1 θ1 = 0,
⎪
⎪
1
⎪
θ
⎪
⎩ ψ(1+2 θ2 ) − 1 i=n Xi −θ0 2 log Xi −θ0 = 0.
i=1
n
θ1
θ1
i=n
θ2
Figure 2 presents the empirical distribution functions for the Cramér–von Mises
statistic at n = 5000 for each function. The simulation was carried out at θ0 =
0, θ1 = 1, and at various values of θ2 . The empirical distribution functions are
condensed at θ2 greater than five. Estimates for other θ2 were found to be strongly
biased. This can be explained by both the irregularity of the family and insufficient
sample sizes.
304
G. Martynov
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
Fig. 2 Empirical distribution of the Cramér–von Mises statistic for different values of θ2 with
θ0 = 0 and θ1 = 1. Sample size is equal to 5000
3.2 Test with Moments Estimation
Classic variants of the parametric Cramér–von Mises tests use maximum likelihood
estimates. These estimates are convenient for the traditional parametric families
discussed in the first section. However, their use for the family of hypernormal
distributions considered in the previous subsection leads, when calculating distribution tables for each specific vector of parameters, to the need for a large number
of calculations. In this section, the maximum likelihood estimate is replaced by the
method of moments. Let us first express the first four moments of the hypernormal
distribution in terms of its parameters. The kth central moment is obtained as
follows:
Eh (x − θ0 ) =
k
=
∞
(x − θ0 )k
1
−∞ 2θ1 (1 + θ )
⎧
⎨$∞
⎩
0
e
x−θ θ2
− θ 0 1
dx,
2
θ
2
− θx
xk
1
dx,
1 e
θ1 (1+ θ )
k = 0, 2, 4, . . . ,
2
0, otherwise,
Cramér–von Mises Test for Parametric Families
305
⎧
k k+1
⎪
θ
⎪
⎨ 1 θ2 , k = 0, 2, 4, . . . ,
= θ 1+ 1
⎪
θ2
⎪ 2
⎩
0, otherwise.
(5)
It used the formula
∞
k −x θ2
x e
0
1
k+1
dx = .
θ2
θ2
Finally, we get the following formula
Eh (x − θ0 )k =
θ1k qk (θ2 ), k = 0, 2, 4, . . . ,
0, otherwise,
where
qk (θ2 ) =
k+1
θ2
.
θ2 1 + θ12
(6)
It follows from the identities
x ≡ θ0 + (x − θ0 ),
x 2 ≡ θ02 + 2θ0(x − θ0 ) + (x − θ0 )2 ,
x 3 ≡ θ03 + 3θ02 (x − θ0 ) + 3θ0(x − θ0 )2 + (x − θ0 )3 ,
x 4 ≡ θ04 + 4θ03 (x − θ0 ) + 6θ02 (x − θ0 )2 + 4θ0 (x − θ0 )3 + (x − θ0 )4
and formula (5) the following equalities for the sample moments
μ1 = θ 0 ,
μ2 = θ02 + θ12 q2 (θ2 ),
μ3 = θ03 + 3θ0 θ12 q2 (θ2 ),
μ4 = θ04 + 6θ02 θ12 q2 (θ2 ) + θ14 q4 (θ2 ).
Replacing μ1 , μ2 , μ3 , and μ4 with the sample moments about the origin m1 , m2 ,
m3 , and m4 , correspondingly, we obtain the following equations for the estimates
θ̃0 , θ̃1 , and θ̃2 of the parameters θ0 , θ1 , and θ2 :
m1 = θ̃0 ,
m2 = θ̃02 + θ̃12 q2 (θ̃2 ),
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G. Martynov
Table 5 Critical values of 0.1 for the Cramér–von Mises statistic with parameters estimated by
the method of moments
5000
T0,1,2.5
=
0.090
5000
5000
T0.5,1,4
= 0.185 T−0.5,1,4
= 0.167
2500
2500
2500
2500
2500
T0,1,3 = 0.084 T0,1,4
= 0.082
T0,1,5
= 0.082 T0,1,7
= 0.082 T0,1,9
= 0.093
5000 = 0.081 T 5000 = 0.090
T0,1.1,4
0,1.5,4
m3 = θ̃03 + 3θ˜0 θ̃12 q2 (θ̃2 ),
m4 = θ̃04 + 6θ̃02 θ̃12 q2 (θ̃2 ) + θ̃14 q4 (θ̃2 ).
In this system of equations, the second and third equations are inconsistent. We put
θ̃12 q2 (θ̃2 ) = (m2 − m21 ) in the fourth equation. We get the equation
m4 = m41 + 6m1 (m2 − m21 ) + (m2 − m21 )2
q4 (θ̃2 )
q22 (θ̃2 )
to find θ̃02 . Finally, we get with using formula (6) the following equation
S(θ̃2 ) ≡ θ̃2
5
θ̃2
1+ 1
6m21
m4 − m41
θ̃2 −
+
= 0.
(m2 − m21 )2
m2 − m21
2 3
(7)
θ̃2
Having calculated the estimate θ̃2 from this equation and having the estimate θ̃0 =
m1 , we obtain the estimate θ1 as
θ̃1 =
=
(m2 − m21 )/q2 (θ̃2 ).
We denote by Tθn0 ,θ1 ,θ2 critical values of the Cramér–von Mises statistic, where n is
the sample size. Some critical values of the significance level 0.1 are presented in
Table 5.
Figure 3 represents the graphs of the left-hand sides of Eq. (7) to find the estimate
θ̃2 of the parameter θ2 .
From the results of this and the previous subsection, it can be concluded that the
verification of the described hypotheses with the applied estimation methods can be
carried out only in the region of the family, remote from its irregular part.
Cramér–von Mises Test for Parametric Families
307
1.5
1
0.5
0
-0.5
-1
-1.5
1
2
3
4
5
6
7
8
9
10
Fig. 3 Graph of the function S(θ̃2 ) depending from sample moments m1 , m2 , and m4 under
conditions that the parameter vector (θ0 , θ1 , θ2 ) takes values (0, 1, 2), (0, 1, 3), (0, 1, 4), (0, 1, 5),
and (0, 1, 7)
References
1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Integrals, 1046 pp. Dover, New
York (1965)
2. Anderson, T.W., Darling, D.A.: Asymptotic theory of certain “Goodness of Fit” criteria based
on stochastic processes. Ann. Math. Stat. 23(2), 193–212 (1952)
3. Durbin, J.: Weak Convergence of the sample distribution function when parameters are
estimated. Ann. Stat. 1(2), 279–290 (1973)
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Soc. Ser. B 37(2), 216–237 (1975)
5. Gikhman, I.I.: One conception from the theory of ω2 -test [in Ukrainian]. Nauk. Zap. Kiiv Univ.
13, 51–60 (1954)
6. Haywood, J., Khmaladze, E.: On distribution-free goodness-of-fit testing of exponentiality. J.
Econ. 143, 5–18 (2005)
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on distance methods. Ann. Math. Stat. 30, 420–447 (1955)
8. Khmaladze, E.V.: A martingale approach in the theory of parametric goodness-of-fit tests.
Theor. Prob. Appl. 26(2), 240–257 (1981)
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Statistical Science, pp. 715–718. Springer, Berlin
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nonparametric goodness-of-fit tests in testing composite hypotheses. Commun. Stat. Theory
Methods 39(3), 460–471 (2010)
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Debrezen. 76(3), 341–346 (2010)
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13. Martynov, G.: Weighted Cramer–von Mises test with estimated parameters. Commun. Stat.
Theory Methods 40(19–20), 3569–3586 (2011)
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15. Neuhaus, G.: Asymptotic properties of the Cramér–von Mises statistic when parameters are
estimated. In: Proc. Prague Symp. Asymptotic Stat., vol. 2, pp. 257–297. Charles University,
Prague (1973/1974)
16. Roenko, A.A., Lukin, V.V., Djurović, I., Simeunović, M.: Estimation of parameters for
generalized Gaussian distribution. In: 6th International Symposium on Communications,
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Berlin (1996)
CVaR Hedging in Defaultable
Jump-Diffusion Markets
Alexander Melnikov and Hongxi Wan
Abstract This paper deals with the problem of conditional value-at-risk (CVaR)based optimal partial hedging in a defaultable jump-diffusion market. The minimal
superhedging costs of claims with a zero recovery rate are derived. Meanwhile, we
convert the CVaR minimization problem into a static optimization problem in the
corresponding default-free market, and the solution of such a problem is given with
the help of the Neyman-Pearson lemma. Furthermore, our method is implemented
to derive minimal values of CVaR and optimal hedging strategies of defaultable
equity-linked life insurance contracts, whose payoffs are equal to the maximum of
two risky assets conditioned by the occurrence of default events.
Keywords Partial hedging · Conditional value-at-risk · Jump-diffusion model ·
Equity-linked life insurance contracts · Default
1 Introduction
Hedging and risk management are crucial topics in financial mathematics. In a
complete market, any contingent claim can be replicated with sufficient initial
wealth. However, if a market is incomplete, the initial costs of superhedging are
often too high. In this case, a hedger usually allocates initial capitals less than the
superhedging costs while accepts the possibility of shortfall. Such a hedging strategy
is called partial hedging. Föllmer and Leukert are pioneers in this filed. They studied
quantile hedging and efficient hedging [3, 4] in semimartingale financial market
models. In their papers, explicit solutions in complete markets were provided with
the help of the classical Neyman-Pearson lemma, while solutions in incomplete
markets were given according to the convex duality approach. Another reason that
A. Melnikov · H. Wan ()
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB,
Canada
e-mail: melnikov@ualberta.ca; hongxi@ualberta.ca
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_17
309
310
A. Melnikov and H. Wan
makes partial hedging interesting is that although it has some downside risk, it
brings opportunities to gain benefits. Some financial institutions such as insurance
companies indeed exploit risk to make profits. The recent book of Melnikov and
Nosrati [8] discussed several partial hedging methods and their applications in
pricing and hedging insurance contracts. In this paper, we consider the partial
hedging problem in defaultable markets, which may be incomplete. There is a list
of references in this area. For instance, Nakano [13] solved problems of optimal
quantile hedging and efficient hedging with a linear loss function for claims with a
single default time. Later, Melnikov and Nosrati [7] focused on the efficient hedging
problem with more general loss functions for claims with several independent
default times and provided a closed form solution in the special case that recovery
rates were zeros. In our paper, we would discuss partial hedging by employing
a coherent risk measure called conditional value-at-risk (CVaR) which provides
information about the average loss that exceeds the value-at-risk (VaR) level.
Melnikov Smirnov [11] studied partial hedging with such a measure and provided a
semi-explicit solution in complete markets.
Aforementioned papers dealt with option pricing and hedging in Brownian
market models. However, a growing number of evidences show that pure diffusion
models are not accurate enough to represent real-life assets’ dynamics. It is not
rare to observe jumps in stock prices when some significant financial or political
announcements are published. In order to address this drawback, a jump-diffusion
model was proposed by Merton [12]. Melnikov and Skornyakova [10] and Kirch
and Melnikov [5] discussed quantile hedging and efficient hedging problems,
respectively, in a two-factor jump-diffusion model. However, to our knowledge, a
jump-diffusion market model with default has not been well studied, and CVaR
hedging problems have not been discussed in this literature.
Our main objective in this paper is to derive a hedging strategy that minimizes
CVaR of the hedging loss subject to a constraint on the initial wealth in a jumpdiffusion defaultable market. The paper is organized as follows. In Sect. 2, we
introduce our financial model. Several useful properties regarding the default time
are listed. Most importantly, similar to the discussion about martingales in a
defaultable Brownian market in Bielecki and Rutkowski [1], we derive densities of
martingale measures in our model. Furthermore, utilizing properties of equivalent
martingale measures, we show that the minimal superhedging costs of a defaultable
claim with a zero recovery rate coincide with the perfect hedging costs of the
corresponding non-defaultable claim. In Sect. 3, the explicit form of the optimal
CVaR hedging strategy in our incomplete market is derived. We prove that the CVaR
minimization problem of a zero-recovery rate defaultable claim can be converted to
a problem of finding the optimal randomized test in the default-free complete market
and the optimal strategy is then given by the option decomposition of a modified
claim. In Sect. 4, a numerical example is provided to illustrate the implementation
of our method in life insurance contracts that have stochastic guarantees. Section 5
gives a conclusion for the paper.
CVaR Hedging in Defaultable Jump-Diffusion Markets
311
2 Model Setup and Preliminaries
Let (, G, P ) be a standard probability space. Consider a financial market with
terminal time T ∈ (0, ∞) consisting one riskless asset (St0 )t ∈[0,T ] and two risky
assets, (St1 )t ∈[0,T ] and (St2 )t ∈[0,T ] , described by a two-factor jump-diffusion model:
dSt0 = rSt0 dt, S00 = 1,
dSti = Sti− (μi dt + σi dWt − υi dNt ), S0i > 0,
i = 1, 2,
(1)
where r ≥ 0 is the risk-free interest rate. Constants μi ∈ R, σi > 0, and
υi < 1 are drifts, volatilities, and jump parameters, respectively. W and N are
independent Wiener process and Poisson process with a filtration F = (Ft )t ≥0 ,
Ft = σ (Ns , Ws ; s ≤ t). Here, we assume σ1 > σ2 , and the intensity of the Poisson
process is a nonnegative constant λ.
In addition, let a positive random variable τ denote the default time such
that P (τ > t) > 0, ∀t ≥ 0. The default indicator process is defined as
Ht = I{τ ≤t } , t ≥ 0, and the corresponding filtration generated by it is H =
(Ht )t ≥0, Ht = σ (Hs ; s ≤ t). Let us specify that τ is independent of W and
N. Moreover, let G = (Gt )t ≥0 be a joined filtration, i.e., G = H ∨ F. For
simplicity, we assume G = GT . It is worth to mention that since H and F are
independent, any F-martingale is also a G-martingale (see Bielecki and Rutkowski
[1] Section 6.1.1).
In this paper, credit risk is modeled with the help of a hazard process, that is, the
survival probability is defined by
P (τ > t) = e−t ,
(2)
$t
where (t )t ≥0 is the hazard process such that t = 0 βu du and βu is a nonnegative
deterministic function called the hazard rate of the random time τ .
Let us summarize some crucial results regarding the default time and the default
indicator process from Bielecki and Rutkowski [1] for the reader convenience:
1. The process
Mt = Ht −
t ∧τ
0
βu ds = Ht −
t
βu (1 − Hu )du,
(3)
0
is both a H-martingale and a G-martingale (Proposition 5.1.3).
2. The process Qt = (1 − Ht )et , t ≥ 0 follows a G-martingale with the
dynamic dQt = −Qt − dMt . Moreover, for any bounded F-martingale m,
the product Qm and the quadratic covariation [Q, m] are G-martingales
(Lemma 5.1.7).
312
A. Melnikov and H. Wan
3. For any Gt -measurable random variable Y , there is a Ft -measurable random
variable
Ỹ =
E(I{τ >t } Y |Ft )
,
P (τ > t)
(4)
such that Y I{τ >t } = Ỹ I{τ >t } , P − a.s. (Lemma 5.1.2).
Note that the market (1) with default may be incomplete due to the additional
random source τ , so that there is a set of martingale measures. Let us start with
the structure of martingales in our setting.
Theorem 1 For a G-measurable integrable random variable XT , we define a Gmeasurable martingale Xt = E(XT |Gt ), t ∈ [0, T ]. Then, it admits the following
representation:
Xt = X0 +
t
0
ξuW dWu +
t
0
ξuN d N̂u +
t
0
ξuM dMu +
t
0
ξuMN d[M, N̂]u ,
(5)
where N̂t = Nt − λt is the compensated Poisson process and ξ W , ξ N , ξ MN , ξ M
are G-predictable processes.
Proof Since GT = HT ∨ FT , according to Bielecki and Rutkowski [1], it is
sufficient to consider a random variable XT = (1 − Hs )Y for some fixed s ≤ T
and some FT -measurable random variable Y :
XT = (1 − Hs )Y = (1 − Hs )es Ȳ = Qs Ȳ ,
where Ȳ = e−s Y .
Let us define a F-martingale m:
mt = E(Ȳ |Ft ) = E(Ȳ ) +
t
0
ξu dWu +
t
ζu d N̂u ,
0
where ξ and ζ are F-predictable processes. The second equality is because of the
martingale representation property in the filtration F.
Since Q is a process with finite variation, we have
[Q, m]t =
u≤t
Qu mu =
u≤t
−Qu− ζu Mu N̂u = −
t
Qu− ζu d[M, N̂]u .
0
CVaR Hedging in Defaultable Jump-Diffusion Markets
313
Obviously, mT = Ȳ and according to the integration by parts formula, we get
XT = Qs mT = Q0 m0 +
= E(Ȳ ) +
−
T
Qu− dmu +
0
Qu− ξu dWu +
0
T
0
T
T
T
0
mu− I[0,s] (u)dQu + [Q, m]s
Qu− ζu d N̂u −
0
T
0
mu− I[0,s] (u)Qu− dMu
ζu Qu− I[0,s] (u)d[M, N̂]u .
With the choice ξuW = Qu− ξu , ξuN = Qu− ζu , ξtM = −mu− I[0,s] (u)Qu− , ξ MN =
−ζu Qu− I[0,s] (u), (5) is proved.
A simple modification of Theorem 1 implies that the Radon-Nikodym density of a
martingale measure P̃ which is equivalent to P on (, G) has the form
d P̃
|G = 1 +
Ẑt =
dP t
t
Zu− (θu dWu + ψu d N̂u + ku dMu + γu d[M, N̂]u ),
(6)
0
where θ , ψ, k, γ are G-predictable processes. And the solution of (6) is
Ẑt = Et (
·
θu dWu +
0
·
0
ψu d Nˆu +
·
ku dMu +
0
·
γu d[M, N̂]u ),
(7)
0
where Et (·) is the Doleans exponential.
Note that, in order for Ẑ to be a positive martingale, we have to impose the
following restrictions: (a) kt > −1, ψt > −1, kt + ψt + γt > −1, ∀t ∈ [0, T ];
(b) E(Ẑt ) = 1, ∀t ∈ [0, T ].
With the help of Girsanov theorem, we conclude that processes
Wt∗ = Wt −
t
0
θu du, Ñt = Nt −
t
(1 + ψu )λdu,
0
are martingales under the measure P̃ defined by (6).
To derive processes θ , ψ, k, and γ , let us consider discounted value processes
S̃ti = e−rt Sti , (i = 1, 2), which can be rewritten as
d S̃ti = S̃ti− (μi − r)dt + σi dWt − vi dNt
= S̃ti− (μi − r + σi θt − vi (1 + ψt )λ)dt + σi dWt∗ − vi d Ñt , (i = 1, 2).
(8)
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A. Melnikov and H. Wan
They are martingales under the measure P̃ if drift terms vanish, i.e.,
μ1 − r + σ1 θt − v1 (1 + ψt )λ = 0,
μ2 − r + σ2 θt − v2 (1 + ψt )λ = 0,
(9)
(μ1 − r)v2 − (μ2 − r)v1
(μ1 − r)σ2 − (μ2 − r)σ1
− 1.
, ψ=
σ2 v1 − σ1 v2
(σ2 v1 − σ1 v2 )λ
(10)
which implies
θ=
However, processes k and γ cannot be determined uniquely, and hence the market
(1) with default is incomplete. We denote M as the set containing all martingale
measures and L is the set of Radon-Nikodym densities of martingale measures.
In particular, since N̂ and M arepurely discontinuous
martingales, their
quadratic covariation is [M, N̂]t =
Mu N̂u =
Hu Nu , and hence
u≤t
u≤t
[M, N̂]t I{τ >T } is 0, t ∈ [0, T ]. Therefore, we arrive to
ẐT I{τ >T } = ET (θ W + ψ N̂ +
·
= ET (θ W )ET (ψ N̂ +
0
ku dMu )I{τ >T }
·
0
= ET (θ W )ET (ψ N̂ )ET (
ku dMu )I{τ >T }
·
0
ku dMu )I{τ >T }
θ2
= exp θ WT − T + (λ − λ∗ )T + (lnλ∗ − lnλ)NT
2
T ∧τ
T
exp
ln(1 + ku )dHu −
ku βu du I{τ >T } = ZT∗ ZTk I{τ >T } ,
0
0
(11)
where
λ∗ = (1 + ψ)λ =
ZT∗ = exp(θ WT −
(μ1 − r)σ2 − (μ2 − r)σ1
,
(σ2 v1 − σ1 v2 )
θ2
T + (λ − λ∗ )T + (lnλ∗ − lnλ)NT ),
2
ZTk = exp(−
T
ku βu du).
0
CVaR Hedging in Defaultable Jump-Diffusion Markets
315
The second and third equalities are both due to the multiplication rule of Doleans
exponential and the fact [W, M]t = 0, [W, N̂ ]t = 0, [N̂, M]t I{τ >T } = 0, ∀t ≤ T .
∗
∗
Remark 1 The probability measure P ∗ defined by dP
dP = ZT that satisfies ψ > −1,
and σ2 v1 −σ1 v2 = 0 is the unique martingale measure in the non-defaultable market
(1) on the probability space (, FT , P ). Moreover, W ∗ and N are independent
Wiener process and Poisson process (with intensity λ∗ ) under this measure. We
denote E ∗ (·) as the expectation under the measure P ∗ .
With notations introduced above, Sti , i = 1, 2, can be rewritten as
1
Sti = S0i exp σi Wt + (μi − σi 2 )t + Nt ln(1 − υi )
2
1
= S0i exp σi Wt∗ + (r + υi λ∗ − σi 2 )t + Nt ln(1 − υi ) .
2
(12)
A G-strategy is a G-predictable process π := (πt0 , πt1 , πt2 )t ∈[0,T ] such that
T
0
|πt0 |dt < ∞,
T
0
(πti Sti )2 dt < ∞, P − a.s (i = 1, 2),
and the value process corresponding to the strategy π at time t ∈ [0, T ] is
Vt = πt0 St0 + πt1 St1 + πt2 St2 .
(13)
In addition, for a given initial value v ≥ 0, a trading strategy is called self-financing
admissible if its value process satisfies
Vt = v +
t
0
πu0 dSu0 +
t
0
πu1 dSu1 +
t
0
πu2 dSu2 , and Vt ≥ 0, ∀t ∈ [0, T ].
(14)
We denote the set of all admissible self-financing strategies with initial value v as
A(v).
Remark 1 indicates that the default-free market (1) is complete, and hence the fair
price of any contingent claim with payoff C which is nonnegative FT -measurable
is defined as E ∗ (e−rT C), and there is a self-financing strategy π that duplicates it
(see [5] for the form of the duplication strategy).
However, we would like to investigate the hedging problem of a defaultable claim
C 0 = CI{τ >T } in the enlarged filtration G described before. As we know from the
option pricing theory in incomplete markets, the minimal initial superhedging costs
of such a claim are defined as
U0 = sup E P̃ (e−rT C 0 ).
P̃ ∈M
(15)
316
A. Melnikov and H. Wan
Let us denote
Ũt = Ut e−rt = ess supE P̃ (e−rT C 0 |Gt ), t ∈ [0, T ],
(16)
P̃ ∈M
which is a supermartingale with respect to any P̃ ∈ M and represents the discounted
value process of the minimal superhedging strategy of the claim C 0 .
According to the optional decomposition theorem (see Kramkov [6]), there is an
admissible strategy (U0 , π) and a discounted optional consumption process D with
D0 = 0 such that
Ũt = U0 +
t
0
πu1 d S̃u1 +
t
0
πu2 d S̃u2 − Dt .
(17)
In our setting, the minimal superhedging costs U0 can be derived explicitly.
Lemma 1 Assume E ∗ (C) < +∞. In the market (1) with default, the discounted
superhedging value process of C 0 satisfies
Ũt = ess supE P̃ e−rT C 0 |Gt = E ∗ e−rT C|Ft I{τ >t } , ∀t ∈ [0, T ].
(18)
P̃ ∈M
In particular, U0 = Ũ0 = E ∗ (e−rT C), i.e., the minimal superhedging costs equal
to the fair price of the non-defaultable contingent claim C.
Proof For a given martingale measure P̃ with a density ẐT , we know that
Ũt = ess sup E P̃ [e−rT CI{τ >T } |Gt ] = ess sup
P̃ ∈M
E[e−rT ẐT CI{τ >T } |Gt ]
Ẑt
Ẑ∈L
.
Moreover, by (11), we have
$t
$T
E[ẐT CI{τ >T } |Gt ] = e− 0 ku βu du E[ZT∗ e− t ku βu du CI{τ >T } |Gt ].
By (4), the Ft -measurable random variable
% $t
#
$T
E E[ZT∗ e− t ku βu du CI{τ >T } |Gt ]I{τ >t } |Ft e 0 βu du
$t
$T
= E ZT∗ e− t ku βu du CI{τ >T } |Ft e 0 βu du
satisfies
$T
$t
$T
E ZT∗ e− t ku βu du CI{τ >T } |Ft e 0 βu du I{τ >t} = E ZT∗ e− t ku βu du CI{τ >T } |Gt I{τ >t} .
CVaR Hedging in Defaultable Jump-Diffusion Markets
317
Thus, Ũt can be rewritten as
$t
Ũt = ess sup
$T
$t
e− 0 ku βu du E e−rT ZT∗ e− t ku βu du CI{τ >T } |Ft e 0 βu du I{τ >t }
Ẑt
Ẑ∈L
$T
= ess sup
Ẑ∈L
E(e−rT ZT∗ e− t ku βu du CI{τ >T } |Ft ) $ t βu du
e0
I{τ >t } .
Zt∗
In particular, choosing k constant and k 7 −1, we have
$T
e− t kβu du E(e−rT ZT∗ CI{τ >T } |Ft ) $ t βu du
Ũt ≥ lim
e0
I{τ >t} = E ∗ (e−rT C|Ft )I{τ >t} ,
k7−1
Zt∗
(19)
where the second equality is due to the independence of τ and FT .
On the other hand, since kt > −1, we get
$T
E(e−rT ZT∗ e t βu du CI{τ >T } |Ft ) $ t βu du
e0
I{τ >t } = E ∗ e−rT C|Ft I{τ >t } .
Ũt ≤
Zt∗
(20)
Combing (19) and (20), we arrive to
Ũt = E ∗ e−rT C|Ft I{τ >t } .
(21)
3 CVaR Partial Hedging
If a hedger allocates less capitals than the minimum superhedging costs U0 , there is
a possibility of shortfall characterized by L = C 0 − VT , where VT is the value of
a hedging portfolio at T . Conditional value-at-risk (CVaR) is a famous criterion to
measure the hedging loss, and it is widely used in financial institutions.
Definition 1 The CVaR of the loss L at a confidence level α ∈ (0, 1) is defined as
the following:
CV aRα (L) =
1
1−α
1
V aRs (L)ds,
α
where V aRα (L) = inf{s ∈ R : P (L ≤ s) > α}.
(22)
318
A. Melnikov and H. Wan
CV aRα (L) measures the expected loss for a hedging strategy given that the loss
exceeds its α quantile. Rockafellar and Uryasev [14] have indicated that it can also
be represented as
1
E (L − z)+ : z ∈ R .
(23)
1−α
1
Moreover, they showed that V aRα (L) = min y|y ∈ argmin z + 1−α
E (L −
z∈R
z)+
.
Our goal is to find a self-financing admissible strategy with an initial budget
constraint v0 < U0 that minimizes hedging losses under the measure CVaR, i.e.,
CV aRα (L) = inf z +
min CV aRα (L),
(v,π)∈A0
(24)
where A0 = {(v, π)|(v, π) ∈ A(v), v ≤ v0 } is the set of self-financing admissible
strategies with the initial hedging capital no more than v0 .
Melnikov and Smirnov [11] have indicated that we can interchange the order of
two minimization problems:
min CV aRα (L) =
(v,π)∈A0
#
min min z +
(v,π)∈A0 z∈R
#
= min z +
z∈R
%
1
E (C 0 − VT − z)+
1−α
%
min E (C 0 − VT − z)+ .
1
1 − α (v,π)∈A0
(25)
If the inner minimization problem in (25) for each z is solved, then the initial
problem (24) is reduced to a one-dimensional optimization problem over z. Thus,
for a fixed z, let us consider the problem
min E (C 0 − VT − z)+ .
(v,π)∈A0
(26)
We focus on z ≥ 0 because z is corresponding to the V aRα of the hedging loss, and
it is nonnegative when α is close to 1. Consequently, we have
(C 0 − VT − z)+ = ((C 0 − z)+ − VT )+ = (C 0 (z) − VT )+ ,
where C 0 (z) = (C 0 − z)+ = (C − z)+ I{τ >T } = C(z)I{τ >T } with C(z) = (C − z)+ .
Obviously, C 0 (z) is a GT -measurable nonnegative random variable, so it can be
treated as a contingent claim, and thus the problem (26) is equivalent to an optimal
efficient hedging problem of the contingent claim C 0 (z). Föllmer and Leukert [3]
studied this kind of problem and proved that if a random variable ϕ solves
maxE ϕC 0 (z) ,
ϕ∈R
(27)
CVaR Hedging in Defaultable Jump-Diffusion Markets
319
where R = {ϕ : → [0, 1]| GT − measurable, sup E P̃ e−rT C 0 (z)ϕ ≤ v0 },
P̃ ∈M
then the optimal hedging strategy π is obtained from the optional decomposition
(17) for the modified claim ϕ C(z). However, usually the optimal randomized test
does not admit an explicit form in incomplete markets. To address such a difficulty,
we transfer the optimization problem (27), which is in the enlarged filed GT into a
problem in the filed FT .
Lemma 2 If a FT -measurable random variable ϕ̃ ∈ R̃ solves the problem
(28)
maxE ϕC(z) ,
ϕ∈R̃
where R̃ = {ϕ : → [0, 1]| FT − measurable, E ∗ e−rT C(z)ϕ ≤ v0 }, then
ϕ = ϕ̃I{τ >T } is the solution of the problem (27).
Proof Suppose ϕ̃ is the solution for (28). Let us define ϕ = ϕ̃I{τ >T } . Obviously
ϕ ∈ [0, 1] and is GT -measurable. Also, by Lemma 1, we know that
sup E P̃ e−rT C 0 (z)ϕ = sup E P̃ e−rT C(z)ϕ̃I{τ >T } = E ∗ e−rT C(z)ϕ̃ ≤ v0 ,
P̃ ∈M
P̃ ∈M
and hence we have ϕ ∈ R.
$T
On the other hand, for any other ϕ ∈ R, define ϕ0 = E(I{τ >T } ϕ|FT )e 0 βu du
and by (4) we conclude that ϕI{τ >T } = ϕ0 I{τ >T } . Thus, E ∗ (e−rT C(z)ϕ0 ) =
sup E P̃ (e−rT C 0 (z)ϕ) ≤ v0 , which indicates that ϕ0 ∈ R̃. Furthermore,
P̃ ∈M
E ϕC 0 (z) = E ϕC(z)I{τ >T } = E ϕ0 C(z)I{τ >T }
= E ϕ0 C(z) E(I{τ >T } ) ≤ E ϕ̃C(z) E I{τ >T } = E ϕ C 0 (z) ,
(29)
where the last line is due to the independence of FT and τ and the optimal property
of ϕ̃. The Inequality (29) indicates that ϕ is the solution of (27). Lemma 2 is proved.
The solution of the problem (28) is given with the help of the classical NeymanPearson lemma (see Föllmer and Leukert [3]) such that the optimal randomized test
has the form
ϕ ∗ (z) = I{ã(z)<Z ∗ −1 } + (z)I{ã(z)=Z ∗ −1 }
(30)
ã(z) = inf{a ≥ 0, E ∗ (C(z)I{a<Z ∗ −1 } ) ≤ v0 erT },
(31)
T
T
T
(z) =
v0 erT − E ∗ (C(z)I{ã(z)<Z ∗ −1 } )
T
E ∗ (C(z)I{ã(z)=Z ∗ −1 } )
T
.
(32)
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A. Melnikov and H. Wan
Here, (z) = 0 if P (ã(z) = ZT∗ −1 ) = 0, and such a condition is satisfied in our
setting.
Theorem 2
(A) The optimal hedging strategy (v0 , π̂ ) of the problem (24) is given by the
optional decomposition of the modified contingent claim ϕ ∗ (ẑ)C(ẑ)I{τ >T } ,
where ϕ ∗ (z) is given by (30)–(32) and ẑ is the point of minimum of the function
d(z) = z +
1
· E C(z)(1 − ϕ ∗ (z))I{τ >T } ,
1−α
over the interval [0, z∗ ], where z∗ is the solution of the equation
sup E P̃ e−rt C 0 (z) = E ∗ e−rt C(z) = v0 .
P̃ ∈M
In addition, the value d(ẑ) is the minimal CVaR.
(B) The optimal hedging strategy π̂ = (π̂t0 , π̂t1 , π̂t2 )t ∈[0,T ] satisfies
i
π̂ti = π t I{τ ≥t } , (i = 1, 2);
and
π̂t0 =
fort ∈ [0, τ ∧ T ],
π 0t ,
Vτ e−rτ , fort ∈ (τ, T ],
where π = (πt 0 , πt 1 , πt 2 )t ∈[0,T ] is the duplication strategy of the nondefaultable claim ϕ ∗ (ẑ)C(ẑ) and Vt = πt 0 St0 + πt 1 St1 + πt 2 St2 , t ∈ [0, T ]
is its value process.
Proof See Appendix 1.
Remark 2 The reason why we focus on z ≤ z∗ is that, with initial capital v0 , for
z ≥ z∗ , C 0 (z) can be hedged perfectly, which means min E (C 0 (z)−VT )+ = 0
(v,π)∈A0
and d(z) = z. d(z) is increasing after z∗ , and hence it achieves its minimum during
[0, z∗ ].
Remark 3 The optimal hedging strategy can be explained as constructing the
perfect hedging of the modified claim ϕ ∗ (ẑ)C(ẑ) during [0, τ ]. If a default event
happens, hold zero positions in risky assets and deposit all cash into the risk-free
account.
CVaR Hedging in Defaultable Jump-Diffusion Markets
321
4 Applications to Equity-Linked Life Insurance Contracts
One important application of partial hedging is to deal with pricing and hedging of
equity-linked life insurance contracts. In this section, we will calculate the minimal
CVaR and provide the optimal CVaR hedging strategy of defaultable equity-linked
life insurance contracts.
Following actuarial traditions, let a random variable T (x) on an “actuarial”
probability space (, G2 , P2 ) denote the remaining life time of a person of current
age x and T px = P2 (T (x) > T ) be the survival probability for the next T
years of an insured. There are two sources of risk: market risk associated with
underline assets prices and the default time and insurance risk reflected by the
insured mortality. Since usually the insurance risk and the financial market risk
have no effect on each other, we would take a natural assumption that (, G, P )
and (, G2 , P2 ) are independent.
Instead of simple call or put options, we consider a pure endowment life
insurance contract with payoff C 0 = max(ST1 , ST2 )I{τ >T } provided that an insured
is alive at T . This kind of contract has a flexible guarantee S 2 and a potential
for future gains associated with S 1 . It is popularly traded in insurance companies.
According to Brennan and Schwartz [2], the premium for such a contract is defined
as
T Ux =
sup E P̃ (e−rT C 0 )E P2 (I{T (x)>T } ) = T px U0 ,
(33)
P̃ ∈M
where x is the insured’s age, T is the maturity time of the contract, and U0 is
the minimal superhedging costs of the defaultable claim C 0 . Notice that T Ux <
U0 , which means the premium that the insurance company collects would be
less than the minimal superhedging costs and therefore only a partial hedging
strategy can be constructed. In our setting, we assume that the insurance company
constructs the optimal hedging strategy described in Sect. 3 with the premium
v0 = T px U0 .
Theorem 3
(A) The minimal superhedging costs U0 of the defaultable claim C 0
max(ST1 , ST2 )I{τ >T } are
U0 =
∞
#
√
√ %
∗
1
2
s0,n
p0,n
(1 (n) + σ1 T ) + s0,n
(−1 (n) − σ2 T ) .
=
(34)
n=0
(B) The optimal CVaR hedging strategy for C 0 = max(ST1 , ST2 )I{τ >T } is given
by the perfect hedging of the modified contingent claim (max(ST1 , ST2 ) −
ẑ)+ I{Z ∗−1 >ã(ẑ)} during [0, τ ] while holding zero positions in risky assets after
T
default, where ã(z) is the unique solution of e−rT E ∗ (C(z)I{a<Z ∗ −1 } ) = v0 ,
T
322
A. Melnikov and H. Wan
that is,
∞
#
√
√
∗
1
p0,n
s0,n
2 1,3 (z, n) + σ1 T , 2 (a, n) − σ1 sign(θ) T , 1
n=0
√
√
√
2
+ s0,n
3 − 1 (n) − σ2 T , 2 (a, n) − σ2 sign(θ) T , 4 (z, n) + σ2 T , 2
− ze−rT (2 1,3 (z, n), 2 (a, n), 1 + 3 − 1 (n), 2 (a, n), 4 (z, n), 2
%
= v0 .
(35)
ẑ is the point of minimum of the function
d(z) = z +
1
· f (z)
1−α
(36)
over the interval [0, z∗ ], where f (z) = E[C(z)(1 − ϕ ∗ (z))I{τ >T } ], i.e.,
$T
f (z) = e− 0 βu du
∞
√
√
p0,n S01 (1 − v1 )n eμ1 T 2 (5,6 (n) + σ1 T , 7 (n) + σ1 sign(θ) T , 3 )
n=0
√
√
T , 7 (n) + σ2 sign(θ) T , 8 (n) + σ2 T , 4 )
− z 2 (5,6 (n), 7 (n), 3 ) + 3 (−6 (n), 7 (n), 8 (n), 4 ) .
(37)
+ S02 (1 − v2 )n eμ2 T 3 (−6 (n) − σ2
√
Meanwhile, d(ẑ) is the corresponding value of minimal CVaR.
The parameter z∗ is determined from E ∗ (e−rt C(z)) = v0 , that is,
∞
#
√
∗
1
2
s0,n
p0,n
(1,3 (z, n) + σ1 T ) + s0,n
2 (−1 (n)
n=0
√
√
− σ2 T , 4 (z, n) + σ2 T , 5 )
− ze−rT (1,3 (z, n)) + 2 (−1 (n), 4 (z, n), 5 )
%
= v0 .
(38)
Here,(·) is the distribution function of a standard normal random variable, and
J (·, ) denotes the distribution function of J jointly normally distributed random
variables with zero means, unit variances, and the correlation matrix .
∗
i
st,n
= Sti evi λ (T −t) (1 − vi )n ,
s1
1 (n) =
ln( 0,n
2 )+
s0,n
σ22 −σ12
2 T
√
(σ1 − σ2 ) T
2
, 2 (a, n) =
− ln(aZt∗ ) − ( θ2 + λ − λ∗ )T − (ln λ∗ − ln λ)n
√
,
|θ| T
CVaR Hedging in Defaultable Jump-Diffusion Markets
s1
323
s2
σ2
σ2
1
2
ln 0,n
ln 0,n
z + (r − 2 )T
z + (r − 2 )T
, 4 (z, n) =
,
3 (z, n) =
√
√
σ1 T
σ2 T
5 (n) =
ln
S01 (1−v1 )n
σ2
+ (μ1 − 21 )T
z
√
σ1 T
ln
, 6 (n) =
S01 (1−v1 )n
σ 2 −σ 2
+ (μ1 − μ2 + 2 2 1 )T
S02 (1−v2 )n
√
(σ1 − σ2 ) T
2
ln
ln ã(z) − θ2 T + (λ − λ∗ )T + (ln λ∗ − ln λ)n
7 (n) =
, 8 (n) =
√
|θ| T
,
S02 (1−v2 )n
σ2
+ (μ2 − 22 )T
z
√
σ2 T
,
1,3 (z, n) = min{1 (n), 3 (z, n)}, 5,6 (n) = min{5 (n), 6 (n)},
pt,n = exp(−λ(T − t))
1 =
1
−sign(θ)
−sign(θ)
1
3 =
(λ(T − t))n
(λ∗ (T − t))n
∗
, pt,n
,
= exp(−λ∗ (T − t))
n!
n!
1
sign(θ)
sign(θ)
1
⎛
⎞
1
sign(θ)
−1
⎜
⎟
, 2 = ⎝ sign(θ)
1
−sign(θ) ⎠ ,
−1
−sign(θ)
1
⎞
1
−sign(θ) −1
⎟
⎜
, 4 = ⎝ −sign(θ)
1
sign(θ) ⎠ ,
−1
sign(θ)
1
⎛
5 =
1 −1
−1 1
.
Proof See Appendix 2.
Note that distribution functions 2 (x, y, 2± ) and 2 (x, y, z, 3± ) can
Remark 4
be expressed in terms of (·):
2 (x, y, 2+ ) = min(x, y) ;
2 (x, y, 2− ) =
(x) − (−y), if x > −y,
0,
otherwise;
(39)
(40)
3 (x, y, z, 3+ ) =
min(x, y) − (−z), if min(x, y) > −z,
0,
otherwise;
(41)
3 (x, y, z, 3− ) =
(x) − (max(−y, −z)), if x > max(−y, −z),
0,
otherwise;
(42)
324
A. Melnikov and H. Wan
where
2± =
⎛
⎞
1 ±1 −1
1 ±1
±
, 3 = ⎝ ±1 1 ∓1 ⎠ .
±1 1
−1 ∓1 1
Example 1 We consider the Russell 2000 (RUT-I) and the Dow Jones Industrial Average (DJIA) as S 1 and S 2 . Melnikov and Skornyakova [10] estimated
(μi , σi ) (i = 1, 2) for those two risky assets using daily observations and assume
υ1 = υ2 = 0, i.e., the Black–Scholes model. In our paper, we assume underlying
risky assets with same drifts and volatilities as Melnikov and Skornyakova [10] but
with nonzero jump components. Values of parameters are given as the following:
μ1 = 0.0481, σ1 = 0.2232,
μ2 = 0.0417, σ2 = 0.2089, S01 = S02 = 100,
υ1 = −0.05, υ2 = −0.1, λ = 0.1, r = 0.
We assume the hazard rate β is constant. The insured’s age, contract maturities, and
levels of β would differ in this example. According to the most recently published
United States 2015 Life Table (National Vital Statistics Reports volume 67, Number
7), the survival probability T px of a given insured can be found, and then for a fixed
confidence level α = 0.95, utilizing results in Theorem 3, we can derive the minimal
CVaR that can be achieved with the hedging capital T px U0 . Results are displayed
in Table 1. It is observed that the minimal CVaR increases as the insured’s age or
the contract maturity increases. This is because for an older client and a long-term
contract, the survival probability of the insured would decrease so as the premium
which leads to an increase in hedging losses. Moreover, for a given insured and a
contract maturity time, the minimal CVaR decreases as the hazard rate increases.
In addition, the decrease trend is more significant for long-term contracts. This can
be explained by the fact that there is a higher chance that contracts would mature
with a zero payoff if the hazard rate increases and for long-term contracts, there is
an even higher possibility that a default event would occur during contract periods,
which reduces the shortfall risk further.
Table 1 Minimal CV aR0.95 for contracts with different maturities, insured’s age, and levels of β
Age = 20
Age = 30
Age = 40
T = 10 T = 15 T = 20 T = 10 T = 15 T = 20 T = 10 T = 15 T = 20
0.9902 0.9839 0.9764 0.9860 0.9758 0.9609 0.97462 0.9509 0.9166
T px
β=0
0.996
1.6431 2.4136 1.4318 2.4745 4.017
2.5966 5.0477 8.6448
β = 0.01 0.9958 1.6378 2.3884 1.4316 2.4698 3.9978 2.5964 5.045
8.6394
β = 0.015 0.9956 1.6337 2.3684 1.4314 2.4662 3.9818 2.5963 5.0427 8.6335
β = 0.02 0.9954 1.6283 2.3411 1.4312 2.4613 3.9596 2.5961 5.0395 8.624
CVaR Hedging in Defaultable Jump-Diffusion Markets
325
5 Conclusion
This paper focuses on the problem of CVaR-based partial hedging in a defaultable
jump-diffusion model. We first provide the set of martingale measures in this
incomplete market which admit special forms Z ∗ Z k conditioned on {τ > T }, and
then by their properties, the minimal superhedging costs of a defaultable claim
with a zero recovery rate are given that coincide with the initial wealth required
for perfectly hedging the non-defaultable claim. Most importantly, we prove that
the optimal CVaR hedging problem in the defaultable market can be converted to
a problem of finding an optimal randomized test in the corresponding default-free
market. The hedging strategy can be explained as constructing the perfect hedging
of a modified non-defaultable claim during [0, τ ] while investing nothing in risky
assets and depositing all cash into the saving account after the default. Furthermore,
our method is applied to the area of life insurance. Numerical results show that for
a given insured and a contract maturity time, the minimal CVaR decreases as the
hazard rate increases.
Acknowledgments The authors are grateful to anonymous reviewers and the editor for fruitful
suggestions to improve the paper.
This research was supported by the Natural Sciences and Engineering Council of Canada under
Discovery Grant NSERC RES0043487.
Appendix 1: Proof of Theorem 2
For a fixed z, from Lemma 2 and Eqs. (30)–(32), we know that the optimal hedging
strategy of the efficient hedging problem (27) is a superhedging strategy of the
modified claim ϕ ∗ (z)C(z)I{τ >T } , and hence the shortfall is
%
#
(C 0 (z) − VT )+ = C 0 (z) − VT C 0 (z) = C 0 (z) − ϕ ∗ (z)C(z)I{τ >T } = C(z) 1 − ϕ ∗ (z) I{τ >T } .
Thereby,
#
CV aR(L) = min z +
z∈R
%
1
E C(z) 1 − ϕ ∗ (z) I{τ >T } .
1−α
1
With the notation d(z) = z + 1−α
E C(z) 1 − ϕ ∗ (z) I{τ >T } , it is clear that
CV aR(L) = d(ẑ), where ẑ is the point of minimum of the function d(z).
326
A. Melnikov and H. Wan
Furthermore, applying Lemma 1 to the modified claim ϕ ∗ (ẑ)C(ẑ)I{τ >T } , the
discounted value process of the superhedging strategy of such a claim is
Ũt = E ∗ e−rT ϕ ∗ (ẑ)C(ẑ)|Ft I{τ >t }
= E ∗ e−rT ϕ ∗ (ẑ)C(ẑ)|Ft − E ∗ e−rT ϕ ∗ (ẑ)C(ẑ)|Ft I{τ ≤t } .
(43)
Let us define Vt e−rt = E ∗ (e−rT ϕ ∗ (ẑ)C(ẑ)|Ft ). It is the discounted value process
for ϕ ∗ (ẑ)C(ẑ) in the default-free market.$This non-defaultable
claim has a replica$t
t
tion strategy π such that Vt e−rt = v0 + 0 πu1 d S̃u1 + 0 πu2 d S̃u2 .
Applying the integration by parts formula for E ∗ (e−rT ϕ ∗ (ẑ)C(ẑ)|Ft )I{τ ≤t } =
Vt e−rt Ht and because Ht has finite variation, we get
Vt e−rt Ht =
t
0
Hu− πu1 d S̃u1 +
t
0
Hu− πu2 d S̃u2 +
t
0
Vu e−ru dHu .
Notice that Hu− = lim I{τ ≤t } = I{τ <u} and consequently we have
t 'u
Ũt = v0 +
t
0
πu1 I{τ ≥u} d S̃u1 +
t
0
πu2 I{τ ≥u} d S̃u2 − Vτ e−rτ Ht .
(44)
Comparing (44) with the form of option decomposition, we conclude
π̂ti = πt i I{τ ≥t } , (i = 1, 2), Dt = Vτ e−rτ Ht .
(45)
Note that, for τ ≥ t, π̂t0 St0 = (Vt − πt 1 St1 − πt 2 St2 ) = πt 0 St0 , which implies
π̂t0 = πt 0 , while for τ < t, π̂t0 = Vτ e−rτ .
Appendix 2: Proof of Theorem 3
By Lemma 1, the minimal superhedging costs of C 0 = max(ST1 , ST2 )I{τ >T } are
U0 = sup E P̃ (e−rT CI{τ >T } ) = E ∗ (e−rT C)
P̃ ∈M
= e−rT E ∗ (ST1 I{S 1 ≥S 2 } ) + E ∗ (ST2 I{S 2 >S 1 } )
T
= e−rT
∞ #
n=0
T
T
T
E ∗ (ST1 I{S 1 ≥S 2 } |NT = n) + E ∗ (ST2 I{S 2 >S 2 } |NT = n) P ∗ (NT = n)
T
T
T
%
T
(46)
CVaR Hedging in Defaultable Jump-Diffusion Markets
327
Conditioning on {NT = n}, we have
,
σ12
T
r−
2
,
+
σ22
2
n v2 λ ∗ T
∗
T
exp σ2 WT + r −
≥ S0 (1 − v2 ) e
2
+
∗
S01 (1 − v1 )n ev1 λ T exp
{ST1 ≥ ST2 } =
σ1 WT∗ +
= {Y1 ≤ 1 (n)},
where Y1 = −
W∗
√T
T
∼ N(0, 1), 1 (n) =
∗
S 1 (1−v1 )n ev1 λ T
σ 2 −σ 2
1 T
2
∗ )+
2
v
λ
n
S0 (1−v2 ) e 2 T
√
ln( 02
(σ1 −σ2 ) T
.
{ST2 > ST1 } = {Y1 > 1 (n)} = {−Y1 < −1 (n)} = {Y2 < −1 (n)},
W∗
where Y2 = √ T ∼ N(0, 1).
T
According to the multi-asset theorem (see Melnikov and Romanyuk [9]), Eq. (46)
can be rewritten as
⎡
σ2
∞
r− 21 T
∗
∗ ⎣ 1
p0,n
E ∗ eσ1 WT I{Y1 ≤1 (n)}
s0,n e
U0 = e−rT
n=0
σ2
2 (r− 22 )T ∗ σ2 WT∗
+s0,n
e
E (e
I{Y2 <−1 (n)} )
=
∞
#
√
√ %
∗
1
2
p0,n
s0,n
(1 (n) + σ1 T ) + s0,n
(−1 (n) − σ2 T ) ,
n=0
∗
n
T
T
∗ = P ∗ (N = n) = exp(−λ∗ T ) (λ T ) .
where p0,n
T
n!
Part A of this theorem is proved. Let us start to prove part B. Firstly, for a fixed
z, we have
E ∗ C(z)I{a<Z ∗−1 } = E ∗ (ST1 − z)+ I{S 1 ≥S 2 } I{a<Z ∗−1 }
T
T
T
T
+ E ∗ (ST2 − z)+ I{S 1 <S 2 } I{a<Z ∗−1 }
T
T
T
∗
1
= E (ST − z)I{S 1 >z} I{S 1 ≥S 2 } I{a<Z ∗−1 }
T
T
T
T
+ E ∗ (ST2 − z)I{S 2 >z} I{S 1 <S 2 } I{a<Z ∗−1 } .
T
T
328
A. Melnikov and H. Wan
Conditioning on {NT = n}, we get
θ2
∗ −1
∗
∗
∗
ZT > a = exp −θ WT − T − (λ − λ )T − (ln λ − ln λ)n > a
2
= {Y3 < 2 (a, n)},
2
W∗
T
where Y3 = sign(θ ) √T ∼ N(0, 1), 2 (a, n) =
− ln aZt∗ −( θ2 +(λ−λ∗ )+(ln λ∗ −ln λ)n)T
√
.
|θ| T
σ12
)T ] > z}={Y1 < 3 (z, n)},
2
∗
{ST1 > z} = {S01 (1 − v1 )n ev1 λ T exp[σ1 WT∗ + (r −
s1
σ2
1 )T
ln 0,n
z +(r−
√ 2
where 3 (z, n) =
.
σ1 T
+
,
σ2
∗
{ST2 > z} = S02 (1 − v2 )n ev2 λ T exp σ2 WT∗ + r − 2 T > z = {Y1 < 4 (z, n)},
2
s2
σ2
2 )T
ln 0,n
z +(r−
√ 2
where 4 (z, n) =
.
σ2 T
And thus, we have
∞
E C(z)I{a<Z ∗−1 } =
E ∗ C(z)I{a<Z ∗−1 } |NT = n P ∗ (NT = n)
∗
T
n=0
T
+
+ ,
∞
σ12
∗
∗
1
∗
p0,n E
s0,n exp σ1 WT + r −
=
T −z
2
n=0
×I{Y1 <1 (n)} I{Y3 <2 (a,n)} I{Y1 <3 (z,n)}
+
,
σ12
∗
2
∗
+E
s0,n exp σ1 WT + r −
T −z
2
×I{Y2 <−1 (n)} I{Y3 <2 (a,n)} I{Y1 <4 (z,n)}
+
+ ,
∞
σ12
∗
∗
1
∗
=
p0,n E
s0,n exp σ1 WT + r −
T −z
2
n=0
×I{Y1 <1,3 (z,n)} I{Y3 <2 (a,n)}
+
+E ∗
σ2
2
s0,n
exp σ2 WT∗ + r − 2
2
,
T
×I{Y2 <−1 (n)} I{Y3 <0 (a,n)} I{Y1 <4 (z,n)}
2
−z
%
,
(47)
CVaR Hedging in Defaultable Jump-Diffusion Markets
329
where 1,3 (z, n) = min{1 (n), 3 (z, n)}. According to the multi-asset theorem,
(47) becomes
=
∞
σ2
r− 21 T
∗
∗ s 1 e
p0,n
E ∗ eσ1 WT I{Y1 <1,3 (z,n)} I{Y3 <2 (a,n)}
0,n
n=0
− zP ∗ (Y1 < 1,3 (z, n), Y3 < 2 (a, n))
2 e
+ s0,n
σ2
r− 22 T
∗
E ∗ eσ2 WT I{Y2 <−1 (n)} I{Y3 <2 (a,n)} I{Y1 <4 (z,n)}
− zP ∗ (Y2 < −1 (n), Y3 < 2 (a, n), Y1 < 4 (z, n))
=
∞
√
√
∗ s 1 erT 2 ( (z, n) + σ
p0,n
1,3
1 T , 2 (a, n) − σ1 sign(θ) T , 1 )
0,n
n=0
− z2 (1,3 (z, n), 2 (a, n), 1 )
√
√
√
2 erT 3 (− (n) − σ
+ s0,n
1
2 T , 2 (a, n) − σ2 sign(θ) T , 4 (z, n) + σ2 T , 2 )
− z3 (−1 (n), 2 (a, n), 4 (z, n), 2 ) ,
where the correlation matrix 1 and 2 are
1 =
1
−sign(θ )
−sign(θ )
1
⎛
⎞
1
sign(θ )
−1
, 2 = ⎝ sign(θ )
1
−sign(θ ) ⎠ .
−1 −sign(θ )
1
Thus, the equation E ∗ (e−rT C(z)I{a<Z ∗−1 } ) = v0 is (35) and for a fixed z ∈ [0, z∗ ]
T
and 0 ≤ a1 < a2 , we have
E ∗ C(z)I{a1 <Z ∗−1 } ≥ E ∗ C(z)I{a2 <Z ∗−1 } .
T
(48)
T
However, the equality holds if and only if C(z)I{a1 <Z ∗−1 } = C(z)I{a2 <Z ∗−1 } , a.s,
T
T
i.e.,
P ∗ C(z)I{a1 <Z ∗−1 } = C(z)I{a2 <Z ∗−1 } = 1 → P ∗ C(z)I{a1 <Z ∗−1 ≤a2 } = 0 = 1,
T
T
T
which is not true. Hence, E ∗ (e−rT C(z)I{a<Z ∗−1 } ) is a strictly decreasing function
T
regarding to a.
On the other hand, for z ∈ [0, z∗ ], we have
E ∗ C(z)I{0<Z ∗−1 } = E ∗ C(z) ≥ v0 (becauseoftheinitialvalueconstraint)
T
(49)
330
A. Melnikov and H. Wan
and
lim E ∗ C(z)I{a<Z∗−1T } = 0.
(50)
a→∞
Consequently, we can conclude that there is a root for (35) and it is unique.
As for E(C(z)I{ã(z)≥Z ∗−1} I{τ >T } ), we have
T
$T
E(C(z)I{ã(z)≥Z ∗−1} I{τ >T } ) = E(C(z)I{ã(z)≥Z ∗−1} )e− 0 βu du
T
$T
= e− 0
T
βu du
E((ST1 − z)I{S 1 ≥S 2 } I{S 1 >z} I{ã(z)≥Z ∗−1} )
T
T
T
T
2
+ E((ST − z)I{S 2 >S 1 } I{S 2 >z} I{ã(z)≥Z ∗−1} ) .
T
T
T
(51)
T
Conditioning on {NT = n}, n = 0, 1... , and under the measure P , we can present
{ST1 ≥ ST2 }, {ST1 > z},{ã(z) ≥ ZT∗−1 }, {ST2 > ST1 }, {ST2 > z} as follows:
{ST1 > z} =
S01 (1 − v1 )n exp
WT
where Y4 = − √
T
σ2
σ1 WT + μ1 − 1
2
∼ N(0, 1) and 5 (n) =
{ST1 ≥ ST2 } =
ln
T
> z = {Y4 < 5 (n)},
S01 (1−v1 )n
σ2
+(μ1 − 21 )T
z
√
σ1 T
σ2
σ1 WT + μ1 − 1
2
σ2
× exp σ2 WT + μ2 − 2 T
2
S01 (1 − v1 )n exp
.
T
≥ S02 (1 − v2 )n
= {Y4 ≤ 6 (n)},
where 6 (n) =
ln
S 1 (1−v1 )n
σ 2 −σ 2
0
+(μ1 −μ2 + 2 2 1 )T
2
n
S0 (1−v2 )
√
(σ1 −σ2 ) T
.
{ST1 < ST2 } = {−Y4 < −6 (n)} = {Y5 < −6 (n)}.
WT
where Y5 = √
∼ N(0, 1).
T
θ2
{ã(z) ≥ ZT∗−1 } = exp −θ WT + T − (λ − λ∗ )T − (ln λ∗ − ln λ)n < ã(z)
2
= {Y6 ≤ 7 (n)},
CVaR Hedging in Defaultable Jump-Diffusion Markets
331
2
WT
where Y6 = −sign(θ ) √
∼ N(0, 1) and 7 (n) =
T
{ST2 > z} =
σ2
σ2 WT + μ2 − 2
2
S02 (1 − v2 )n exp
ln
ln ã(z)− θ2 T +(λ−λ∗ )T +(ln λ∗ −ln λ)n
√
.
|θ| T
T
> z = {Y4 < 8 (n)},
S02 (1−v2 )n
σ2
+(μ2 − 22 )T
z
√
where 8 (n) =
.
σ2 T
Thus, Eq. (51) can be rewritten as
=e
∞
$
− 0T βu du
#
p0,n E((ST1 − z)I{Y4 <5 (n)} I{Y4 ≤6 (n)} I{Y6 ≤7 (n)} )
n=0
+ E((ST2 − z)I{Y5 <−6 (n)} I{Y6 ≤7 (n)} I{Y4 ≤8 (n)} )
$T
= e− 0 βu du
∞
%
#
√
p0,n S01 (1 − v1 )n eμ1 T 2 (5,6 (n) + σ1 T , 7 (n)
n=0
√
+ σ1 sign(θ ) T , 3 )
− z2 (5,6 (n), 7 (n), 3 )
√
+ S02 (1 − v2 )n eμ2 T 3 (−6 (n) − σ2 T , 7 (n)
√
√
+ σ2 sign(θ ) T , 8 (n) + σ2 T , 4 )
%
− z3 (−6 (n), 7 (n), 8 (n), 4 ) ,
where pt,n = exp(−λ(T − t)) (λ(Tn!−t )) , 5,6 = min{5 (n), 6 (n)}, and
n
3 =
1
sign(θ )
sign(θ )
1
⎛
⎞
1
−sign(θ ) −1
, 4 = ⎝ −sign(θ )
1
sign(θ ) ⎠ ,
−1
sign(θ )
1
and consequently, Eq. (36) is established.
According to Lemma 1, we have
sup E P̃ (e−rT C 0 (z)) = sup E P̃ (e−rT C(z)I{τ >T } ) = E ∗ (e−rT C(z))
P̃ ∈M
=e
P̃ ∈M
−rT
= e−rT
∗
[E ((ST1 − z)I{S 1 >z} I{S 1 ≥S 2 } ) + E ∗ ((ST2 − z)I{S 2 >z} I{S 2 >S 1 } )]
T
T
T
T
T
T
∞
n=0
#
σ2
∗
1 (r− 21 )T ∗ σ1 WT∗
p0,n
e
E (e
I{Y1 ≤1 (n)} I{Y1 <3 (z,n)} )
s0,n
332
A. Melnikov and H. Wan
− zP ∗ (Y1 ≤ 1 (n), Y1 < 3 (z, n))
σ22
∗
2 (r− 2 )T ∗ σ2 WT
e
E (e
I{Y2 <−1 (n)} I{Y1 <4 (z,n)} )
+ s0,n
%
− zP ∗ (Y2 < −1 (n), Y1 < 4 (z, n))
=
∞
#
√
∗
1
s0,n
p0,n
(1,3 (z, n) + σ1 T ) − ze−rT (1,3 (z, n))
n=0
√
√
2
+ s0,n
2 (−1 (n) − σ2 T , 4 (z, n) + σ2 T , 5 )
%
− ze−rT 2 (−1 (n), 4 (z, n), 5 ) ,
1 −1
where 5 =
.
−1 1
+
Since C(z) = (C − z)+ = max(ST1 , ST2 ) − z is decreasing as regards to z,
and hence we have E ∗ (e−rT C(z1 )) ≥ E ∗ (e−rT C(z2 )) for any 0 ≤ z1 < z2 . We can
prove the inequality is strict, because
E ∗ (e−rT C(z1 )) = E ∗ (e−rT C(z2 )) → C(z1 ) = C(z2 ), a.s
→ (C − z1 )+ I{C>z2 } = (C − z2 )+ I{C>z2 } , a.s
→ (z2 − z1 )I{C>z2 } = 0, a.s → P ∗ C > z2 = 0,
which is not true because C = max(ST1 , ST2 )+ ∈ (0, ∞) and the probability that
C is bigger than a constant is positive. Hence, E ∗ (e−rT C(z)) is strictly decreasing
regarding to z. Moreover, we have
E ∗ C(0) = E ∗ C > v0 ,
lim E ∗ C(z) = 0,
z→∞
and consequently, we conclude that there is a root for the equation E ∗ (e−rT C(z)) =
v0 which is (38) and it is unique.
References
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51(1), 182–190 (2012)
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3(1), 125–144 (1976)
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Financ. 26, 1443–1471 (2002)
Out-of-Sample Utility Bounds for
Empirically Optimal Portfolios in a
Single-Period Investment Problem
Dmitry B. Rokhlin
Abstract We provide bounds for estimation and generalization errors of empirically optimal portfolios in a single-period utility maximization problem. The
portfolios are selected after n observations of stock returns. The cases of Lipschitz,
Hölder, and linear utilities are considered separately. It is assumed that the returns
are independent and identically distributed and belong to an Lp or an Orlicz L"p
space. For concave utilities we give estimation error bounds for portfolios produced
by the exponentiated gradient algorithm.
Keywords Portfolio selection · Statistical learning · Empirical utility ·
Generalization bounds · Tails of the returns
1 Introduction
We consider the portfolio selection problem for an investor, whose preferences are
described by an expected utility function. After n observations of (daily) returns, the
investor selects fractions of wealth to be invested in each of d stocks. It is assumed
that returns are independent and identically distributed, but their distribution is
unknown.
These assumptions, in fact, reduce the single-period investment problem to a
standard statistical learning problem, where an expected utility is used instead
of an expected loss. However, the standard results concerning the estimation and
generalization error bounds can be applied only under restrictive assumptions, such
as boundedness of returns, boundedness of the utility function and/or its Lipschitz
continuity. In most of the classical financial problems, these assumptions are not
satisfied. Consider, for example, the lognormal model for stock returns and the
D. B. Rokhlin ()
I.I. Vorovich Institute of Mathematics, Mechanics and Computer Sciences and Regional Scientific
and Educational Mathematical Center of Southern Federal University, Rostov-on-Don, Russia
e-mail: dbrohlin@sfedu.ru
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_18
335
336
D. B. Rokhlin
power utility u(x) = x α , α ∈ (0, 1). In this case the returns are unbounded, the
utility function is unbounded, and only Hölder continuous.
In this paper we provide out-of-sample error bounds in the mentioned singleperiod investment problem for unbounded returns and utility functions under
appropriate assumptions on the tails of the returns. More precisely we assume that
the returns belong to an Lp or an Orlicz L"p space.
We are not aware of an analysis of the described problem in the classical form,
adopted in the present paper. However, we can mention several papers considering
portfolio out-of-sample performance in the utility optimization problems with
various forms of regularization: [3, 5] (distributionally robust formulation), [2]
(regularization in problems with side information), [6, 8] (portfolio constraints), and
[1] (performance-based regularization). We also mention the paper [13], considering
a similar problem with the relative performance criterion.
In Sect. 2 we recall the terminology of learning theory in connection with the
investment problem, and present a general out-of-sample bound. The cases of Lipschitz, Hölder (Sect. 3), and linear (Sect. 4) utilities are considered separately, since
they require somewhat different technique. In Sect. 5 we consider the portfolios
produced by the exponentiated gradient (EG) algorithm, assuming additionally that
the utility is concave.
For Lipschitz and Hölder utilities, we obtain error bounds of order n−1/2 (recall
that n is the number of available observations), if the returns are at least square
integrable. Constants depend on the number d of stocks. They improve when the
tails become lighter. For non-square integrable returns, the obtained error bounds
decrease slower than n−1/2 . For linear utilities the decreasing rates are faster. In
particular, they are exponential for returns with finite moment-generating functions.
For concave utilities we obtain standard O(n−1/2 ) bounds for portfolios constructed
by the EG algorithm.
In Sects. 3 and 4, the proofs are based on the notion of Rademacher complexity
and various probabilistic inequalities. In Sect. 5 we use the known regret bound for
the online mirror descent algorithm and the online-to-batch conversion scheme.
2 A General Error Bound
Let (sk1 , . . . , skd ) be the prices of d risky assets at time moments k = 0, . . . , n + 1.
We assume that these prices are strictly positive and their (total) daily returns rki =
i
ski /sk−1
, k = 1, . . . , n + 1, are independent and identically distributed. At time n
an investor selects a portfolio (γn1 , . . . , γnd ), based on the price history (r1 , . . . , rn ).
The quantity γni ≥ 0 represents the number of units of i-th stocks to be bought.
Investor’s wealth is distributed between d stocks in accordance with fractions:
j j d
d
γn Sn
∈ = {z ≥ 0 :
zj = 1}.
νn =
Xn
j =1
j =1
Out-of-Sample Utility Bounds for Empirically Optimal Portfolios
337
Without loss of generality, put Xn = 1. At time n + 1,
Xn+1 = $γn , Sn+1 % = $νn , rn+1 %,
where $a, b% is the usual scalar product in Rd .
We assume that the investor’s preferences are described by a utility function:
u : [0, ∞) ,→ [0, ∞),
u(0) = 0
which is nondecreasing and continuous. Consider the single-period expected utility
maximization problem:
U (ν) = Eu($ν, rn+1 %) → max .
ν∈
(1)
Lemma 1 Let
sup Euα ($ν, rn+1 %) < ∞
for some α > 1.
(2)
ν∈
Then, the problem (1) is solvable.
Proof Let νj be a maximizing sequence:
u($νj , rn+1 %) → sup Eu($ν, rn+1 %),
j → ∞.
ν∈
Without loss of generality, we can assume that νj → ν ∗ ∈ . From (2) it follows
that the sequence (u($νj , rn+1 %))∞
j =1 is uniformly integrable [15, Chap. 2, Sect. 6,
Lemma 3]. Hence,
Eu($νj , rn+1 %) → Eu($ν ∗ , rn+1 %),
j → ∞,
(see [15, Chap. 2, Sect. 6, Theorem 4]) and ν ∗ is an optimal solution.
Denote by 0
νn an optimal solution of the empirical utility maximization problem:
0n (ν) = 1
U
n
n
u($ν, rk %) → max .
k=1
ν∈
0n (ν) − U (ν). Using the inequalities
Consider the empirical process Gn (ν) = U
0n (0
0n (ν ∗ ) ≤ U
νn ),
U
U (0
νn ) ≤ U (ν ∗ ),
338
D. B. Rokhlin
where U (0
νn ) = E (f (0
νn , rn+1 )|r1 , . . . , rn )) is the conditional expectation, we
obtain
0n (ν ∗ ) + U
0n (0
νn ) ≤ U (ν ∗ ) − U
νn ) − U (0
νn )
U (ν ∗ ) − U (0
0n (ν ∗ ) + sup Gn (ν),
≤ U (ν ∗ ) − U
ν∈
0n (0
0n (0
νn ) − U (ν ) ≤ U
νn ) − U (0
νn ) ≤ sup Gn (ν).
U
∗
ν∈
Put Gn = E supν∈ Gn (ν). Taking the expectation, we get
0n (0
max{U (ν ∗ ) − EU (0
νn ), EU
νn ) − EU (0
νn )} ≤ Gn .
Following the terminology of the learning theory [16], we can say that the quantity
Gn provides estimation and generalization error bounds (in expectation). In Sects. 3
and 4, we present non-asymptotic bounds for Gn under various assumptions.
3 Lipschitz and Hölder Utilities
A well-known tool, suitable for bounding Gn , is the Rademacher complexity [11,
16]. For a set A ⊂ Rn , its Rademacher complexity R(A) is defined as follows:
1
ak εk ,
R(A) = E sup
n a∈A
n
k=1
where εk are independent Rademacher random variables: P(εk = 1) = P(εk =
−1) = 1/2. Furthermore, consider the class of functions F = {u($ν, ·%) : ν ∈ },
which are defined on Rn+ . Put Rn = (r1 , . . . , rn ),
F ◦ Rn = {(u($ν, r1 %), . . . u($ν, rn %) : ν ∈ } ⊂ Rn .
The quantities
n
1
0
εk u($ν, rk %)Rn ,
R(F ◦ Rn ) = E sup
n
ν∈
R(F ◦ Rn ),
R(F ◦ Rn ) = E0
k=1
are called the empirical Rademacher complexity and the Rademacher complexity
of F with respect to the random sequence Rn . The symmetrization argument [16]
gives the bound
Gn ≤ 2R(F ◦ Rn ).
(3)
Out-of-Sample Utility Bounds for Empirically Optimal Portfolios
339
Recall (see, e.g., [17]) that a random variable X with mean EX = μ is called:
• Sub-Gaussian with parameter σ 2 if
2 2
Eeλ(X−μ) ≤ eσ λ /2 ,
λ∈R
• Sub-exponential with parameters (σ 2 , α) if
2 2
Eeλ(X−μ) ≤ eσ λ /2 ,
|λ| <
1
α
p
For "p (x) = ex − 1, p ≥ 1, denote by
X"p = inf t > 0 : E"p
X
t
≤1
the "p -Orlicz norm. The space L"p of functions with finite "p -Orlicz norm is
called an Orlicz space. Sub-Gaussian and sub-exponential random variables are
precisely the elements of L"2 and L"1 , respectively: see their characterization in
[17] (Theorems 2.6 and 2.13). A more general class of Orlicz spaces is induced
by N-functions [12], that is, even convex functions " : R ,→ R+ , satisfying the
conditions
"(0) = 0,
lim "(x)/x = 0,
x→0
lim "(x)/x = +∞.
x→∞
We will use two inequalities for the expected maximum of finitely many random
variables. The first one is reformulated from [17, Lemma 5.37], and the second is
taken from [4, Theorem 2.5].
Lemma 2 Let Yi ≥ 0, Yi " ≤ 1, i = 1, . . . , d, where " is a strictly increasing
N-function. Assume that " −1 is concave. Then,
E(IA max Yj ) ≤ P(A)" −1
1≤j ≤d
d
P(A)
for any measurable set A.
Lemma 3 Suppose that ψj (λ) = ln EeλXj ≤ ψ(λ), j ∈ {1, . . . , d}, where ψ is
a convex and continuously differentiable function defined on [0, b), and ψ(0) =
ψ (0) = 0. Then,
E max Xj ≤ ψ ∗−1 (ln d),
1≤j ≤d
ψ ∗−1 (y) := inf
λ∈(0,b)
y + ψ(λ)
.
λ
340
D. B. Rokhlin
For sub-exponential returns r1i with parameters (σi2 , αi ), we will use the notation
μi := Er1i ,
ρ 2 = max (σi2 + μ2i ),
α = max αi .
1≤i≤d
1≤i≤d
Also put rk∗ = max{rk1 , . . . , rkd }, k = 1, . . . , n.
Theorem 1 Let the utility function u be globally Lipschitz continuous:
|u(x1) − u(x2 )| ≤ K|x1 − x2 |.
(a) If r1i ∈ L∞ , then
;
Gn ≤ 2K
2 ln d
max r i L∞ .
n 1≤i≤d 1
(b) If r1i ∈ Lp , p ∈ [2, ∞), then
;
Gn ≤ 2K
2 ln d 1/p
d
max r1i Lp .
1≤i≤d
n
(c) If r1i ∈ Lp , p ∈ (1, 2), then
1/q p 1/p
q
Gn ≤ 2K
+
q
p
×
2 ln d
n
1/(2q)
j
d 1/p max r1 Lp ,
1≤i≤d
1
1
+ = 1.
p q
(d) If r1i are sub-exponential with parameters (σi2 , αi ), then
√
2K
2α ln d
.
Gn ≤ √ max ρ 2 ln d, √
n
n
In particular, if r1i are sub-Gaussian with parameters σi2 , then
;
Gn ≤ 2Kρ
2 ln d
.
n
Proof By the contraction lemma [16, Theorem 9], we have
0
R(F ◦ Rn ) ≤ K 0
R ({$ν, ·% : ν ∈ } ◦ Rn ) .
(4)
Out-of-Sample Utility Bounds for Empirically Optimal Portfolios
341
Furthermore, can be replaced by its extreme points e1 = (1, 0, . . . , 0), . . . , ed =
(0, . . . , 0, 1) (see the same reference):
0
R ({$ν, ·% : ν ∈ } ◦ Rn ) = 0
R ({$ei , ·% : i = 1, . . . , d} ◦ Rn ) = 0
R(r 1 , . . . , r d ),
R({r 1 , . . . , r d }). So, we can replace the right-hand side
where 0
R(r 1 , . . . , r d ) := 0
in (3) by the Rademacher complexity of the trajectories r i = (r1i , . . . , rni ) of stock
returns:
Gn ≤ 2KR(r 1 , . . . , r d ).
(5)
By the Massart lemma [16, Lemma 8] [11, Theorem 3.7],
C
√
D n
D
2
ln
d
1
d
0
max E (rki )2 .
R(r , . . . , r ) ≤
1≤i≤d
n
k=1
In the remaining part of the proof, we give the estimates for 0
R(r 1 , . . . , r d ), which
should be combined with (5) to obtain all assertions of the theorem.
(a) For uniformly bounded returns, we have
;
0
R(r 1 , . . . , r d ) ≤
2 ln d
max max r i L∞ =
n 1≤i≤d 1≤k≤n k
;
2 ln d
max r i L∞ .
n 1≤i≤d 1
(6)
(b) For p ∈ [1, ∞), by the Hölder inequality,
n
1/p
n
(rki )2 ≤
(rki )2p
n1/q ,
k=1
k=1
1
1
+ = 1.
p q
Using also the inequality E (ξ γ ) ≤ (Eξ )γ , γ ∈ (0, 1), ξ ≥ 0, we get
C
n
1/(2p)
D n
D i
1/(2q)
i 2p
2
E
E max
(r ) ≤ n
E max
(r )
1≤i≤d
k
1≤i≤d
k=1
≤n
1/(2q)
E max
n
1≤i≤d
= n1/(2q) n1/(2p)
k
k=1
1/(2p)
(rki )2p
≤n
1/(2q)
k=1
d
i=1
d
1/(2p)
n
i 2p
E
(rk )
i=1
k=1
1/(2p)
E(r1i )2p
≤ n1/2 d 1/(2p) max r1i L2p .
1≤i≤d
342
D. B. Rokhlin
(c) For M > 0 we have
1
1
E max
εk rki I{rk∗ ≤M} + E max
εk rki I{rk∗ >M}
n 1≤i≤d
n 1≤i≤d
n
R(r 1 , . . . , r d ) ≤
n
k=1
k=1
≤ R(r 1 I{r ∗ ≤M} , . . . , r d I{r ∗ ≤M} ) +
1
E
n
n
rk∗ I{rk∗ >M}
k=1
= R(r 1 I{r ∗ ≤M} , . . . , r d I{r ∗ ≤M} ) + E(r1∗ I{r1∗ >M} ).
(7)
Since rki I{rk∗ ≤M} ≤ M, we can apply (6) to estimate the first term in the righthand side:
√
2 ln d
1 ∗
d
∗
R(r I{r ≤M} , . . . , r I{r ≤M} ) ≤ √ M.
(8)
n
To estimate the second term, put Yi = r1i /r1i Lp and apply Lemma 2 with
"(x) = x p , p ∈ (1, 2):
E(r1∗ I{r1∗ >M} ) ≤ E
I{r1∗ >M} max Yi max r1i Lp
1≤i≤d
≤ P(r1∗ > M)
1≤i≤d
d
∗
P(r1 > M)
1/p
max r1i Lp
1≤i≤d
= d 1/p P(r1∗ > M)1/q max r1i Lp .
(9)
1≤i≤d
By the Markov inequality,
P (r1∗ > M) ≤
E(r1∗ )p
d max1≤i≤d r1i Lp
≤
Mp
Mp
p
(10)
From (7) to (10), it follows that
√
d
2 ln d
p
R(r , . . . , r ) ≤ √ M + p/q max r1k Lp .
1≤k≤d
M
n
1
d
Optimization over M > 0 after simple calculations gives assertion (c).
(d) Finally, assume that r1i are (σi2 , αi )-sub-exponential, and consider the logmoment-generating functions of r1i − μi :
ψi (λ) = ln Eeλ(r1−μi ) ≤
i
σi2 λ2
,
2
|λ| <
1
.
αi
Out-of-Sample Utility Bounds for Empirically Optimal Portfolios
343
We have
1 λr i 1 −λr i
eλμi ψi (λ) e−λμi ψi (−λ)
Ee k + Ee k ≤
e
e
+
2
2
2
2
λμi
e
e−λμi
2 2
2 2
2 2
+
≤ eσi λ /2
≤ eσi λ /2 eλ μi /2 , |λ| < 1/αi .
2
2
i
Eeλεk rk =
In the last step, we used an elementary inequality: see [17, Example 2.3].
For the random variables $ε, r i % = nk=1 εk rki , we get
ln Ee
λ$ε,r i %
=
n
i
ln Eeλεk rk =
k=1
n(σi2 + μ2i ) 2
λ ,
2
|λ| <
nρ 2 λ2
,
2
1
;
α
1
.
αi
Hence,
ln Eeλ$ε,r % ≤ ψ(λ) :=
i
|λ| <
see the notation (4). By Lemma 3,
E sup $ε, r i % ≤ ψ ∗−1 (ln d),
1≤i≤d
where
ψ
∗−1
nρ 2
y
y + ψ(λ)
= inf
+
λ .
(y) = inf
λ∈(0,1/α)
λ∈(0,1/α) λ
λ
2
By an elementary calculation in the sub-Gaussian case (α = 0), we get
ψ ∗−1 (y) = ρ 2ny,
and in the general sub-exponential case,
ψ
∗−1
√
ρ 2ny,
√
√
2y ≤ ρ n/α,
(y) =
√
√
2y ≥ ρ n/α.
αy + nρ 2 /(2α),
It easily follows that
ψ ∗−1 (y) ≤ max{ρ 2ny, 2αy}.
344
D. B. Rokhlin
Thus,
1
1
E max $ε, r i % ≤ ψ ∗−1 (ln d)
n 1≤i≤d
n
√
2α ln d
1
.
≤ √ max ρ 2 ln d, √
n
n
R(r 1 , . . . , r d ) =
This gives (d).
Theorem 2 Assume that u is globally Hölder continuous:
|u(x) − u(y)| ≤ K|x − y|α ,
(a)
If r1i ∈ L∞ , then
;
Gn ≤ CK
(b)
Gn ≤ CK
d − 1 α/p
max r1i αLp .
d
αn
1≤i≤d
If r1i ∈ Lp , 1 ≤ p < 2α, then
Gn ≤ (CK)1−α/p
×
(d)
d −1
max r i α ∞ .
αn 1≤i≤d 1 L
If r1i ∈ Lp , max{2α, 1} ≤ p < ∞, then
;
(c)
α ∈ (0, 1).
d−1
αn
p−α
α
α/p
+
α
p−α
(p−α)/p (p−α)/(2p)
d α/p max r1i αLp .
1≤i≤d
If r1i ∈ L"p , p ≥ max{2α, 1}, then
;
Gn ≤ CK
d −1
(1 + ln d)α/p max r1i α"p .
1≤i≤d
αn
Here, C is an absolute constant.
Proof Similar to [13] note that the process
1
εj u($ν, rj %)
n
n
Zn (ν) =
j =1
Out-of-Sample Utility Bounds for Empirically Optimal Portfolios
345
is sub-Gaussian:
λ(Zn (ν)−Zn (ν )) λ2 ρ 2 (ν,ν )/2
E e
Rn ≤ e
with respect to the data-dependent pseudometric
⎛
ρ(ν, ν ) =
1⎝
n
⎞1/2
n
2
u($ν, rj %) − u($ν , rj %) ⎠
,
j =1
defined on . A set N ⊂ is called -dispersed if ρ(ν, ν ) ≥ for ν, ν ∈ N with
ν = ν . Let D(, ρ, ) be the -packing number of (, ρ):
D(, ρ, ) = sup{|N| : N is an −dispersed}.
Here, |N| is the cardinality of N. The conditional expectation of the supremum of
Zn is bounded by the Dudley entropy integral [4, Corollary 13.2]:
d/2
0
R(F ◦ Rn ) = E sup Zn (ν)Rn ≤ 12
ln D(, ρ, ) d,
(11)
0
ν∈
where d is the diameter of .
By the Hölder continuity of u, we get
K
ρ(ν, ν ) ≤
n
n
1/2
|$ν − ν , rk %|
2α
k=1
≤ KAn ν − ν αl1 ,
al 1 =
K
≤
n
d
n
1/2
(rk∗ )2α ν − ν 2α
1
k=1
|aj |,
j =1
n
∗ 2α 1/2 . In [13] it is shown that a similar estimate together
where An = n1
k=1 (rk )
with (11) and an estimate of the -packing number of with respect to the metric,
induced by · l 1 (see [7, Proposition C.1]), imply the inequality
;
0
R(F ◦ Rn ) ≤ C1 K
d −1
An ,
α
C1 = 12
1;
ln
0
5
dz.
z
(12)
The assertions of the theorem follow from (3), (12), and subsequent estimates.
(a) For uniformly bounded returns,
An ≤ n−1/2 max r1i αL∞ .
1≤i≤d
(13)
346
D. B. Rokhlin
(b) For r1i ∈ Lp , p ≥ max{2α, 1}, we have
1/2
n
=
1
1 =
1
2α/p
∗ 2α
E
EAn ≤
(rk )
= √ E(r1∗ )2α ≤ √
E(r1∗ )p
n
n
n
k=1
α/p
1
d α/p
d max E(r1i )p
≤√
= √ max r1i αLp .
1≤i≤d
n
n 1≤i≤d
(c) For r1i ∈ Lp , 1 ≤ p < 2α, we use the same direct approach, as for (c) in
Theorem 1. Consider the estimate
1
E sup
εk u($ν, rk I{rk∗ ≤M} %) + E(u(r1∗ )I{r1∗ >M} ).
n ν∈
n
R(F ◦ Rn ) ≤
k=1
The first term can be bounded using (12) and (13):
1
E sup
εk u($ν, rk I{rk∗ ≤M} %) ≤ C1 K
n ν∈
n
k=1
;
d −1
max M α .
αn 1≤i≤d
Let us apply Lemma 2 to "(x) = x p/α and
Yi = (r1i )α /r1i αp
to bound the second term:
E(u(r1∗ )I{r1∗ >M} ) ≤ KE((r1∗ )α I{r1∗ >M} )
α/p
d
≤ KP(r1∗ > M)
max r1i αLp
1≤i≤d
P(r1∗ > M)
= Kd α/p P(r1∗ > M)1−α/p max r1i αLp .
1≤i≤d
Applying also the Markov inequality (10), we obtain
E(u(r1∗ )I{r1∗ >M} ) ≤ Kd
p
max1≤i≤d r1i Lp
.
M p−α
Thus,
;
R(F ◦ Rn ) ≤ C1 K
p
max1≤i≤d r1i Lp
d −1 α
M + Kd
.
αn
M p−α
Minimizing over M > 0, we get assertion (c).
Out-of-Sample Utility Bounds for Empirically Optimal Portfolios
347
(d) For r1i ∈ L"p , p ≥ max{2α, 1}, we have
(r1i )p
≤2
t p/(2α)
(r1i )p
2α
≤ 2 = r1i 2α
= inf τ : E exp
"p .
τp
(r1i )2α "p/(2α) = inf t > 0 : E exp
Let us apply Lemma 2 to Yi = (r1i )2α /r1i 2α
"p and "(x) = "p/(2α) (x) =
ex
p/(2α)
− 1:
−1
i 2α
E max (r1i )2α ≤ E( max Yi ) · max r1i 2α
"p = "p/(2α) (d) · max r1 "p
1≤i≤d
1≤i≤d
1≤i≤d
1≤i≤d
= (ln(1 + d))2α/p · max r1i 2α
"p .
1≤i≤d
The inequality
1 =
(ln(1 + d))α/p
EAn ≤ √ E(r1∗ )2α ≤
√
max r1i α"p
1≤i≤d
n
n
finishes the proof.
4 Linear Utility
Let u(x) = x and assume that
μ1 > μ2 ≥ · · · ≥ μd ,
μi = Er1i .
Then, the solution of (1) reduces to the choice of the asset with the largest expected
return:
Eu($ν ∗ , rn+1 %) = $ν ∗ , μ% = μ1 .
(14)
Put r i = n1 nk=1 rki . An optimal solution of the empirical utility maximization
problem (2) coincides with e1 = (1, 0, . . . , 0) on the event An = {r 1 >
max2≤i≤d r i } and with another basis vector ej on its compliment Acn . Thus,
Eu($ν̂n , rn+1 %) = E$ν̂n , μ% ≥ E μ1 IAn + μd IAcn = μ1 + (μd − μ1 )P(Acn ).
(15)
348
D. B. Rokhlin
From (14) to (15), we get
νn ) = E$ν ∗ , rn+1 % − E$ν̂n , rn+1 % ≤ (μ1 − μd )P(Acn ).
U (ν ∗ ) − EU (0
(16)
We will use the following corollary from the Marcinkiewicz-Zygmund inequality: see [9, Corollary 8.2].
Lemma 4 Let Yk be i.i.d. random variables with EYk = 0 and E|Yk |p < ∞ for
some p ≥ 1. Then, there exists a constant Bp depending only on p, such that
Bp np/2 E|Y1 |p , p ∈ [2, ∞),
E|Y1 + · · · + Yn |p ≤
Bp nE|Y1 |p ,
p ∈ [1, 2].
Theorem 3 Assume that the utility function is linear: u(x) = x.
(a) If Eeλ(r1−r1 ) < ∞, i = 2, . . . , d for some positive λ, then
U (ν ∗ ) − EU (0
νn ) ≤ (d − 1)(μ1 − μd ) exp −n min ψi∗ (μ1 − μ2 ) ,
i
1
2≤i≤d
where
ψi∗ (t) = sup{λt − ψi (λ)},
ψi (λ) = ln Eeλ((r1−r1 )−(μi −μ1 )) .
i
1
λ≥0
(b) If r i − r 1 ∈ Lp for some p ∈ (1, ∞), then
Bp (d − 1)(μ1 − μd )
p
max r1i − r11 − (μi − μ1 )Lp ,
p
γ
2≤i≤d
(μ1 − μ2 ) n
p/2,
p ∈ [2, ∞),
U (ν ∗ ) − EU (0
νn ) =
γ =
p − 1, p ∈ (1, 2].
Proof
(a) Put Xi = r i − r 1 − (μi − μ1 ). We have
P(Acn ) = P
i
1
max r − r ≥ 0
2≤i≤d
=P
max
2≤j ≤d
≤P
r j − r 1 − (μ2 − μ1 ) ≥ μ1 − μ2
d
max Xi ≥ μ1 − μ2 ≤
P(Xi ≥ μ1 − μ2 )
2≤i≤d
i=2
≤ (d − 1) max P(Xi ≥ μ1 − μ2 ).
2≤i≤d
(17)
Out-of-Sample Utility Bounds for Empirically Optimal Portfolios
349
Furthermore, since
ψXi (λ) = ln EeλXi = ln
n
<
Eeλ(rk −rk −(μi −μ1 ))/n = nψi (λ/n),
i
1
k=1
ψX∗ i (t) = sup(tλ − ψXi (λ)) = sup(tλ − nψi (λ/n)) = nψi∗ (t),
λ≥0
λ≥0
by the Chernoff inequality [4, Chap. 2], we get
P(Xi ≥ t) ≤ exp(−ψX∗ i (t)) = exp(−nψi∗ (t)).
(18)
Assertion (a) is implied by (16)–(18).
(b) Similar to (17), applying additionally the Markov inequality, we get
d
d−1
max E|Xi |p
(μ1 − μ2 )p 2≤i≤d
i=2
n
p
d −1
i
1
=
max
E
(r
−
r
−
(μ
−
μ
))
.
i
1
k
k
(μ1 − μ2 )p np 2≤i≤d P(Acn ) ≤
P(|Xi | ≥ μ1 − μ2 ) ≤
k=1
Assertion (b) now follows from (16) and Lemma 4.
5 Exponentiated Gradient Algorithm for Concave Utilities
In this section we assume that u is concave and differentiable. The second
assumption is not essential; it only simplifies the notation. Let us construct a
sequence of portfolios (ν1 , . . . , νn ) by the exponentiated gradient (EG) algorithm
[10]:
ν1 = (1/d, . . . , 1/d),
(19)
i
i
νk+1
= νki eηyk
j ηy j
d
k
j =1 νk e
,
yk = u ($νk , rk+1 %)rk+1 ,
k = 1, . . . , n − 1,
where η is a positive parameter.
Theorem 4 Assume that u is concave and differentiable and
E(u (r1,∗ )r1∗ )2 ≤ B 2 < ∞,
r1,∗ = min r1i ,
1≤i≤d
r1∗ = max r1i .
1≤i≤d
(20)
350
D. B. Rokhlin
Let (νk )nk=1 be the portfolio weights constructed by the EG algorithm (19) and (20)
=
with η = B1 lnnd . Then, for ν n = n1 nk=1 νk , we have
;
∗
U (ν ) − EU (ν n ) ≤ 2B
ln d
.
n
(21)
Let fk : S ,→ R, k = 1, . . . , n be a sequence of differentiable convex functions
defined on a convex set S. Consider the online mirror descent (OMD) algorithm
with some link function g : Rd ,→ S (see [14, Chap. 2]):
θ1 = 0,
wk = g(θk ),
θk+1 = θk − zk ,
zk = fk (wk ),
k = 1, . . . , n − 1.
The assertion of Theorem 4 is implied by the following result [14, Theorem 2.15]:
Lemma 5 Let : S ,→ R be a η1 -strongly convex function with respect to a norm
· . Let (wk )nk=1 be a sequence constructed by the OMD algorithm with the link
function
g(θ ) = arg max($w, θ % − (w)).
w∈S
Then, for all w ∈ S,
n
n
(fk (wk ) − fk (w)) ≤ (w) − min (w ) + η
zk 2∗ ,
w ∈S
k=1
(22)
k=1
where · ∗ is the dual norm.
Proof
of Theorem 4 For S = and the entropic regularization (w) =
1 d
i
i
i=1 w ln w , the OMD algorithm reduces to the EG algorithm of the form
η
w1 = (1/d, . . . , 1/d),
wki e−ηzk
i
i
wk+1
=
j −ηzj
d
k
j =1 wk e
,
zk = fk (wk ),
k = 1, . . . , n − 1.
Furthermore, is η1 -strongly convex with respect to the l 1 -norm [14, Example 2.5],
(v) ≤ 0, and
min (v ) = −
v ∈S
ln d
.
η
Out-of-Sample Utility Bounds for Empirically Optimal Portfolios
351
Hence, (22) shapes to the form
n
ln d
zk 2l ∞ .
+η
η
n
(fk (wk ) − fk (w)) ≤
k=1
k=1
Let us apply this inequality to fk (ν)
−u ($ν, rk+1 %)rk+1 ,
=
−u($ν, rk+1 %). We have zk
=
i
∗
| ≤ |u (rk+1, ∗ )rk+1
|,
zk l ∞ = max |u ($ν, rk+1 %)rk+1
1≤i≤d
since u is nonincreasing. It follows that
n
ln d
∗
+η
(u (rk+1,∗ )rk+1
)2
η
n
[u($ν, rk+1 %) − u($νk , rk+1 %)] ≤
k=1
(23)
k=1
for the sequence (νk )nk=1 defined by the exponentiated gradient algorithm (19) and
(20).
The Inequality (23) is true for an arbitrary sequence of returns rk . Now, recall
that rk are i.i.d., and apply the online-to-batch conversion scheme [14, Chap.5]. We
have
Eu($ν, rk+1 %) = Eu($ν, rn+1 %),
Eu($νk , rk+1 %) = EE(u($νk , rk+1 %)|r1 , . . . , rk ) = EE(u($νk , rn+1 %)|r1 , . . . , rk )
= Eu($νk , rn+1 %).
Here, we used the law of total expectation and the fact that νk depends only on
(r1 , . . . , rk ). From (23) it follows that
1
ln d
+ ηB 2
Eu($νk , rn+1 %) +
n
nη
n
Eu($ν, rn+1 %) ≤
k=1
≤ Eu($ν n , rn+1 %) +
ln d
+ ηB 2 ,
nη
whereB 2 ≥ E(u (r1,∗ )r1∗ )2
by the concavity of u. Optimizing over η > 0, we get (21).
352
D. B. Rokhlin
=
If u is globally K-Lipschitz, then |u (x)| ≤ K, and we can take B = K E(r1∗ )2 .
In this case from (21), we easily get the estimates
;
U (ν ∗ ) − EU (ν n ) ≤ 2K
;
∗
U (ν ) − EU (ν n ) ≤ 2K
ln d
max r i L∞ ,
n 1≤i≤d 1
ln d 1/p
d
max r1i Lp ,
1≤i≤d
n
p ∈ [2, ∞),
which coincide with the assertions (a) and (b) of Theorem 1 up to an absolute
constant. In general, B depends on the joint distribution of r1,∗ and r1∗ .
It would be interesting to extend the results of the paper to the problems with
side information.
Acknowledgement The research is supported by the Russian Science Foundation, project 17-1901038.
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A Guaranteed Deterministic Approach
to Superhedging: Optimal Mixed
Strategies of the Market and Their
Supports
Sergey N. Smirnov
Abstract A guaranteed deterministic formulation for the super-replication problem
with discrete time is considered. The aim is to ensure the coverage of the contingent
claim under the option contract for all admissible scenarios. These scenarios are
given by means of a priori given compacts, depending on the prehistory of prices;
the increments of the price at every point in time must lie in the corresponding
compacts. We assume that there are no transaction costs. The problem statement is
game theoretical in nature and leads to Bellman-Isaacs equations for both pure and
mixed strategies of the “market.” In the present paper, we examine “smoothness”
properties of optimal mixed strategies of the “market” and their supports.
Keywords Super-replication · Option · Trading constraints · No arbitrage ·
Bellman–Isaacs equations · Mixed strategies · Game equilibrium ·
Semicontinuity · Set-valued mapping · Continuous selection
1 Introduction
This work complements a series of papers [1–8] developing a guaranteed deterministic approach to superhedging. The main premise of our approach, developed in this
series of papers, is based on a specific assumption concerning a priori information
regarding price movements. The model of the market represents “uncertain” price
dynamics with discrete time and can be considered as purely deterministic: the
problem setting does not use any probability or family of probabilities. Formally,
it can be classified as a specific pathwise (or pointwise) approach addressing
uncertainty in market modeling by describing the set of market scenarios. We should
stress here that we adopt an alternative interpretation to the common robust approach
to pricing of contingent liabilities. Our interpretation is game theoretic: we deal with
S. N. Smirnov ()
Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University,
Moscow, Russia
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_19
355
356
S. N. Smirnov
a (deterministic) dynamic two-player zero-sum game of “hedger” against “market.”
A family of probabilities appears as a secondary notion, thanks to the introduction
of mixed strategies of the “market.” A review of the relevant literature can be found
in [1] in Russian and in [8] in English. Herein, we restrict our focus to a brief outline
of the content of the above series of papers, which is necessary for the purposes of
this paper.
In [1], a guaranteed deterministic approach is presented in detail; the financial
market model and trading restrictions are described, and the problem of superhedging contingent claims on options is set. The main idea of the proposed approach
is to set “uncertain” price dynamics by assuming a priori information on price
movements1 at time t; namely, the increments Xt of discount prices2 lie in a
priori given compact sets Kt (·), where the dot denotes the history of prices up to
and including time t − 1, t = 1, . . . , N. vt∗ (·) denotes the infimum of the portfolio
value at time t (for a known prehistory) that guarantees, under a certain a feasible
hedging strategy, the satisfaction of current and future liabilities, regarding possible
payouts on the American option. The corresponding Bellman-Isaacs equations for
discounted prices arise directly from economic sense by choosing at step t the
“best” feasible hedging strategy h ∈ Dt (·) for the “worst” scenario y ∈ Kt (·)
of (discounted) price increments for the given functions gt (·), describing potential
payouts on the option. Thus, we obtain the following recurrence relations3 for
t = N, . . . , 1:
∗
(x̄N ) = gN (x̄N ),
vN
vt∗−1 (x̄t −1 ) = gt −1 (x̄t −1 )∨
inf
sup
h∈Dt (x̄t−1 ) y∈Kt (x̄t−1 )
∗
vt (x̄t −1 , xt −1 + y) − hy ,
(BI)
where x̄t −1 = (x0 , . . . , xt −1 ) describes the prehistory up to present time t, symbol
∨ is the maximum, and hy = $h, y% is the scalar product of vector h by vector y.
The conditions for the validity of (BI) are formulated in Theorem 3.1 of [1].
It is convenient to (formally) consider that g0 = −∞ (there are no payment
liabilities at the initial time); gt ≥ 0 for t = 1, . . . , N in the case of an American
option. Set Dt (·) is assumed to be convex and 0 ∈ Dt (·). Set-valued mappings
x ,→ Kt (x) and x ,→ Dt (x) as well as functions x ,→ gt (x) are assumed to be
defined for all x ∈ (Rn )t and t = 1, . . . , N. Therefore, the functions x ,→ vt∗ (x) are
given by Eqs. (BI) for all x ∈ (Rn )t . In Eqs. (BI), functions vt∗ and the corresponding
t = Xt − Xt−1 , where Xt is the vector of
(discounted) prices of n risky assets at time t.
2 We assume that a risk-free asset has a fixed price equal to unity.
3 Such a kind of relations, essentially a dynamic programming procedure, is used by a number of
papers, considering models with transaction costs, to obtain constructive results and to solve the
problem numerically, e.g., [9, 10] and [11].
1 The increments are taken “backward,” i.e., X
A Guaranteed Deterministic Approach to Superhedging . . .
357
supremums and infimums take values in an extended set of real numbers R ∪
{−∞, +∞} = [−∞, +∞], i.e., two-point compactification4 of R.
A trajectory of asset prices (x0 , . . . , xt ) = x̄t on time interval [0, t] = {0, . . . , t}
is deemed possible if x0 ∈ K0 , x1 ∈ K1 (x0 ), . . . , xt ∈ Kt (x0 , . . . , xt −1 ); t =
0, 1, . . . , N. The set of possible trajectories of asset prices for time interval [0, t] is
denoted by Bt :
Bt = {(x0, . . . , xt ) : x0 ∈ K0 , x1 ∈ K1 (x0 ), . . . , xt ∈ Kt (x0 , . . . , xt −1 )}.
(1.1)
One of the important assumptions is boundedness above of the payout functions
on the corresponding sets of possible trajectories, i.e.,
Ct = sup gt (x) < ∞.
(B)
C = ∨N
t =1 Ct .
(1.2)
x ∈Bt
Denote
Functions vt∗ are also bounded from above by C due to this assumption.
For ease of notation, we make an “additive” change in the latter variable of
functions vt∗ by assuming
w(x̄t −1 , y) = wt (x1 , . . . , xt −1 , y) = v ∗ (x1 , . . . , xt −1, xt −1 + y).
(T)
Further, we use Bellman-Isaacs equations written in terms of functions wt , t =
N, . . . , 0, as follows:
vt∗−1 (·) = gt −1 (·) ∨ inf
sup
h∈Dt (·) y∈Kt (·)
wt (·, y) − hy ,
t = N, . . . , 1.
Here and below, the dot indicates “current” variables; in this case, the argument is
x̄t −1.
Below, we assume that the assumptions of Theorem 3.1 employed in [1] and the
assumptions listed in point (1) of Remark 3.1 in [1] are fulfilled.
In [2], different concepts of “no arbitrage” are studied for the deterministic
market model, namely, no guaranteed arbitrage, no arbitrage opportunities, and no
guaranteed arbitrage with unlimited profit. The notion of the robustness (structural
stability) of “no arbitrage” property is introduced, and the criteria for robustness are
obtained.
For the convenience of the reader, we give several definitions of the notions
introduced in [2].
4 Neighborhoods of points −∞ and +∞ are [−∞, a), a ∈ R and (b, +∞], b ∈ R, respectively.
358
S. N. Smirnov
By the (deterministic) arbitrage opportunity (AO) at the time step t, we mean the
following:
1. There exists a strategy h∗ ∈ Dt (·), such that h∗ y ≥ 0 for all y ∈ Kt (·).
2. There exists a price movement y ∗ ∈ Kt (·), such that h∗ y ∗ > 0.
By the (deterministic) sure arbitrage (SA) at the time step t, we mean the
following: there exists a strategy h∗ ∈ Dt (·), such that h∗ y > 0 for all y ∈ Kt (·).
We say that there is a (deterministic) sure arbitrage with unlimited profit (SAUP)
at the time step t if the function5 h ,→ min{hy, y ∈ Kt (·)} takes arbitrarily large
values for h ∈ Dt (·).
Using these three notions of “arbitrage,” we can define the corresponding “no
arbitrage” properties on a time interval; any “no arbitrage” propriety on a time
interval tantamount to “no arbitrage” propriety at every time step of this interval.
We get so the following “no arbitrage” properties:
– No deterministic arbitrage opportunity (NDAO)
– No deterministic sure arbitrage (NDSA)
– No deterministic sure arbitrage with unlimited profit (NDSAUP)
The structural stability in our context means that the qualitative behavior of
a price dynamics Kt (·), namely, some “no arbitrage” property, is unaffected by
sufficiently small (with respect to the Pompeiu-Hausdorff metric) perturbations of
Kt (·); such a “no arbitrage” property is called robust (or coarse). In what follows
we need two robust “no arbitrage” properties:
– Robust no deterministic arbitrage opportunity (RNDAO)
– Robust no deterministic sure arbitrage with unlimited profit (RNDSAUP).6
In [3] we investigate the properties of the semicontinuity and continuity of
solutions of the Bellman-Isaacs equations (BI).
Proposition 2.1 in [3] provides a sufficient condition for the compactness of Bt ,
namely, the upper semicontinuity of compact-valued mappings x ,→ Kt (·), t =
1, . . . , N, and a sufficient condition for the validity of property (B), namely, the
upper semicontinuity of Kt (·) and the upper semicontinuity of gt (·), t = 1, . . . , N.
If additionally the multivalued mappings Dt (·) are lower semicontinuous, then the
5 It can be interpreted as the guaranteed profit for a strategy h ∈ D (·). Note that min{hy, y ∈
t
Kt (·)} is attained due to the compactness of Kt (·) for some (worst scenario) yh∗ ∈ Kt (·); it is
positive for the sure arbitrage strategy h∗ ∈ Dt (·).
6 Suppose that the initial market model satisfies NDSAUP condition and consider a perturbed model
with increments Xt lying in compacts K̃t (x) for t = 1, . . . , N. Coarse NDSAUP property, or
RNDSAUP, means that for any t ∈ {1, . . . , N} and any prehistory x̄t of the prices, there exists
t (x̄t ) > 0, such that if the Pompeiu-Hausdorff distance between every perturbed compact K̃t (x̄t )
and initial compact Kt (x̄t ) is not greater then t (x̄t ), for all t ∈ {1, . . . , N} and x̄t , then the
perturbed market model still satisfies NDSAUP condition. A geometric criterion of RNDSAUP
property is presented in [12].
A Guaranteed Deterministic Approach to Superhedging . . .
359
functions x̄t −1 ,→ vt∗ (x̄t −1 ), defined by the relations (BI), are upper continuous,
according to Theorem 2.5 of [3].
The main result of [3], Theorem 3.2, related to the continuity of solutions (BI) is
formulated below.
Theorem 1.1 For t = 1, . . . , N, let the numeric functions x̄t ,→ gt (x̄t ) be
continuous, let the compact-valued mappings x̄t −1 ,→ Kt (x̄t −1) be continuous,
and let the multivalued mappings x̄t −1 ,→ Dt (x̄t −1 ) be lower semicontinuous and
closed. If the robust condition of no sure arbitrage with unlimited profit (RNDSAUP)
is satisfied, then the functions x̄t −1 ,→ vt∗ (x̄t −1 ), defined by the relations (BI), are
continuous.
In [4], under the conditions of above-stated theorem and no trading constraints7
(i.e., when Dt (·) ≡ Rn ), we estimate the modulus of the continuity of BellmanIsaacs functions, in particular, in the case of Lipschitz functions.
In [5] we introduce the mixed extension of pure strategies of the “market” and
study the existence of game equilibrium and its properties. We use some of these
concepts and results in this paper.
The transition from pure market strategies Xt ∈ Kt (·) on the time step t to their
mixed extension, i.e., to a family Pt (·) of distributions having supports contained
in Kt (·) and such that every distribution concentrated in a single point y ∈ Kt (·)
belongs to Pt (·), is justified in Lemma 2.1 of [5]; the following equality holds:
sup
wt (·, y) − hy = sup
y∈Kt (·)
wt (·, y) − hy Q(dy).
(1.3)
Q∈Pt (·)
We shall use the following notations: P∗ (E) is the class of all probability measures
on E with finite support; P(E) is the class of all probability measures on a
measurable space (E, A) with some (fixed) σ -algebra A of subsets of E; and Pn (E)
is the class of all probability measures, concentrated in at most n + 1 points of E.
Note that each of the families P∗ (Kt (·)), P(Kt (·)), or Pn (Kt (·)) can be chosen as a
mixed extension.
Following [5], denote
ρt (·) =
inf
sup
wt (x̄t −1 , y) − hy Q(dy),
(1.4)
wt (x̄t −1 , y) − hy Q(dy),
(1.5)
h∈Dt (·) Q∈Pt (·)
ρt (·) = sup
inf
Q∈Pt (·) h∈Dt (·)
7 When there are no trading constraints, the condition RNDSAUP is tantamount to the robust
condition of no arbitrage opportunities RNDAO and the multifunction Dt (·) ≡ Rn is lower
semicontinuous and closed.
360
S. N. Smirnov
and assume that integrals in (1.4) and (1.5) are well-defined with respect to the
measure Q. The expression for ρt (·) appears in the Bellman-Isaacs equations for
mixed strategies; following [5], we can rewrite them in reduced form:
∗
vN
(·) = gN (·),
vt∗−1 (·) = gt −1 (·) ∨ ρt (·), fort = 1, . . . , N.
(1.6)
Note that under assumption (B), functions ρt (·) and vt∗ (·) are bounded from above
by constant C; see [5].
Proposition 3.1 $from [5] states that if functions wt and a class of measures Pt (·)
such that integrals wt (·, y)Q(dy) are defined8 for any measure Q ∈ Pt (·), then
ρt (·) = sup
wt (·, y)Q(dy) − σDt (·)
yQ(dy) ,
(1.7)
Q∈Pt (·)
where σA is the support function of the set A.
Note that ρt (·) > −∞ if and only if the condition of no sure arbitrage with
unbounded profit NDSAU P is fulfilled (see Remark 3.2, Item (1), in [5]). If the
condition of no sure arbitrage NDSA is fulfilled, then ρt (·) ≥ 0 (see Remark 3.5,
Item (3), of [5]).
The inequality ρt (·) ≥ ρt (·) always holds; the equality ρt (·) = ρt (·) represents
the game equilibrium. The aim of introduction of the mixed extension is just to
achieve the game equilibrium under some appropriate assumptions. In the case of
game equilibrium, we can separate the pricing problem from the hedging one, thanks
to (1.7), and deal the Bellman equations9 instead of the Bellman-Isaacs equations.
Note that, when there are no trading constraints, the expression (1.7) takes a rather
simple form;10 see Item (5) of Remark 3.2 in [5]:
ρt (·) =
Q∈Pt (·),
sup
$
wt (·, y)Q(dy);
(1.8)
y Q(dy)=0
A sufficient condition for the game equilibrium is upper semicontinuity of
solution of (BI):11 in this case a maximum in (1.4) and (1.5) is attained for some
mixed strategy Q∗t (·), which we call optimal;12 using a theorem proven in [13], it
8 To avoid additional requirements on the functions v ∗ , one can use, for example, the class
t
P∗ (Kt (·)). On the other hand, one can require the vt∗ to be universally measurable (or have some
smoothness property, e.g., semicontinuity).
9 Where hedging strategies are no longer present
10 The supremum is taken, therefore, over the class of distributions, such that (discounted) prices
of risky assets are martingales. In the case of no trading constraints this result has an important
economic interpretation, explaining the origin of risk neutrality.
11 In its turn, a sufficient condition for this is mentioned above (a result from [3]).
12 It can be interpreted as the most unfavorable (for the “hedger”) strategy of the “market.”
A Guaranteed Deterministic Approach to Superhedging . . .
361
can be shown that there exists an optimal mixed strategy concentrated in at most
n + 1 points, where n is the number of risky assets.
Our paper [6] focuses on the game equilibrium of a case involving no trading
constraints under the assumption of no arbitrage opportunities. The main result
of [6] is Theorem 1, which is essentially a generalization of the result of V. N.
Kolokoltsov, namely, Theorem 12.21 in [14].
In [7], a two-step method for solving the Bellman-Isaacs equations is proposed
for the game equilibrium case (equilibrium conditions are given in [5] and [6]). It
is shown that the most unfavorable mixed market strategies can always be found in
the class of distributions concentrated in at most n + 1 points.13 A connection with
the general problem of moments is established; this method can be applied to other
problems involving hedging contingent claims on options.
In [8], we study the superhedging problem under the framework of the guaranteed deterministic approach for a case of convex payoff functions and a market
model, where compact sets of possible price increments that describe the uncertainty
of price movements are convex. In the framework of the model with convexity
assumptions, we also obtain some delicate properties for optimal mixed strategies
of the “market” and clarify the behavior of their supports.14 We provide a twodimensional15 example that brings to light some particularities of this behavior.
In the present paper, we obtain conditions for the upper semicontinuity of setvalued mappings that map a point of the price history into the class of optimal mixed
strategies of the “market”; the main result is Theorem 2.2. We also derive sufficient
conditions for the existence of continuous branches (selections) for supports of
optimal “market” strategies that contain no more than n + 1 points.
2 Smoothness Property of Optimal Mixed Strategies
of the “Market” and Its Supports
Proposition 2.1 Let X be a compact topological space with Borel σ -algebra BX ,
let Y be a separable metric space with σ -algebra BY , let P(Y ) be the set of
probability measures on (Y, BY ) equipped with the weak topology, and let S : X ,→
K(Y ) be an upper semicontinuous set-valued mapping,16 where K(Y ) is the class
13 If an optimal mixed strategy Q∗ (·) exists, it can be chosen such that it is concentrated in at
t
most n + 1 points. Therefore, if we fix some initial price and consider a market with conditional
distributions PXt |X̄t−1 =x̄ = Q∗t (x̄), we obtain a complete market (in the usual sense).
14 The general case is considered in the present paper. Note that initially we planned to submit this
paper to the journal “Mathematical Games Theory and its Applications” (Mat. Teor. Igr Prilozh.,
in Russian), so in [8] there is a reference to this paper.
15 We consider a rainbow option whose payoff depends on two underlying risky assets.
16 By compactness, the (upper and lower) semicontinuity and h-semicontinuity coincide (see [15],
Propositions 2.61 and 2.66).
362
S. N. Smirnov
of nonempty compact subsets of Y , and
M(x) = {Q ∈ P(Y ) : supp(Q) ⊆ S(x)}.
(2.1)
Then, set-valued mapping x ,→ M(x) is upper semicontinuous.
Proof Fix an arbitrary ε > 0 and choose an arbitrary point x0 ∈ X. Due to
the upper semicontinuity of set-valued map x ,→ K(x), there is a neighborhood
Uε (x0 ) of point x0 such that, for all x ∈ Uε (x0 ), the inclusions17 S(x) ⊆ {y ∈
Y : ρ(y, S(x0 )) ≤ ε} = Fε take place. Consider the xα → x0 , Qα ∈ M(xα ).
From Corollary 2.20 of [15], the image (in the multivalued sense) S(X) is compact,
and therefore, the image of M(X) is also compact in the weak topology on P(Y ),
which is metrizable; see [16]. Hence, subnet Qαβ is weakly converging to measure
Q0 ∈ P(Y ). Moreover, from the Alexandroff theorem [17], we have the inequality18
lim inf Qαβ (Fεc ) ≥ Q0 (Fεc ) = 0 since the set Fεc is open; thus, supp(Q0 ) ⊆ Fε .
β→∞
Since ε > 0 is arbitrary, we obtain supp(Q0 ) ⊆ S(x0 ), i.e., Q0 ∈ M(x0 ).
As mapping M(·) takes compact values, this mapping is upper semicontinuous
according to Proposition 2.19 of [15].
The following statement was proved in [18, Proposition 2]: if X is a topological
space (its compactness is not required), S : X ,→ F (Y ) is a lower semicontinuous
set-valued mapping, where F (Y ) is the class of all closed subsets, and then, M(·)
given by relation (2.1) is h-lower semicontinuous. Thus, we have the following
statement:
Proposition 2.2 Suppose that the conditions of Proposition 2.1 hold and, in
addition, set-valued mapping S : X ,→ K(Y ) is lower semicontinuous. Then, setvalued mapping M(·) given by relation (2.1) is continuous.
Lemma 2.1 Let X and Y be Hausdorff topological spaces and let X be compact.
If multivalued mapping K : X ,→ K(Y ) is upper semicontinuous, then for n ≥ 1,
set-valued mapping x ,→ Pn (K(x)), x ∈ X, is upper semicontinuous.
Proof Let us consider nets xα → x∗ , Qα ∈ Pn (K(xα )). Note that Qα can be
represented as
Qα =
n+1
pαi δyαi ,
i=1
n+1
pαi = 1, pαi ≥ 0, yαi ∈ K(xα ) ⊆ K(X).
i=1
17 Here, ρ(y, A) = inf{ρ(y, z), z ∈ A}.
18 Here, Ac is the complement of set A (in the corresponding space).
A Guaranteed Deterministic Approach to Superhedging . . .
363
From Corollary 2.20 of [15], the image of K(X) is compact since K(·)
is compact. Therefore, based on the upper semicontinuity of K(·), from
Proposition 2.19 of [15], there is a subnet for which yαi β → y∗i ∈
K(x∗ ), i = 1, . . . , n + 1, and due to the compactness of the simplex
n+1
i
Sn = {p = (p1 , . . . , pn+1 ) ∈ Rn+1 :
p = 1, pi ≥ 0, i = 1, . . . , n + 1}, it
i=1
is possible to extract one more subnet, such that pαβγ = (pα1βγ , . . . , pαn+1
) → p∗ ∈
βγ
Sn . Thus, Qαβγ →
n+1
i=1
p∗i δy∗i ∈ Pn (K(x∗ )). Applying Proposition 2.19 from [15]
again, we obtain the upper semicontinuity for x ,→ Pn (K(x)).
Let (Y, ρ) be a separable metric space. Introduce linear space BL(Y ) of bounded
Lipschitz functions f : Y ,→ R with norm
f BL = f L + f ∞ ,
f ∞ = sup{|f (y)| : y ∈ Y },
|f (y) − (y )|
: y, y ∈ Y, y = y .
f L = sup
ρ(y, y )
M(Y ) is the normalized vector space of alternating measures on space Y with Borel
σ -algebra BY equipped with the Dudley norm:
μD = sup f dμ : f BL ≤ 1 .
The set of all normalized (probability) measures P(Y ) ⊆ M(Y ) is a closed convex
subset of M(Y ) with metric ρD generated by norm · D ; i.e., ρD (π1 , π2 ) =
π1 −π2 D . According to Theorem 11.3.3 of [19], for separable metric space Y , the
weak convergence of a sequence of probability measured from P(Y ) is equivalent
to convergence in the Dudley metric; i.e., metric ρD generates a weak topology on
P(Y ).
Lemma 2.2 Let X be a Haudorff topological space, let (Y, ρ) be a separable metric
space, and let F : X ,→ K(Y ) be a lower semicontinuous mapping. Then, setvalued mapping x ,→ Pn (F (x)), where n ≥ 0, is h-lower semicontinuous19 with
respect to the Dudley metric.
19 According to Proposition 2.66 from [15], the h-lower semicontinuity of a set-valued mapping
implies its lower semicontinuity.
364
S. N. Smirnov
Proof Let Q0 ∈ Pn (K(x0 )), where x0 is an arbitrary point of X. Then, measure Q0
can be represented in the following form:
Q0 =
n+1
q i δy i , y0i ∈ F (x0 ),
0
i=1
q i ≥ 0, i = 1, . . . , n + 1,
n+1
q i = 1.
i=1
Since F takes compact values, the lower semicontinuity implies h-lower semicontinuity according to Proposition 2.68 of [15]. Therefore, for any ε > 0,
there is a neighborhood Vε (x0 ) of point x0 such that for any x ∈ Vε (x0 ),
F (x0 ) ⊆ {y ∈ Y : ρ(y, F (x)) < ε}. Hence, for every y0i , i = 1, . . . , n + 1, there is
n+1
i
q δy i ∈ Pn (F (x)), we
y i ∈ F (x), such that ρ(y0i , y i ) < ε. Assuming that Q =
i=1
0
find that ρD (Q0 , Q) < ε; i.e., Pn (F (·)) is h-lower semicontinuous at point x0 .
From Lemma 3.1 of [3], trading constraints Dt (·) in the Bellman-Isaacs equations (BI) can be replaced without changing the values of the function vt∗ (·) by the
compact sets20
D̂ta (·) = {h ∈ Dt (·) : σKt (·) (−h) ≤ a},
where a ≥ C and constant C is defined by Formula (1.2). Without a loss of
generality, we can assume21 that C > 0. From Proposition 3.2 of [5], we have an
(game) equilibrium in the class of measures with a finite support contained in Kt (·).
However, we then make assumptions about Dt (·), Kt (·), and gt (·), guaranteeing the
continuity of vt∗ (·) so that we can apply Proposition 3.3 from [5], according to which
equilibrium holds for the class of all measures with support contained in Kt (·).
Lemma 2.3 Suppose that all the conditions of Theorem 1.1 holds. Then, for t =
1, . . . , N, functionals Jt (x, Q), x ∈ Bt −1 , Q ∈ P(Kt (Bt −1 )), defined by relation
Jt (x, Q) =
wt (x, y)Q(dy) − σD̂ a (x)
t
yQ(dy) ,
(2.2)
are continuous (in all variables) on the compact set Bt −1 × P(Kt (Bt −1 )), where
P(Kt (Bt −1 )) is equipped with a weak topology.22
20 Here, σ
A denotes the support function of set A, i.e., σA (x) = sup $x, y%.
y∈A
21 For an American option, C ≥ 0; if C = 0, then the payoff function is trivial, i.e., g (·) ≡ 0,
t
which is not of interest from the point of view of economic interpretation.
22 A weak topology on space P(Y ) of probability measures defined on Borel σ -algebra B
metrizable when Y is a separable metric space; see [19, Theorem 11.3.3].
Y
is
A Guaranteed Deterministic Approach to Superhedging . . .
365
Proof Fix t ∈ {1, . . . , N} and consider sequences x n ∈ Bt −1 ⊆ (Rn )t , Qn ∈
P(Kt (Bt −1 )), such that x n → x ∗ and Qn → Q∗ (weakly); obviously, (x ∗ , Q∗ ) ∈
Bt −1 × P(Kt (Bt −1 )). From Theorem 2.7 of[3], solution vt∗ (·) of the Bellman-Isaacs
equations (BI) is continuous; i.e., function (x, y) ,→ wt (x, y) is continuous (in all
variables) on set Bt −1 × Kt (Bt −1 ), which is compact by Proposition 2.1 of [3] and
Remark 2.1, point (4), from [3], so that function wt (·, ·) is uniformly continuous on
Bt −1 × Kt (Bt −1 ). In particular, the family of functions y ,→ wt (x, y), x ∈ Bt −1 , is
equicontinuous and bounded.
Based on this observation,
we verify the continuity of the first term in difference
$
(2.2), i.e., Jt1 (x, Q) = wt (x, y)Q(dy) on Bt −1 × P(Kt (Bt −1 )):
Jt1 (x n , Qn ) − Jt1 (x ∗ , Q∗ ) =
wt (x , y)Q (dy) −
n
+
n
∗
n
wt (x , y)Q (dy)
[wt (x n , y) − wt (x ∗ , y)]Q∗ (dy).
The first term (in square brackets) tends to zero from Corollary 11.3.4 of [19]. The
second term tends to zero since the integrand$tends to zero (uniformly). For the
continuity of expression Jt2 (x, Q) = σD̂ a (x) yQ(dy) , note that y is bounded
t
$
on compact Kt (Bt −1 ), i.e.,
sup y = A1 < ∞; hence, yQ(dy) ≤
y∈Kt (Bt−1 )
$
yQ(dy) ≤ A1 . In addition, since set-valued map x ,→ D̂ta (x) is continuous, it
follows that sup {h : h ∈ D̂ta (x)} = A2 < ∞, based on Proposition 6.4 of [20].
x ∈Bt−1
From Propositions 9.10 and 9.11 provided by Polovinkin [20], we have
Jt2 (x n , Qn ) − Jt2 (x ∗ , Q∗ ) = σD̂ a (x n )
t
yQn (dy) − σD̂ a (x ∗ )
yQ∗ (dy)
t
= σD̂ a (x n )
t
yQ (dy) − σD̂ a (x n )
t
∗
yQ (dy) − σD̂ a (x ∗ )
t
∗
yQ (dy)
t
+ σD̂ a (x n )
n
∗
yQ (dy)
= A2 yQn (dy) − yQ(dy)
+ A1 hρ (D̂ta (x n ), D̂ta (x ∗ )) → 0.
Thus,
Jt (x n , Qn ) = Jt1 (x n , Qn )−Jt2 (x n , Qn ) → Jt1 (x ∗ , Q∗ )−Jt2 (x ∗ , Q∗ ) = Jt (x ∗ , Q∗ ).
366
S. N. Smirnov
Now we are ready to prove the main result of this section.
Theorem 2.2 Let class Pt (·) be the maximal mixed extension of the class of
pure strategies.23 Suppose that all the conditions of Theorem 1.1 holds. Then,
opt
the class of all optimal mixed strategies Pt (·) is a nonempty convex, and setopt
opt
24
valued mappings x ,→ Pt (x) and x ,→ Pt (x) ∩ Pn (Kt (x)) = ∅ are upper
semicontinuous.
Proof We choose mixed extension Pt (x) = P(Kt (x)) ⊆ P(Kt (Bt −1 )) in
expressions (2.2) for x ∈ Bt −1 ; by virtue of point (4) in Remark 2.1 from [3] and
Proposition 2.1 of [3], set Kt (Bt −1 ) is compact, and functions gt (·) are bounded
from above on compact sets Bt , t = 1, . . . , N. From Theorem 2.5 of [3], function
vt∗ (·) in (1.6) is upper semicontinuous and hence is bounded on compact Bt ,
t = 1, . . . , N. From Proposition 3.3 of [5], there is an equilibrium for the chosen
mixed extension, i.e., ρt (·) = ρt (·), where ρt (·) and ρt (·) are given by expressions
(1.4) and (1.5), and the integrals in these expressions are defined. As mapping
x ,→ Dt (·) is closed, sets Dt (·) are closed (see Remark 2.12 of [15]), condition
(2.1) holds, and we find in accordance with point (1) of Remark 3.1 from [3] that set
D̂ta (·) = {h ∈ Dt (·) : σKt∗ (·) (−h) ≤ a}, where Kt∗ (·) = conv(Kt (·)), is compact.
From Lemma 3.1 from [3], Dt (·) can be replaced with D̂ta (·) without changing
the value of ρt (·); according to point (2) of Proposition 3.3 from [5], there is an
equilibrium with saddle point (h∗t (·), Q∗t (·)). From Proposition 3.1 of [5], we have
ρt (·) = ρt (·) =
sup
J (x, Q),
(2.3)
Q∈P(Kt (x))
where J (x, Q) is given by Formula (2.2). Based on Proposition 2.2 on the continuity
of set-valued mapping x ,→ P(Kt (x)), Proposition 3.3 from [3] on the continuity
of set-valued mapping x ,→ D̂ta (x), and Lemma 2.3 on the continuity of functional
J = J (x, Q), x ∈ Bt −1 , Q ∈ P(Kt (Bt −1 )), we may apply the Berge theorem on the
opt
maximum (see Theorem 2.4 of [3]). According to this theorem, for the set Pt (x) ⊆
P(Kt (x)) of measures for which the maximum of the functional J on P(Kt (x))
opt
is reached in (2.3), the corresponding set-valued mapping x ,→ Pt (x) is upper
opt
semicontinuous on Bt −1 . The convexity of Pt (x) follows from the convexity of the
functional J (x, Q) with respect to variable Q and taking into account the convexity
of class P(Kt (x)).
From Lemma 2.1 on the upper semicontinuity of mapping x ,→ Pn (Kt (x)) and
opt
Proposition 2.45 from [15], which is applicable since Pt (x) ∩ Pn (Kt (x)) = ∅
23 Remind that, in accordance with the terminology of [5], the mixed extension involves only the
measures with support contained in Kt (·) and all the measures concentrated at point y ∈ Kt (·)
belong to it. The maximal mixed extension consists of all the measures with support contained in
Kt (·).
24 Class Pn (X) of all probability measures concentrated at most n + 1 points of space X is not
convex if X contains more than n + 1 points.
A Guaranteed Deterministic Approach to Superhedging . . .
367
opt
according to Theorem 3.1 of [7], we find that set-valued mapping x ,→ Pt (x) ∩
Pn (Kt (x)) is upper semicontinuous.
Corollary 1 If the hypotheses of Theorem 2.2 hold and if for any x ∈ Bt −1 , set
opt
Pt (x) ∩ Pn (Kt (x)) is a singleton (in particular, if the maximizer in (2.3) is unique
opt
or set Pt (x) contains only one point), i.e.,
Pt (x) ∩ Pn (Kt (x)) = {Q∗ (x, ·)},
opt
(2.4)
then transition kernel Q∗ is Feller25 and Q∗ (x, ·) ∈ Pn (Kt (x)).
This is a consequence of the following elementary statement.
Lemma 2.4 If X and Y are topological spaces, F : X ,→ 2Y is a set-valued
mapping26 such that set F (x) is a singleton for all x ∈ X, i.e., there is a
representation F (x) = {f (x)}) for some single-valued function f : X ,→ Y , then
the upper semicontinuity27 of F (·) is equivalent to the continuity of f .
The statement follows directly from Definition 2.1 (a) and Proposition 2.5 (c) in
[15], according to which the upper semicontinuity of F (·) is equivalent to the fact
that for any closed set C ⊆ Y , a set of form
{x ∈ X : F (x) ∩ C = ∅} = {x ∈ X : f (x) ∈ C} = f −1 (C)
is closed.
The following simple example shows that the maximizer in (2.3) may be
nonunique:
Example 1 Consider a one-step, one-dimensional (n = 1) model for the case of no
trading constraints. Fix x0 and put K1 (x0 ) = [a, b], a < b. Assume that there is
interval [x , x ], such that:
(1) 0 ∈ (x , x ) ∩ (a, b).
(2) The restriction of the payoff function g1 (·) to [x , x ] is affine, g1 (x) = αx + β,
x ∈ [x , x ].
(3) Outside the interval [x , x ], the payoff function satisfies the inequality g1 (x) <
αx + β.
Then, the concave hull g1∗ (·) of function g1 (·) coincides with it on [x , x ], and
$any measure Q with support supp(Q) ⊆ [x , x ], which has a zero mean, i.e.,
xQ(dx) = 0, is the optimal mixed strategy (see [7, Sect. 4]).
,→ Q∗ (x, ·) is continuous; according to Proposition 1 of [18], Q∗ is a transition
kernel.
26 Here, 2Y denotes the set of all subsets of Y .
27 A similar statement holds if the upper semicontinuity for F (·) is replaced with lower semicontinuity.
25 Mapping x
368
S. N. Smirnov
3 Support Properties for the Case of the Unique Optimal
Market Strategy Concentrated at Most n + 1 Points
opt
We further assume that Pt (x) ∩ Pn (Kt (x)) is a singleton and contains measure
Q∗t (x, ·); by Corollary 1, mapping x ,→ Q∗t (x, ·) is continuous. Denote the support
of measure Q∗t (x) by
S(x) = supp(Q∗t (x, ·)).
(3.1)
According to Proposition 2 of [18], set-valued mapping x ,→ S(x) is lower
semicontinuous due to the Feller property of Q∗t .
The most “good” behavior from this set-valued mapping S(·) defined by (3.1) is
obtained when, for all x,
|S(x)| = n + 1,
(3.2)
where |A| is the number of points in set A. Condition (3.2), where S(·) is given by
(3.1), is called the condition of the maximum power of the supports.
Proposition 3.1 Let X be a topological space, Y be a Hausdorff topological space,
and F : X ,→ 2Y be a lower semicontinuous set-valued mapping whose values are
nonempty finite sets. Then:
(1) Function x ,→ |F (x)|, x ∈ X, is lower semicontinuous.
(2) If function x ,→ |F (x)| takes a constant value, then F is continuous.
Proof Fix arbitrary x ∈ X and let F (x) = {y1 , . . . , yk }.
If k = 1, then it is obvious that |F (x )| ≥ |F (x)| = 1 for all x ∈ X so that
property (1) holds at point x. For assertion (2), in the case of k = 1, by following
one of the equivalent definitions (see [15, Proposition 2.5]), the set-valued mapping
F is lower semicontinuous if, for any open G ⊆ Y , set {x ∈ X : F (x) ∩ G = ∅} is
open. However, {x ∈ X : F (x) ∩ G = ∅} = {x ∈ X : f (x) ∈ G} = f −1 (G), and
the continuity of function f is equivalent to the fact that the inverse images of open
sets are open.
Now consider the case of k ≥ 2. Let V be an arbitrary open set such that F (x) ⊆
V . Since F (x) = {y1 , . . . , yk } ⊆ Y and space Y is Hausdorff, for each pair (i, j ) ∈
{1, . . . , k}×{1, . . . , k}, i = j , there are neighborhoods of points yi and yj , Vj (yi ) ⊆
V and Vi (yj ) ⊆ V , respectively, such that Vj (yi ) ∩ Vi (yj ) = ∅. Assuming V (yi ) =
∩j ∈{1,...,k},j =i Vj (yi ), i = 1, . . . , k, we find that the new neighborhoods V (yi ) of
points yi , i = 1, . . . , k, are pairwise disjoint. Since F is lower semicontinuous, for
i ∈ {1, . . . , k}, there are neighborhoods Ui (x) of point x such that F (x ) ∩ V (yi ) =
∅ for all x ∈ Ui (x). When U (x) = ∩ki=1 Ui (x), then, for an arbitrary x ∈ U (x),
we have F (x ) ∩ V (yi ) = ∅ for all i ∈ {1, . . . , k}. Thus, each k pairwise disjoint
set V (yi ), i ∈ 1, . . . , k, contains at least one point of set F (x ) for each x ∈ U (x).
Hence, |F (x )| ≥ |F (x)|, which implies that (1). If we additionally assume the
A Guaranteed Deterministic Approach to Superhedging . . .
369
constancy of function x ,→ |F (x)|, then set F (x ) for x ∈ U (x) contains exactly
k points so that each neighborhood V (yi ), i = 1, . . . , k, contains exactly one point
of set F (x ). Therefore, F (x ) ⊆ ∪ki=1 V (yi ) ⊆ V for all x ∈ U (x); i.e., F is upper
semicontinuous at point x. Thus, F is continuous at point x; i.e., assertion (2) is
also proved for k ≥ 2.
Remark 3.1
(1) The proof of assertion (2) in Proposition 3.1 for k = 1 does not use the
Hausdorff property. In addition, under the hypotheses of Proposition 3.1, for
the case of k = 1, the lower semicontinuity of F can be replaced with upper
semicontinuity, i.e., the continuity of single-valued function f is also equivalent
to this property; see Lemma 2.4.
(2) In the proof of assertion (2) of Proposition 3.1 for k ≥ 2, it is actually proved
that the lower semicontinuity of set-valued mapping F (with a fixed finite
number of elements F (x)) at a point implies its continuity at this point (this
is obvious for the case of k = 1); see the definitions in [15, Remark 2.4].
(3) When Y is a metric space, assertion (2) is obtained from [21, Lemma
2]; this lemma states28 the h-continuity of F ; however, the upper (lower)
semicontinuity of compact-valued mappings is equivalent to h-upper (lower)
semicontinuity, and hence, continuity is equivalent to h-continuity (see [15,
Theorem 2.68]).
We say that set-valued mapping F such that |F (x)| = k ≥ 1 for all x is
decomposed into branches that do not coincide with each other everywhere if
F (x) = {f1 (x), . . . , fk (x)}, where fi , i = 1, . . . , k, are (single-valued) functions
fi : X ,→ Y such that fi (x) = fj (x) for i = j , x ∈ X. The following result has
recently been proved by Zhukovskiy in [22]:
Theorem 3.1 (Zhukovskiy) Let X be a linearly and simply connected29 topological space, let Y be a Hausdorff space, and let F : X ,→ 2Y be a continuous
set-valued mapping whose values are finite sets containing exactly k ≥ 1 elements.
Then, F is decomposed into k everywhere distinct continuous branches.
28 For definitions of the lower semicontinuity and h-lower semicontinuity of set-valued mappings,
see Definition 2.3, Proposition 2.5, and Definition 2.60 in [15]. For a general case, h-lower
semicontinuity implies lower semicontinuity (Proposition 2.66), while upper semicontinuity
implies h-upper semicontinuity (see [15], Definition 2.60 and Proposition 2.61).
29 A simply connected space is a linearly connected space whose fundamental group is trivial
(consisting of unity); i.e., closed paths in this space are homotopic to zero.
370
S. N. Smirnov
Remark 3.2
(1) Zhukovskiy [22] generalizes the results of [23, Lemma 1], where the additional
compactness requirement is imposed on spaces X and Y involved in Theorem 3.1. Lemma 1 from [23], in turn, is based30 on Lemma 4 of [24]: if X
is a simply connected compact topological space, Y is a compact metric space,
and a continuous set-valued mapping F : X ,→ 2Y has a property whereby, for
every x ∈ X, set F (x) has exactly k ≥ 1 connected components and then the
graph of this set-valued mapping, i.e., set {(x, y) ∈ X × Y : y ∈ F (x)}, has
exactly k connected components.
(2) The example is given in Remark 2 of [22], namely, set-valued mapping F :
X ,→ Y , where31 X = Y = C \ {0}, defined by relation F (x) = {y ∈ C \ {0} :
y 2 = x} for x ∈ C \ {0}, shows that the simple connectivity of space Y is
essential.
(3) In [21], Theorem 2 gives sufficient conditions for a set-valued mapping whose
values are finite sets with a fixed number of elements to be decomposed
into branches that do not coincide everywhere; however, in our opinion, this
condition is difficult to check in practice.
The above results can be summarized by the following statement:
Proposition 3.2 Let X be a linearly and simply connected space, let Y be a
Hausdorff space, and let F : X ,→ 2Y be a set-valued mapping whose values
are finite sets containing exactly k ≥ 1 elements. Then, F is decomposed into k
everywhere distinct continuous branches if and only if F is lower semicontinuous.
Proof This sufficiency follows from assertion (2) of Proposition 3.1 and Theorem 3.1.
Conversely, suppose that F is decomposed into continuous branches that do
not coincide everywhere, i.e., F (x) = {f1 (x), . . . , fk (x)}, where (single-valued)
functions fi are continuous, i = 1, . . . , k. From Proposition 14.2 of [20], the
set-valued mapping32 F (x) = cl ∪ki=1 fi (x) = {f1 (x), . . . , fk (x)} is lower
semicontinuous.
Proposition 3.3 If the hypotheses of Theorem (2.2) and Corollary 1 (on the
uniqueness of the maximizer from Pn (Kt (·))) hold, compact sets K0 and Kt (x),
x ∈ Bt −1 , are convex, and set-valued mapping x ,→ S(x) given by (2.4) satisfies
the condition for the maximum cardinality of support (3.1), and then, this mapping
is continuous and can be decomposed into everywhere distinct continuous branches.
30 When [23] refers to Lemma 4 from [24], the compact metric space under the hypotheses of
the lemma is replaced by Schirmer [23] with a compact Hausdorff topological space (without
explanation).
31 Here, C denotes the set of complex numbers.
32 Here, cl(A) denotes the closure of set A.
A Guaranteed Deterministic Approach to Superhedging . . .
371
In fact, in this case, sets Bt , t = 0, . . . , N, defined by (1.1) are convex and
therefore linearly and simply connected; further, as noted above, x ,→ S(x) is lower
semicontinuous due to the Feller property of kernel Q∗t (·, ·); see Proposition 2 from
[18].
Note that condition (3.1) for the maximum cardinality of the support may not be
fulfilled if Kt (·) is nondegenerate (full-size), i.e., when affine hull Kt (·) coincides
with Rn ; in other words, supports of optimal mixed strategies can degenerate and be
concentrated at less than n + 1 points of Kt (·); and it suffices to invoke Example 2
from [7], where n = 1. A more interesting example, where n = 2, is presented
in [8].
Acknowledgement This work was supported by the Russian Foundation for Basic Research
(Project No. 19-01-00613a).
References
1. Smirnov, S.N.: A guaranteed deterministic approach to superhedging: financial market model,
trading constraints and Bellman-Isaacs equations. Mat. Teor. Igr Prilozh. 10(4), 59–99 (2018)
[in Russian]
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Some Properties of Regularly Varying
Functions and Series in the Orthant
A. L. Yakymiv
n → R is called regularly varying at infinity in
Abstract A function f : R+
+
n
R+ with auxiliary functions of one variable r1 (t), . . . , rn (t) such that r1 (t) →
n the relation
∞, . . . , rn (t) → ∞ (t → ∞), if for all x ∈ G = int R+
f (r1 (t)x1 , . . . , rn (t)xn )/f (r1 (t), . . . , rn (t)) → λ(x1 , . . . , xn ) ∈ (0, ∞)
is valid as t → ∞. According to the fundamental one-dimentional J. Karamata’s
result (1930), the last relation holds locally uniformly in x (i.e., for n = 1 and
r1 (t) ≡ t). It is also known that, generally speaking, this is not valid in the
multidimensional case. In this article, we give some sufficient conditions on the
function f , under which this relation also holds locally uniformly in x ∈ G. As
an application, the limiting properties of the multiple power series distributions are
investigated.
Keywords Regularly varying functions and series in the orthant · Multiple power
series distributions · Limit theorem
1 Introduction
One-dimensional regularly varying functions were introduced by J. Karamata in [9].
A measurable function f : R+ → R+ is called regularly varying at infinity if there
exists
f (tx)
= λ(x) ∈ (0, ∞)
t →+∞ f (t)
lim
(1.1)
A. L. Yakymiv ()
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
e-mail: arsen@mi-ras.ru
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_20
373
374
A. L. Yakymiv
(t is a positive variable). It follows from (1.1) that λ(x) = x α for some real α [9].
It also follows from (1.1) that, in fact, this relation holds uniformly in x ∈ [a, b] for
arbitrary fixed 0 < a < b < ∞ [9]. Notation: f ∈ RV Fα . If in (1.1) we replace
t by an arbitrary function r(t) : R+ → R+ with limt →+∞ r(t) = +∞, then we
obtain:
lim
t →+∞
f (r(t)x)
= λ(x) ∈ (0, ∞).
f (r(t))
(1.2)
In general, (1.1) does not follows from (1.2). For the monotone functions f , this is
often true:
Lemma 1 [12] Let a function f be monotone and suppose that
lim
t →+∞
f (r(t + 1))
= 1.
f (r(t))
If (1.2) holds for x = x1 and x = x2 such that 0 < xi = 1, i = 1, 2 with irrational
ln x1 / ln x2 then (1.1) is also valid.
This lemma is equivalent to the theorem 1.9 from Seneta [12]. History and related
results are also available in [12, chapter 1.9].
2 Regular Variation in the Orthant
2.1 Uniform Convergence Theorem
n
Put G = int R+
.
n
Definition 1 [5, 11] A measurable function f : R+
→ R+ is said to be regularly
varying at infinity in G with the auxiliary functions r1 (t), . . . , rn (t) : R+ → R+
such that
lim r1 (t) = ∞, . . . , lim rn (t) = ∞,
t →∞
t →∞
if there exists
lim
t →∞
f (r1 (t)x1 , . . . , rn (t)xn )
= λ(x) < ∞, ∀x ∈ G.
f (r1 (t), . . . , rn (t))
(2.1)
fore some positive function λ(x).
Let f be a nonnegative function defined in G. We say that f is nondecreasing
(respectively nonincreasing) if it is componentwise nondecreasing (respectively
Some Properties of Regularly Varying Functions and Series in the Orthant
375
nonincreasing). Notation: f ∈ M1 (G) (respectively f ∈ M2 (G)). Also put
M(G) = M1 (G) ∪ M2 (G).
Theorem 1 Let f ∈ M(G) and ri ∈ RVαi with αi > 0, ∀i = 1, . . . , n. Suppose
that h(t) is some positive function of the variable t > 0 and
f (r1 (t)x1 , . . . , rn (t)xn )
→ λ(x1 , . . . , xn ) < ∞
h(t)
(2.2)
as t → ∞ for all x ∈ G. Then
(1) Function λ(x) is continuous in G and relation (2.2) holds uniformly in x ∈ K,
for an arbitrary compact K ⊂ G.
(2) The relation
f (r1 (tx1 ), . . . , rn (txn ))
→ λ(x α ) = λ x1α1 , . . . , xnαn
h(t)
(2.3)
also holds uniformly in x ∈ K, for an arbitrary compact K ⊂ G.
(3) λ ≡ 0 or λ(x) is positive for each x ∈ G.
(4) For all x ∈ G, a > 0 and some 2 ∈ R, the next equality holds:
λ a α1 x1 , . . . , a αn xn = a 2 λ(x).
(2.4)
Notation For validity of assertions of theorem 1, it is sufficient to require (2.2) for
some dense set in G. Also, it is sufficient to require (2.2) for {x : x ∈ G, x > y},
n
with any fixed y ∈ R+
.
Here and below, the inequality x > y is understood in the componentwise sense.
2.2 Some References
Theorem 1 with notation 1 generalizes the corresponding results from de Haan,
Resnick [4], de Haan, Omey and Resnick [5, Theorem 2.1] and Omey [10, 11].
An approach used here differs from the methods of these works and goes back to
the author’s paper [14]. Note that there are a number of different generalizations of
the concept of one-dimensional regularly varying function to the multidimensional
case. In particular, small reviews in this direction are given in the author’s book
[15, section 1.9] and the article [19]. For extension of this notion to multivariate
generalized functions, see, for example, Vladimirov, Drozhzhinov and Zavyalov
[13]. The brief survey is available in Drozhzhinov [6]. In the one-dimensional
case, there are also a number of generalizations of Karamata’s theory. In particular,
such generalizations and their applications in various fields are contained in the
encyclopedia of N.G. Bingham, C.M. Goldie and J.L. Teugels [1]. We note here only
two last generalizations: of Bingham-Ostaszewski [2] and of Cadena-Omey [3].
376
A. L. Yakymiv
3 Auxiliary Assertions
To prove Theorem 1, we formulate and prove several lemmas. Hereinafter, the
expression
lim f (y) = b
y↓x
lim f (y) = b
y↑x
means that for each ε > 0 there is a δ > 0 such that for an arbitrary y ∈ G with
|yx| < δ, y > x (respectively |yx| < δ, y < x), the inequality |f (y) − b| < ε
Lemma 2 [15, Lemma 1.2.2] Let f ∈ M1 (G) (f ∈ M2 (G)).
(a) Then, two following limits exist:
lim f (y) = f (x− ),
y↑x
lim f (y) = f (x+ ),
y↓x
for each x ∈ G where
f (x− ) = sup(f (y), y ∈ G, y < x),
f (x+ ) = inf(f (y), y ∈ G, x < y)
(respectively f (x− ) = inf(f (y), y < x, y ∈ G), f (x+ ) = sup(f (y), x < y, y ∈
G)) and
f (x− ) ≤ f (x) ≤ f (x+ )
(respectively f (x− ) ≥ f (x) ≥ f (x+ )).
(b) f is continuous in G iff f (x+ ) = f (x− ).
Set for i = 1, 2
Mi+ (G) = {f : f ∈ Mi (G), f (x+ ) = f (x) ∀x ∈ G}, M + (G)=M1+ (G) ∪ M2+ (G).
As usual, for fk ∈ Mi+ (G), (i = 1 or i = 2), the expression
fk ⇒ f
(k → ∞),
means that fk (x) → f (x) for an arbitrary continuous point x of the limiting
function f (·).
Some Properties of Regularly Varying Functions and Series in the Orthant
377
Lemma 3 [15, lemma 1.2.3] Suppose that either fk , f ∈ M1+ (G) or fk , f ∈
M2+ (G) for k ∈ N. If fk ⇒ f and fk ⇒ g as k → ∞ then f (x) = g(x) for
x ∈ G.
Lemma 4 Suppose that either ft , f ∈ M1 (G) or ft , f ∈ M2 (G) for t ∈ R+ . If
ft ⇒ f (t → ∞) then, for an arbitrary x, x(t) ∈ G with x(t) → x
f (x− ) ≤ lim inf ft (x(t)) ≤ lim sup ft (x(t)) ≤ f (x+ )
t →∞
t →∞
in the case when ft , f ∈ M1 (G) and
f (x+ ) ≤ lim inf ft (x(t)) ≤ lim sup ft (x(t)) ≤ f (x− )
t →∞
t →∞
in the case when ft , f ∈ M2 (G).
Proof of Lemma 4 Let ft , f ∈ M1 (G) (t ∈ R+ ). Let 2 > 1 be such that 2x is a
continuity point of the function f . For this 2 > 1, there is some t0 = t0 (2) with
ft (x(t)) ≤ f (2x).
for t ≥ t0 . Therefore,
lim sup ft (x(t)) ≤ lim ft (x2) = f (x2)
t →∞
t →∞
Since the left side of last inequality does not depend on 2, then
lim sup ft (x(t)) ≤
t →∞
lim
2↓1, 2∈P
f (x2) = f (x+ ),
due to Lemma 2. Here P is a set of all 2 such that f is continuous at x2. (We use
the fact that the set R+ \ P is at most countable.) Let 2 < 1 be such that 2x is a
continuity point of the function f . For this 2 < 1, there is some t1 = t1 (2) with
ft (x(t)) ≥ ft (2x).
for t ≥ t0 . Thus,
lim inf ft (x(t)) ≥ lim ft (x2) = f (x2).
t →∞
t →∞
Since the left side of last inequality does not depend on 2, then
lim inf ft (x(t)) ≥
t →∞
lim
2↑1, 2∈P
f (x2) = f (x− ).
due to Lemma 2. The proof for the case ft , f ∈ M2 (G) (t ∈ R+ ) is similar.
378
A. L. Yakymiv
Lemma 5 Let either ft , f ∈ M1 (G) or ft , f ∈ M2 (G) for t ∈ R+ . Suppose that
ft ⇒ f (t → ∞) and f is continuous in G. Then, for any fixed compact K ⊂ G
ft (x) → f (x)
as t → ∞ uniformly in x ∈ K.
Proof of Lemma 5 Suppose the contrary. Let there exist ε > 0 and the sequences
tk ↑ ∞ and xk ∈ K with
|ftk (xk ) − f (xk )| > ε
(3.1)
for any k ∈ N. Since xk ∈ K for arbitrary k ∈ N, then without loss of generality
we will assume that xk → x ∈ K as k → ∞. Put x(t) = xk , for t ∈ [tk , tk+1 ). We
have:
|ft (x(t)) − f (x(t))| ≤ |ft (x(t)) − f (x)| + |f (x(t)) − f (x)| → 0
(3.2)
as t → ∞. The first term on the right-hand side of this inequality tends to zero due
to Lemma 4 and the continuity f in x. The second term tends to zero since f is
continuous in x. The relation (3.2) contradicts to the relation (3.1) for t = tk with
sufficiently large k. This contradiction proves Lemma 5.
4 Proof of the Theorem 1
1/α1
Suppose that there exists x > 0 with λ(x) > 0. Put y = x 1/α ≡ (x1
Consider a function
s(a) = λ(a α y α ) = λ(a α x),
1/αn
, . . . , xn
).
(4.1)
for a ∈ R+ where
a α y α = a α1 y α1 , . . . , a αn y αn .
Note that the function λ(·) is monotone in G and α > 0. Thus, the function s(a) is
also monotone for a > 0 according to (4.1). Let A ⊆ (0, ∞), where A is the set of
continuity points of the function s(a). As it is well-known, the set B = (0, ∞) \ A
of discontinuity points of the function s(a) is at most countable.
Note that the function s(·) is continuous in the point a > 0 iff the function λ(·)
is continuous in the point a α y α . Really, if a ∈ A, then
lim
b↓a, b∈A
=
lim
S(b) = S(a+ ) = S(a) =
b↓a, b∈A
λ(b α y α ) = λ((a α y α )+ )
(4.2)
Some Properties of Regularly Varying Functions and Series in the Orthant
379
and
lim
b↑a, b∈A
=
lim
S(b) = S(a− ) = S(a) =
b↑a, b∈A
λ(b α y α ) = λ((a α y α )− )
(4.3)
according to Lemma 2. From (4.2) and (4.3), we obtain:
λ((a α y α )+ ) = S(a) = λ((a α y α )− ).
Therefore, the function λ(·) is continuous in the point a α y α according to Lemma 2.
The inverse assertion is clear. Let a ∈ A. Then, according to the above, the function
λ(·) is continuous in the point a α y α . Therefore, since ri ∈ RVαi ∀i = 1, . . . , n, we
have
f (r1 (aty1 ), . . . , rn (atyn ))
=
h(t)
f
=
f
r1 (aty1 )
rn (atyn )
r1 (t)
, . . . , rn (t)
=
r1 (t)
rn (t)
a α1 y1α1 + o(1) r1 (t), . . . , a αn ynαn + o(1) rn (t)
→
h(t)
→ λ a α1 y α1 , . . . , a αn y αn = λ(a α y α )
(t → ∞).
(4.4)
according to Lemma 4. Consider a function g(t) = f (r1 (ty1 ), . . . , rn (tyn )), t > 0.
Fix a > 0 with λ(a α y α ) > 0. It follows from (4.4) that
g(bt)
λ(bα y α )
→
g(at)
λ(a α y α )
(t → ∞),
(4.5)
for any fixed b ∈ A. For ν ∈ A/a and g1 (t) = g(at), we have:
λ(b α y α )
g1 (νt)
→ ϕ(ν) =
(t → ∞).
g1 (t)
λ(a α y α )
(4.6)
due to (4.5). According to lemma 1.6 [12], it follows from (4.6) that function g1 (t) is
regularly varying at infinity. Thus, function g(t) is also regularly varying at infinity.
Therefore, for an arbitrary a > 0 and for some 2 ∈ R
λ(a α x)
g(at)
λ(a α y α )
→ a2 =
=
.
g(t)
λ(y α )
λ(x)
(4.7)
380
A. L. Yakymiv
as t → ∞. Fix any vector z ∈ G. Let λ(·) is nonincreasing in G. For this z, there is
a > 0 such that z > a α x. Taking into account (4.7), we have:
λ(z) ≥ λ(a α x) = λ(x) · a 2 > 0,
(4.8)
since λ(x) > 0. Now, suppose that the function λ(·) is nonincreasing in G. Then
there is a > 0 that z > a α x and the relation (4.8) also holds. Therefore, the function
λ(·) is positive and continuous in G. Thus, the relation (2.4) is valid for any x ∈ G.
Moreover, by virtue of Lemma 5, the relation (2.3) holds uniformly in x ∈ K for an
arbitrary compact set K ⊂ G. Without loss of generality, assume that, the functions
r1 (t), . . . , rn (t) increase in t. Then, using Lemma 5 and relation (4.4) with x =
ay, we obtain that the relation (2.2) holds uniformly in x ∈ K, for every compact
K ⊂ G. If λ(·) ≡ 0, then (2.2) and (2.3) also take place locally uniformly in x, as it
follows immediately from Lemma 5. Theorem 1 is proved.
Proof of the notation 1. Without loss of generality, suppose that f ∈ M+ (G). Let
(2.2) holds for some dense set D in G. For x ∈ G \ D, define
λ(x) =
lim
y↓x, y∈D
λ(y)
(see Lemma 1). Since λ(x) is uniquely determines in G, then there is only one limit
point for the set of functions ft (x) = f (r(t)x)/ h(t), x ∈ G as t → ∞ in weak
sense (see definition after Lemma 3). Thus, (2.2) holds for any x belonging to the
continuity set of the limiting function λ(·). The last proof repeats the proof of the
Theorem 1.
Let x ∈ G be fixed. Since ν −αi xi → +∞ as ν ↓ 0 then there exists ν > 0 with
z = ν −α x > y. Thus, for τ = νt, we have:
f (ν α zr(ν −1 τ ))
f (xr(t))
=
h(t)
h(ν −1 τ )
∼
f (ν α (ν −α + ε(τ ))r(τ )z) 2
ν → ν 2 λ(z) ∈ (0, ∞).
h(τ )
Here 2 is the index of regular variation of the function h(·) and vector-function
ε(τ ) → 0 as τ → ∞. This completes the proof of notation 1.
5 Limit Properties of the Multiple Power Series Distributions
5.1 Some Notations
Introduce the following notations. Let the vectors x = (x1 , . . . , xn ) and
y = (y1 , . . . , yn ) belong to R n . Denote xy = (x1 y1 , . . . , xn yn ) and x/y =
Some Properties of Regularly Varying Functions and Series in the Orthant
381
(x1 /y1 , . . . , xn /yn ) (the last in the case, when yk = 0 ∀k = 1, . . . , n). Put
exp(x) = (exp(x1 ), . . . , exp(xn )), ln x = (ln x1 , . . . , ln xn ), P{η ≤ x} =
P{η1 ≤ x1 , . . . , ηn ≤ xn }, P{η < x} = P{η1 < x1 , . . . , ηn < xn }, where η
is random vector from R n . The notation x ↑ 1 means that x → 1, x ∈ (0, 1)n .
n = {x : x = (x , . . . , x ) ∈ R n , x ≥ 0 ∀k =
Here 1 = (1, . . . , 1). Set R+
1
n
k
n
n , x ∈ Z = N ∪ 0 ∀k = 1, . . . , n}.
1, . . . , n}, Z+ = {x : x = (x1 , . . . , xn ) ∈ R+
k
+
n , x = (x , . . . , x ) ∈ R n we use an abbreviation
Also for α = (α1 , . . . , αn ) ∈ R+
1
n
+
xα =
n
<
xkαk ,
k=1
assuming that 00 = 1. Let (ηk , k ∈ N) be a sequence of random vectors (r.v.) from
d
R n . Further, the notation ηk → η means the weak convergence of the corresponding
distributions with P{η ∈ R n } = 1.
5.2 Multiple Power Series Distributions
n ) be a multiple
First, we give the necessary definitions. Let (a(i) ≥ 0, i ∈ Z+
sequence with
B(x) =
a(i)x i < ∞
n
i∈Z+
n is excluded.
for x = (x1 , . . . , xn ) ∈ [0, 1)n . The trivial case a(i) ≡ 0, i ∈ Z+
Definition 2 For x ∈ [0, 1)n a random vector (r.v.) ξx has a multiple power series
distributions if
P{ξx = i} =
a(i)x i
,
B(x)
(5.1)
n.
for some B(x) and for any i ∈ Z+
It is clear that P{ξx ∈ R n } = 1. History of this notion and some bibliographic
references are given in encyclopedias [7, 8], also see the articles [17, 18]. Let the
sequence of vectors b = b(k) ∈ (0, ∞)n , k ∈ N be given with bj = bj (k) →
∞, ∀j = 1, . . . , n as k → ∞.
Definition 3 We say that B(x) regularly varies as x ↑ 1 along the sequence b =
b(k), iff
B(exp(−λ/b))
→ Ψ (λ) ∈ (0, ∞),
B(exp(−1/b))
(5.2)
382
A. L. Yakymiv
for an arbitrary fixed λ = (λ1 , . . . , λn ) > 0 as k → ∞.
(Notations λ/b and exp(−λ/b) are defined in Sect. 5.1.) Let (a(i) > 0, i ∈
n
, ij > C > 0, ∀j = 1, . . . , n).
Z+
Definition 4 [16] We say that the sequence a(i) is weakly one-sided oscillatory at
infinity along the sequence b = b(k) if for every j = 1, . . . , n and for any sequence
zj = zj (k) > 1, zj = 1 + o(1) one of the next inequalities
lim inf a(r1 , . . . , rj −1 , zj rj , rj +1 , . . . , rn )/a(r) ≥ 1;
k→∞
lim sup a(r1 , . . . , rj −1 , zj rj , rj +1 , . . . , rn )/a(r) ≤ 1.
k→∞
n
hold for every fixed y = (y1 , . . . , yn ) ∈ int R+
. Here r = r(k) = (r1 (k), . . . , rn (k))
is an arbitrary function of k with
r1 ∼ y1 b1 , . . . , rn ∼ yn bn .
n
Hereinafter, we define a(x) = a([x]) for x ∈
/ Z+
. The simplest examples of such
n
sequences are monotone in each variable sequences (a(i) ≥ 0, i ∈ Z+
). The
following assertion has been proved in [17] (we formulate it as a lemma).
Lemma 6 Let B(x) regularly varies as x ↑ 1 along the sequence b = b(k), and an
n
arbitrary vector u ∈ int R+
is fixed. Then, for x = exp(−u/b)
d
ξx (1 − x) → η = η(u),
(k → ∞)
(5.3)
where r.v. η(u) has Laplace transform Ψ ((λ + 1)u)/Ψ (u). It follows from (4.2) that
n . Let this
the function Ψ (λ) is Laplace transform of some σ -finite measure ν in R+
n
measure ν be absolutely continuous in int R+ with continuous density ϕ(·). Then the
n with density
distribution of r.v. η(u) is also absolutely continuous in int R+
ϕ(y/u)e−(y,1)
,
ψu (y) = &n
j =1 uj Ψ (u)
n
∀y ∈ int R+
.
(5.4)
n
) is weakly one-sided
Suppose in addition that the sequence (a(i) ≥ 0, i ∈ Z+
oscillatory at infinity along the sequence b = b(k). Then
P{ξx = [y/(1 − x)]}
&n
→ ψu (y) (k → ∞)
j =1 (1 − xj )
n.
uniformly in y ∈ K, for any compact K ⊂ int R+
(5.5)
Some Properties of Regularly Varying Functions and Series in the Orthant
383
Here we extend this lemma to the form of necessary and sufficient conditions for
the fulfillment of relations (4.2) and (4.3). Namely, the following limit theorem is
also valid.
Theorem 2 I. The next assumptions are equivalent.
1. Function B(x) regularly varies as x ↑ 1 along the sequence b = b(k), i.e. (4.2)
holds.
2. The relation (4.3) holds for some r.v. η(u).
II. Let some of this assumptions hold. Also suppose that the measure ν from the
n with continuous and positive density
Lemma 6 is absolutely continuous in int R+
n ) is weakly one-sided
ϕ(·). In this case, (4.5) valids iff the sequence (a(i), i ∈ Z+
oscillatory at infinity along the sequence b = b(k).
n and λ ≥ u. Put x = exp(−u/b) and
Proof Let (4.3) holds. Fix vectors u ∈ int R+
ν = (λ − u)/(b(1 − x)). We have:
ν = (λ − u)/(b(1 − x)) → (λ − u)/u
as k → ∞. It follows from (4.3) and (4.6) that
B(exp(−λ/b))
B(exp(−(ν(1 − x)b + u)/b))
=
B(exp(−u/b))
B(exp(−u/b))
=
B(exp(−ν(1 − x) − u/b))
B(exp(−u/b))
= E exp(−(ν, ξx (1 − x)))
→ E exp(−(λ − u)/u), η(u)) ≡ Ψu (λ) ∈ (0, ∞).
Applying the last relation for λ = 1, we obtain that
B(exp(−1/b))
→ Ψu (1) ∈ (0, ∞).
B(exp(−u/b))
Thus,
B(exp(−λ/b))
B(exp(−1/b))
=
B(exp(−λ/b)) B(exp(−1/b))
B(exp(−u/b)) B(exp(−u/b))
→
Ψu (λ)
∈ (0, ∞).
Ψu (1)
(5.6)
384
A. L. Yakymiv
The relation (4.2) follows from the last relation for any λ ≥ u. Taking into account
notation 1 and Theorem 1, we obtain (4.2) for any λ ∈ G. The converse assertion is
proved in [17]. Further, let (4.2) and (4.5) hold. Put for z ∈ N n
p(z) = P{ξx = [y/(1 − x)] =
a(z)
exp(z, ln x),
B(x)
where x = exp(−1/z). Therefore,
a(z) =
B(exp(−1/z))
B(exp(−1/z))
=
exp(n).
p(z) exp(z, −1/z)
p(z)
Thus, for an arbitrary fixed y ∈ G and for r ∈ N n , r > C1, r ∼ yb, we have:
a(r)
p([yb])
B(exp(−1/r))
=
·
a(yb)
p(r)
B(exp(−1/[yb]))
=
p([yb])
B(exp(−1/r))B(exp(−1/b))
·
p(r)
B(exp(−1/[yb]))B(exp(−1/b))
=
p([yb]) Ψ (b/r) + o(1)
p([yb])
∼
,
p(r) Ψ (1/y) + o(1)
p(r)
since b/r → 1/y and (4.2) takes place locally uniformly in λ ∈ G (Theorem 1).
Further, suppose that ϕ(y) > 0. In this case, it follows from (4.5) that
p([yb]) ∼
n
<
ϕ(y)
exp(n)
(1 − exp(−1/[yi bi ])) ·
Ψ1 (1)
i=1
and
p([r]) ∼
n
<
(1 − exp(−1/[ri ])) ·
i=1
ϕ(y)
exp(n).
Ψ1 (1)
Thus,
a(r)
→ 1.
a(yb)
Therefore, the sequence a(i) weakly oscillates at infinity along the sequence b =
b(k), and somewhat more was proved than required. The converse assertion is
proved in [17]. The theorem is proved.
At the end, the author expresses his deep gratitude to reviewers for valuable
comments.
Some Properties of Regularly Varying Functions and Series in the Orthant
385
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New York (1992)
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(1930)
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Leuven, Belgium. (Dutch) (1982)
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74. Series of Monografphies. EHSAL, Brussel, Belgium (1989)
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Functions. Kluwer, Dordrecht (1988)
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586 (2020)
Influence of the Configuration of Particle
Generation Sources on the Behavior of
Branching Walks: A Case Study
E. B. Yarovaya
Abstract We consider a supercritical continuous-time branching random walk on
a multidimensional lattice with finite number of particle generation sources of the
same intensities without any restrictions on the variance of jumps of the underlying
random walk. The effect of “limit coalescence” of eigenvalues is revealed for an
arrangement of sources under which the pairwise distances between them go off to
infinity. The effect of the arrangement of particle generation sources on the order
of appearance of positive eigenvalues in the spectrum of the evolutionary operator
with receding sources is revealed.
Keywords Branching random walks · Continuous time · Multidimensional
lattices · A supercritical case · Green functions · Receding particle generation
sources
1 Introduction
Consider a continuous-time branching random walk (BRW) on the lattice Zd ,
d ≥ 1, assuming that initially the system contains one particle at a point x and that
branching (birth and death of particles) takes place at a finite number of highlighted
lattice points (the branching sources x1 , x2 , . . . , xN ). BRWs with finite number of
branching sources and finite variance of jumps of random walk were studied, for
example, in [16]; the case of infinite variance was considered in [19]. In [19, 21] it
was shown that the structure of the spectrum of the evolution operator of the mean
population size of particles depends not only on the intensity of birth and death of
particles at sources, but also on their position. A limit theorem for a supercritical
BRW in the case of an arbitrary configuration of branching sources with infinitely
increasing pairwise distances between sources was announced by the author in [20].
E. B. Yarovaya ()
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
e-mail: yarovaya@mech.math.msu.su
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_21
387
388
E. B. Yarovaya
Note that in this study the variance of jumps of the ransom walk were subjected
to no constraints. In the actual fact, our proof depends for the most part on the use
of Green functions of the random walk—these functions can be represented as the
Laplace transforms of the of transition probabilities of the random walk. It should
be noted that the finiteness or infiniteness of the variance of jumps has an effect only
on the asymptotic behavior of the transition probabilities, and, as a corollary, on the
behavior of the Green function [8]. Due to limitations on the size of [20], the author
was unable to present there the proof of the main result. So the main purpose of the
present article is to give the proof of the limit theorem from [20].
Continuous-time branching random walks on Zd have been intensively studied in
the past few decades (see, for instance, [1, 2, 11, 12, 14, 15] and the references given
there). In [18], apparently for the first time it was shown that the number of positive
eigenvalues for the operator describing the evolution of the mean population size of
particles and their multiplicity depends, rather than just on the source intensity, on
their spatial configuration also; a detailed proof of the corresponding results may be
found in [19]. The paper [18] also introduces the concept of a weakly supercritical
BRW with discrete spectrum containing a unique positive eigenvalue. BRWs of such
kind are important because the uniqueness of a positive eigenvalue considerably
facilitates the study of the propagation of particle fronts (see, for example, [7]).
2 Model of a Branching Random Walk
By a branching random walk (BRW) we mean a stochastic process that combines a
random walk of particles with their branching (birth or death) at certain points on
the lattice Zd (called branching sources). Let us give precise definitions.
A random walk of particles on Zd , which underlies the process, is defined in
terms of the matrix of transitionintensities A = (a(x, y))x,y∈Zd , which features the
following regularity property: y∈Zd a(x, y) = 0 for all x, where a(x, y) ≥ 0 for
x = y and a(x, x) < 0.
In this case, following the axioms from Ch. III, § 2 in [6], the probabilities
p(h, x, y) of transition of a particle from a point x = 0 to an arbitrary point y
in a small time h can be written as
p(h, x, y) = a(x, y)h + o(h)
for y = x,
p(h, x, x) = 1 + a(x, x)h + o(h).
(1)
(2)
According to, for example, [6, Ch. III], from (1), (2) it follows that the transition
probabilities p(t, x, y) satisfy the system of difference/differential equations (the
Kolmogorov backward equations)
∂p(t, x, y) =
a(x, x )p(t, x , y),
∂t
p(0, x, y) = δ(x − y),
x
where δ(·) is the discrete Kronecker δ-function on Zd .
(3)
Influence of the Configuration of Particle Generation Sources on the Behavior of BRWs
389
It is assumed that “branching” may occur at some points of the lattice x1 , . . . , xN ,
which are called sources. The branching mechanism at each source xi is controlled
by a Galton–Watson continuous-time process (see [9, Ch. I, § 4], [4, Ch. III]), which
is defined by the infinitesimal generating function
f (u, xi ) =
∞
bn (xi )un ,
0≤u≤1
(4)
n=0
(this function naturally
depends on the source xi ); here bn (xi ) ≥ 0 for n = 1,
b1 (xi ) < 0 and n bn (xi ) = 0. It is assumed that f (r)(1, xi ) < ∞ for all r ∈ N. In
what follows, the quantities
bn (xi )
−1
βi = f (1, xi ) =
nbn (xi ) = (−b1 (xi ))
n
(−b1 (xi ))
n
n=1
will be of special importance; here the last sum is the mean number of offspring in
the source xi .
Now we combine both walking and branching mechanisms to obtain a BRW. Let
μt (y) be the number of particles at time t at a point y. Then the condition that at
the initial time t = 0 the system consists of one particle at a point x is equivalent to
the equality μ0
(y) = δ(x − y). Here, the total number of particles on the lattice is
given by μt = y∈Zd μt (y).
If at a time t a particle is at a nonsource point, then it walks according to the
above description. Hence, it remains only to find the evolution of the particle in the
case when walking and branching are both possible (i.e., at sources).
So, let us assume that at time t the source xi contains μt (xi ) > 0 particles. Then,
for each of them the following alternative holds: in a small time h either the particle
with probability
p(h, xi , y) = a(xi , y)h + o(h)
moves at some point y = xi , or it does not execute any transition, and either
produces with probability
p∗ (h, xi , n) = bn (xi )h + o(h)
n = 1 descendants, which stay at the point xi (we assume that the particle itself is
counted here; in the case n = 0 the particle is said to perish), or with probability
1−
y =xi
a(xi , y)h −
bn (xi )h + o(h)
n=1
the particle undergoes no changes. Thus, the residence time of a particle in a source
is exponentially distributed with the parameter −(a(xi , xi ) + b1 (xi )). Besides, each
new particle evolves by the same law irrespective of the other particles and of the
entire past history.
390
E. B. Yarovaya
Let m1 (t, x, y) := Ex μt (y) be the mean number of particles at time t at a point y
and let m1 (t, x) := Ex μt (y) be the mean total number of particles on Zd at time t
provided that μ0 (y) ≡ δ(x −y). According to [16, 17], the variation of the quantities
m1 (t, x, y) and m1 (t, x) is described by the equations
∂t m1 (t, x, y) =
a(x, x )m1 (t, x , y) +
βi δ(x − xi )m1 (t, x, y),
(5)
i=1
x
∂t m1 (t, x) =
N
a(x, x )m1 (t, x ) +
N
βi δ(x − xi )m1 (t, x)
(6)
i=1
x
with the initial conditions m1 (0, x, y) = δ(x − y) and m1 (0, x) = 1, respectively.
These equations are quite involved, and so in what follows we shall study the BRW
under additional assumptions which are natural in applications.
3 Assumptions
First, we assume that the intensities a(x, y) are symmetric and spatially homogeneous; i.e., a(x, y) = a(y, x) = a(0, y − x). In this case, for brevity, the common
value of the quantities a(x, y), a(y, x), a(0, y − x) will be denoted by a(x − y);
i.e., a(x − y) := a(x, y) = a(y, x) = a(0, y − x).
Second, we assume that a random walk is irreducible (which in terms of the
d
matrix A means its irreducibility):
k for each z ∈ Z there exists a family of vectors
d
z1 , . . . , zk ∈ Z such that z = i=1 zi and a(zi ) = 0 for i = 1, . . . , k.
Third, we assume that the intensities of the branching sources are as follows:
β1 = · · · = βN := β.
4 Methods of Investigation
One way to deal with Eqs. (5) and (6) is to represent these equations as differential
equations in some Banach spaces. To be able to utilize this approach,
we introduce
on the set of functions u(x), x ∈ Zd , the operator (Au)(x) = x a(x − x )u(x )
and (for each of the sources x1 , . . . , xN ), the operators (xi u)(x) = δ(x − xi )u(x).
Moreover, we also consider the operator
Hβ1 ,...,βN = A +
N
i=1
βi x i ,
Influence of the Configuration of Particle Generation Sources on the Behavior of BRWs
391
which, in the case of equal intensities β, assumes the form
Hβ = A + β
N
x i .
i=1
According to, for example, [16, 17], these operators are bounded linear operators
on l p (Zd ), p ∈ [1, ∞]. Interpreting, for each t ≥ 0 and y ∈ Zd , the functions
m1 (t, ·, y) and m1 (t, ·) as elements of one of the spaces l p (Zd ), we can (see, for
example, [16]) write (5) as a linear differential equation in l 2 (Zd )
d
m1 (t, x, y) = (Hβ m1 (t, ·, y))(x)
dt
with the initial condition m1 (0, x, y) = δ(x − y), and (6), as a linear differential
equation in l ∞ (Zd ),
d
m1 (t, x) = (Hβ m1 (t, ·))(x)
dt
with the initial condition m1 (0, x) ≡ 1.
5 Green Functions
Various properties of transition probabilities p(t, x, y) of a random walk can be
conveniently expressed in terms of the so-called Green function, which is defined as
the Laplace transform of the transition probability p(t, x, y):
Gλ (x, y) :=
∞
e−λt p(t, x, y) dt,
λ ≥ 0;
(7)
0
it can be expressed also in an alternative form (see, for example, [14, §2.2]):
1
Gλ (x, y) =
(2π)d
ei(θ,y−x)
dθ,
[−π,π]d λ − φ(θ )
x, y ∈ Zd ,
λ ≥ 0,
(8)
or, which is the same,
Gλ (x, y) =
1
(2π)d
cos (θ, y − x)
dθ,
λ − φ(θ )
[−π,π]d
x, y ∈ Zd ,
λ ≥ 0,
(9)
392
E. B. Yarovaya
where φ(θ ) the Fourier transform of the transition intensities a(z),
φ(θ ) =
a(z)ei(θ,z),
θ ∈ [−π, π]d .
z∈Zd
The function G0 (x, y) has the following transparent probabilistic sense for the
random walk underlying the BRW: this is the average time that a particle spends
at the point y as time t → ∞ when the process emerges from the point x. If we
consider a discrete-time random walk without branching, then the function G0 (x, y)
is the mean number of times when a particle hits the point y for large time (see, for
example, [10]).
In terms of the function Gλ (x, y) one can express the asymptotical behavior of
the mean number of particles m1 (t, x, y) as t → ∞ (see, for example, [14]). In
the same paper, it was shown that the analysis of BRW depends substantially on
whether the quantity G0 := G0 (0, 0) is finite or not.
Definition 1 For G0 := ∞ (respectively, G0 < ∞), we say that the random walk
is recurrent (nonrecurrent).
If the following condition is fulfilled,
|z|2 a(z) < ∞,
(10)
z
then the variance of random walk jumps be finite (here and in what follows |z|
denotes the Euclidean norm of a vector z), and G0 = ∞ in the dimension d = 1 or
d = 2 and G0 < ∞ for d ≥ 3 (see, for example, [3, 14]). If for all z ∈ Zd with
sufficiently large norm, for example, the asymptotic equality
a(z) ∼
z
|z|
,
|z|d+α
H
α ∈ (0, 2),
(11)
is satisfied, where H (·) is a continuous positive symmetric function on the sphere
Sd−1 := {z ∈ Rd : |z| = 1}, then G0 = ∞ in the case d = 1, α ∈ [1, 2); in the
cases d = 1, α ∈ (0, 1) or d ≥ 2, α ∈ (0, 2) the quantity
G0 is finite (see [13]).
As distinct from (10), under condition (11) the series z |z|2 a(z) is divergent, and
therefore, the variance of jumps is infinite.
Remark 1 The approach of this section, which is based on interpreting the BRW
evolution equations as differential equations in Banach spaces, can be also applied in
more general settings (of course, with more or less significant changes), namely, for
the description of evolution of the leading moments of population sizes of particles,
the description of evolution of the population size of particles, and so on (see, for
example, [14] and [16]).
Influence of the Configuration of Particle Generation Sources on the Behavior of BRWs
393
6 Receding Sequence of Configurations of Branching Sources
In the general case, consider the sequence of configurations of branching sources
(k)
S (k) = {x1(k), x2(k) , . . . , xN
},
where N ≥ 2, of equal intensities β.
For each configuration S (k) , we denote by βc0 (S (k) ) the minimal value of the
source intensity such that, for β > βc0 (S (k) ), the spectrum of the corresponding
operator
H(k)
β =A+β
N
i=1
x (k)
i
contains positive eigenvalues. According to Theorem 4 in [21], for β > βc0 (S (k) )
the operator H(k)
β may have at most N positive eigenvalues (counting multiplicity)
λ0 (β, S (k) ) > λ1 (β, S (k) ) ≥ · · · ≥ λN−1 (β, S (k) ) > 0,
(12)
the eigenvalue λ0 (β, S (k) ) having multiplicity 1.
By Theorem 5 in [21], for each k and the above β, λ > 0 is an eigenvalue of the
(k)
operator Hβ if and only if
det (λ, S (k) ) − β −1 I = 0,
(13)
where the matrix (λ, S (k) ) is such that
⎤
(k) (k)
(k) (k)
Gλ (x1 , x1 ) · · · Gλ (x1 , xN )
⎢
(k) (k)
(k) (k) ⎥
⎢ G (x , x1 ) · · · Gλ (x2 , xN ) ⎥
(λ, S (k) ) = ⎢ λ 2
⎥,
⎦
⎣
...
...
...
(k) (k)
(k) (k)
Gλ (xN , x1 ) · · · Gλ (xN , xN )
⎡
(14)
and Gλ (x, y) are the Green functions (7) of the transition probabilities p(t, x, y),
which by (8) have also the alternative representation
Gλ (x, y) =
1
(2π)d
ei(θ,y−x)
dθ,
[−π,π]d λ − φ(θ )
where φ(θ ) is the Fourier transform of the transition probabilities a(z).
Note that since a random walk is symmetric and homogeneous, for each i =
1, . . . , N we have
Gλ (xi(k) , xi(k)) = Gλ (0, xi(k) − xi(k)) = Gλ (0, 0) = Gλ ,
394
E. B. Yarovaya
and hence, for each λ > 0 and k, the matrix (λ, S (k) ) is symmetric.
Let λ∗ (β) > 0 be the root of the equation
Gλ = β −1 .
(15)
It is known (see, for example, [14]) that the function λ∗ (β) is defined and strictly
positive for β > G−1
0 .
Remark 2 If the configuration S (k) is a singleton (i.e., N = 1), the eigenvalue
λ0 (β, S (k) ) for β > βc0 = G−1
0 can be found from Eq. (15).
For each configuration S (k) , by
0 ≤ βc0 (S (k) ) < βc1 (S (k) ) ≤ · · · ≤ βcN−1 (S (k) ),
we denote the critical values β such that a new positive eigenvalue of the operator
Hβ emerges from 0 as β increases and crosses βci (S (k) ) (see, for example, [19]).
If the branching sources are located at the vertices of simplexes on Zd , d ≥ 1,
then 0 ≤ βc0 (S (k) ) ≤ G−1
< βc1 (S (k) ) = · · · = βcN−1 (S (k) ), see [18, 21].
0
(k)
In this case, the operator Hβ may have at most two geometrically different
positive eigenvalues: a simple eigenvalue λ0 (β, S (k) ) > λ∗ (β) and an eigenvalue
λ1 (β, S (k) ) < λ∗ (β) of multiplicity N − 1.
Of course, this is no longer true in the general setting. For each configuration of
sources, by S (k) we denote the minimal distance between sources by
(k)
(k) ρ (k) := min xi − xj .
i =j
Definition 2 If ρ (k) → ∞ as k → ∞, then the sequence of configurations S (k) will
be called receding.
7 The Main Results
Theorem 1 Let N ≥ 2 and let {S (k) } be a receding sequence of configurations.
Then
lim βcj (S (k) ) = G−1
0 ,
k→∞
j = 0, . . . , N − 1,
(16)
and moreover, for each k = 1, 2 . . . ,
(k)
0 < βc0 (S (k) ) < G−1
0 < βcN−1 (S )
0 = βc0 (S (k) ) < βc1 (S (k) ) ≤ · · · ≤ βcN−1 (S (k) )
if G0 < ∞,
if
G0 = ∞.
(17)
(18)
Influence of the Configuration of Particle Generation Sources on the Behavior of BRWs
395
Furthermore, for each β > G−1
0 , there exists k(β) such that, for k ≥ k(β), all the
(k)
quantities λj (β, S ), j = 1, . . . , N, are defined, and besides,
lim λj (β, S (k) ) = λ∗ (β),
k→∞
j = 0, . . . , N − 1,
(19)
and in addition, for each k ≥ k(β),
λ0 (β, S (k) ) > λ∗ (β) > λN−1 (β, S (k) ).
(20)
Before proceeding with the proof of the theorem, we shall prove one auxiliary
result. For each pair of indexes i, j = 1, . . . , N, we introduce the function
(k) (k)
Gλ (xi , xj ) =
1
(2π)d
(k)
e
[−π,π]d
(k)
i(θ,xi −xj )
λ − φ(θ )
dθ
(21)
and consider
= {λ : Gλ < ∞}.
It is clear that = [0, ∞) for G0 < ∞ and = (0, ∞) for G0 = ∞.
Lemma 1 The following results hold:
for each k = 1, 2, . . . , the functions Gλ (xi(k) , xj(k) ) are nonnegative and
monotone decreasing in λ for λ ∈ ;
(ii) for each λ ∈ ,
(i)
Gλ (xi(k), xj(k) ) → 0
as k → ∞.
Proof Assertion (i) of the lemma follows from the alternative representation of the
(k) (k)
function Gλ (xi , xj ),
(k)
(k)
Gλ (xi , xj ) =
∞
0
e−λt p(t, xi , xj ) dt,
(k)
(k)
which holds for λ ∈ .
Note that the function φ(θ ) in (21) is continuous nonpositive
and 2π-periodic
in each coordinate θ = (θ1 , . . . , θd ) (provided that the series z a(z) = 0 is
absolutely convergent). Hence λ − φ(θ ) ≥ λ ≥ 0 for each λ ∈ . In this case,
the integrand in (21) is a continuous function of θ for λ > 0 and this function is
absolutely summable for λ = 0 and G0 < ∞. Hence, for each λ ∈ , the value
of the integral (21) tends to zero as k → ∞ by the well-known Riemann–Lebesgue
theorem from the Fourier analysis (see [5, Thm. 31]). This completes the proof of
assertion (ii). Lemma 1 is proved.
396
E. B. Yarovaya
Let us proceed with the proof of the theorem.
Proof (of Theorem 1) Let β > G−1
0 . Then the solution λ∗ = λ∗ (β) of Eq. (15) is
strictly positive.
(k)
We denote by γi (λ), i = 0, . . . , N −1, the eigenvalues of the matrix (λ, S (k) )
arranged in the decreasing order. According to [19, Lemma 3], each of the functions
γi(k) (λ), i = 0, . . . , N − 1, is strictly decreasing and tends to zero as λ → ∞.
Moreover, Theorem 2 in [19] says that, for each k, the quantities γi(k) (λ) are related
as follows:
(k)
(k)
(k)
0 ≤ γN−1 (λ) ≤ · · · ≤ γ1 (λ) < γ0 (λ),
λ ∈ .
(22)
For λ > 0, the first of these inequalities can be strengthened. Indeed, in this case, the
function (λ−φ(θ )−1 in (21) is continuous and strictly positive, Hence it is separated
from zero. Now according to [19, Lemma 2], the matrix (λ, S (k) ) is symmetric and
positive definite. Hence all its eigenvalues are strictly positive, and therefore,
(k)
0 < γN−1
(λ) ≤ · · · ≤ γ1(k) (λ) < γ0(k) (λ),
λ > 0.
(k)
We further note that, for each λ ∈ , the number γ0 (λ) is the leading (maximal)
positive eigenvalue of the matrix (λ, S (k) ) with the corresponding eigenvector
(k)
(k)
u(k) (λ) = {u1 (λ), . . . , uN (λ)} with positive coordinates. In this case,
(λ, S (k) )u(k) (λ) = γ0(k) (λ)u(k) (λ).
(23)
The matrix (λ, S (k) ) has positive entries, its diagonal entries are equal to Gλ , and
the vector u(k) (λ) has positive coordinates, and hence,
(k) (k) (k)
, xN ) u (λ) < (λ, S (k) )u(k) (λ).
Gλ u(k)(λ) = diag Gλ (x1(k) , x1(k) ), . . . , Gλ (xN
By (23) this implies that
(k)
Gλ < γ0 (λ),
λ ∈ .
(24)
(k)
We finally note that the sum of the numbers γi (λ), i = 0, . . . , N − 1, which
constitute the entire set of eigenvalues of the matrix (λ, S (k) ), is equal to the trace
of this matrix, i.e.,
(k)
(k)
(k)
γ0 (λ) + γ1 (λ) + · · · + γN−1 (λ) = NGλ .
(25)
Since by (24) the greatest of the terms in the sum (25) (which is γ0(k) (λ)) is strictly
(k)
greater than Gλ , the minimal term (which is γN−1
(λ)) should be strictly smaller
Influence of the Configuration of Particle Generation Sources on the Behavior of BRWs
397
than Gλ :
(k)
λ ∈ .
γN−1 (λ) < Gλ ,
(26)
(k)
(k)
Using inequalities (24) and (26) and since the functions γ0 (λ), γN−1 (λ) and Gλ
are strictly monotone, it follows that the solutions of the equations
γ0(k) (λ) =
1
,
β
(k)
γN−1
(λ) =
1
,
β
Gλ =
1
,
β
which control the quantities λ0 (β, S (k) ), λN−1 (β, S (k) ) and λ∗ (β), respectively, are
related by the strict inequalities (20).
Let ε > 0 be arbitrary. Then by the monotonicity of the function Gλ , there exists
δ > 0 such that Gλ∗ −ε > Gλ∗ + δ, Gλ∗ +ε < Gλ∗ − δ.
Given λ = λ∗ − ε, we choose a sufficiently large k = k1 (ε) so as to have
(k)
γ0 (λ∗ − ε) ≤ Gλ∗ −ε + δ,
(k)
γN−1 (λ∗ − ε) ≥ Gλ∗ −ε − δ
for k ≥ k1 (ε)
(this can be done by assertion (ii) of Lemma 1). Hence by (22) we have, for all
i = 0, . . . , N − 1,
Gλ∗ −ε − δ ≤ γi(k) (λ∗ − ε) ≤ Gλ∗ −ε + δ,
for k ≥ k1 (ε).
Similarly, taking λ = λ∗ + ε, we choose k = k2 (ε) ≥ k1 (ε) so as to satisfy the
inequalities
γ0(k) (λ∗ + ε) ≤ Gλ∗ +ε + δ,
(k)
γN−1
(λ∗ + ε) ≥ Gλ∗ +ε − δ
for k ≥ k1 (ε)
(this can be done by assertion (ii) of Lemma 1). Hence for all i = 0, . . . , N − 1 by
(22) we have
(k)
Gλ∗ +ε − δ ≤ γi (λ∗ + ε) ≤ Gλ∗ +ε + δ,
for k ≥ k1 (ε)
We now note that by monotonicity and by the choice of the numbers k1 (ε) and
(k)
k2 (ε), for k ≥ k2 (ε) ≥ k1 (ε) the functions γ0(k) (λ) and γN−1
(λ), and hence all
the functions γi(k) (λ), i = 0, . . . , N − 1, do not assume the value β −1 outside the
interval (this value β −1 is assumed only inside this interval).
So, we have proved that the roots λki (β) of the equations
(k)
γi (λ) = 1/β,
i = 1, . . . , N − 1,
satisfy, respectively, the inequalities
λ∗ − ε ≤ λ(k)
i (β) ≤ λ∗ + ε
for k ≥ k2 (ε) ≥ k1 (ε).
398
E. B. Yarovaya
(k)
Now by noting that λi (β, S (k) ) = λi (β), we conclude that for each i = 1, . . . , N −
1, the eigenvalues λi (β, S (k) ) tend to λ∗ (β) as k → ∞. This proves assertion (19)
of the theorem.
We now set
βi(k) (λ) =
1
(k)
γi (λ)
i = 0, . . . , N − 1.
,
(27)
(k)
Each of the functions βi (λ), i = 0, . . . , N − 1, is defined on the set λ ∈ , is
strictly increasing, and tends to infinity as λ → ∞, because, as was pointed out
(k)
above, each of the functions γi (λ), i = 0, . . . , N − 1, is strictly increasing and
tends to zero as λ → ∞.
Now let G0 < ∞. In this case, inequalities (24) and (26) are satisfied for λ = 0,
which implies that (17):
(k)
0 < βc0 (S (k) ) = β0 (0) =
1
(k)
γ0 (0)
<
1
1
(k)
< (k)
= βN−1 (0) = βcN−1 (S (k) ).
G0
γ
(0)
N−1
(k)
If G0 = ∞, then according to [19, Lemma 5], the functions γi (λ), i = 0, . . . , N −
1, satisfy the estimates
|γ0 (λ) − (N − 1)Gλ | ≤ γk∗ ,
(k)
0 ≤ γN−1 (λ) ≤ . . . ≤ γ1 (λ) ≤ γk∗ .
(k)
(k)
with some γk∗ for each k and all λ ∈ . Hence, passing to the limit for the quantities
βi(k)(λ) = (k)1 , i = 0, . . . , N − 1, we get the required relations (18). In particular,
γi (λ)
1
βc0 (S (k) ) = lim β0(k) (λ) = lim
λ↓0 γ (k) (λ)
0
λ↓0
= 0.
Let us verify equality (16). Given an arbitrary ε > 0, we denote by λε the value of
the parameter λ > 0, for which
Gλ ε =
1
G−1
0 +ε
(note that even if G0 = ∞, the quantity G−1
0 + ε is strictly positive, and hence,
λε > 0). In this case, since Gλ is monotone, we also have the strict inequality
Gλε /2 >
1
G−1
0 +ε
.
Influence of the Configuration of Particle Generation Sources on the Behavior of BRWs
399
Further,
1
G−1
0 +ε
>
1
G−1
0 + 2ε
,
and hence by assertion (ii) of Lemma 1 one can specify k3 = k3 (ε) such that the
inequality
(k)
Gλε /2 > γN−1
(λε /2) >
1
G−1
0 + 2ε
will be satisfied for k ≥ k3 (ε). Hence by definition of βcN−1 (S (k) ) we have
(k)
G−1
0 ≤ βcN−1 (S ) = lim
1
≤ G−1
0 + 2ε.
λ↓0 γ (k) (λ)
N−1
(28)
Since ε > 0 is arbitrary, the last inequalities (which hold, as is easily seen, both for
G0 < ∞ and for G0 = ∞) prove (16) for j = N − 1.
It remains to verify equality (16) for all remaining j . There are two cases to
consider: G0 = ∞ and G0 < ∞.
In the case G0 = ∞, we have already shown that
lim βcN−1 (S (k) ) = 0.
(29)
k→∞
Moreover, by (22) and (27) we have
0 ≤ βc0 (S (k) ) ≤ βc1 (S (k) ) ≤ · · · ≤ βcN−1 (S (k) ),
(30)
and hence
lim βcj (S (k) ) = 0,
k→∞
j = 0, . . . , N − 1,
verifying equality (16) in the case G0 = ∞.
Assume now that G0 < ∞. In this case, as was noted in (25),
(k)
(λ),
NGλ = γ0(k) (λ) + γ1(k) (λ) + · · · + γN−1
(k)
(k)
and moreover, by (22) γN−1 (λ) is not greater than the remaining γi (λ). Hence
(k)
(k)
NGλ ≥ γ0 (λ) + (N − 1)γN−1 (λ),
400
E. B. Yarovaya
and besides by (23)
(k)
γ0 (λ) > Gλ .
Therefore,
(k)
(λ)
βN−1
1
(k)
> β0 (λ) ≥
(k)
Gλ
NGλ β
(λ) − (N − 1)
N−1
Now, making λ ↓ 0 in these inequalities and taking into account that
(k)
lim βN−1 (λ) = βci (S (k) )
λ↓0
we get
βcN−1 (S (k) )
1
> βc0 (S (k) ) ≥
G0
NG0 βcN−1 (S (k) ) − (N − 1)
Finally, making k → ∞ and using the above equality (29), we get that
lim βc0 (S (k) ) = G−1
0 .
k→∞
Now the remaining equalities in (16) follow from the two last equalities together
with inequalities (30).
To conclude the proof it remains to note that the existence of the required k(β)
for β > G−1
0 follows from the limit relation (16).
Theorem 1 is proved.
Remark 3 Note that in the case of a simplicial configurations, it follows from
[18, 21] that in the case G−1
< ∞ for each k there is only one critical value
0
β
=
βc0 (S (k) ), whereas the remaining critical values
to the left of the point G−1
0
βc1 (S (k) ), . . . , βcN−1 (S (k) ) lie to the right of G−1
0 ,
(k)
(k)
βc0 (S (k) ) < G−1
0 ≤ βc1 (S ), . . . , βcN−1 (S ).
(31)
Theorem 1 (see inequalities (17)) indicates only the location of the critical values βc0 (S (k) and βcN−1 (S (k) ) with respect to G−1
0 , but it does not make more
clear the positions of the remaining critical values βc1 (S (k) ), . . . , βcN−1 (S (k) );
here it can be asserted
(by their definition) only that they lie in the interval
βc0 (S (k) ), βcN−1 (S (k) ) .
The next example shows that, for receding configurations, inequalities (31) may
in general fail to hold, and moreover, the critical value βc1 (S (k) ) may lie to the left
of G−1
0 .
Influence of the Configuration of Particle Generation Sources on the Behavior of BRWs
401
Example 1 Assume that there is some BRW on Zd , d ≥ 2, for which G0 < ∞.
We claim that in this case there always exists a sequence of divergent 4-point
configurations S (k) such that
(k)
(k)
0 < βc0 (S (k) ) < βc1 (S (k) ) < G−1
0 < βc2 (S ) ≤ βc3 (S ).
(32)
Given some nonzero linearly independent vectors with integer coordinates u
and v, we define, for each k = 1, 2, . . . , a 4-point configuration S (k) of points
(k) (k) (k) (k)
x1 , x2 , x3 , x4 by
x1(k) = 0,
x2(k) = ku,
x3(k) = l (k) v,
x4(k) = l (k) v + ku,
where the integer quantities l (k) will be specified later.
It is clear that the sequence of configurations {S (k) } is receding as l (k) → ∞;
moreover, the points of each configuration S (k) form a parallelogram.1
To shorten notation, we set
(k)
(k)
(k)
(k)
(k)
(k)
(k)
(k)
(k)
(k)
(k)
(k)
(k)
(k)
(k)
(k)
a (k) = 0 (x1 , x2 ) = 0 (x2 , x1 ),
b(k) = 0 (x1 , x3 ) = 0 (x3 , x1 ),
c(k) = 0 (x1 , x4 ) = 0 (x4 , x1 ),
d (k) = 0 (x2 , x3 ) = 0 (x3 , x2 ).
By construction
x1(k) − x2(k) = x3(k) − x4(k)
x1(k) − x3(k) = x2(k) − x4(k),
and hence
(k)
(k)
(k)
(k)
(k)
(k)
(k)
(k)
0 (x3 , x4 ) = 0 (x4 , x3 ) = a (k),
0 (x2 , x4 ) = 0 (x1 , x3 ) = b(k).
Moreover,
a (k), b (k) , c(k) , d (k) > 0
1 The condition of linear independence of the vectors u and v can be replaced by the condition
that the quantities l (k) grow faster than k, for example, l (k) ≥ k 2 . In this case, the sequence of
configurations {S (k) } is also receding.
402
E. B. Yarovaya
In this case, the matrix (0, S (k) ) (see (14)) assumes the form
⎡
⎤
G0 a (k) b(k) c(k)
⎢ a (k) G0 d (k) b(k) ⎥
⎥
(0, S (k) ) = ⎢
⎣ b(k) d (k) G0 a (k) ⎦ .
c(k) b(k) a (k) G0
(33)
Now let us define the integer numbers l (k) . Note that under the condition G0 < ∞ all
integrals (21) are finite for λ = 0; i.e., the corresponding integrands are summable.
In this case, by assertion (ii) of Lemma 1,
a (k) → 0,
k → ∞,
and so,
b(k), c(k) , d (k) → 0,
l (k) → ∞, k → ∞.
Now, for each k, we choose a sufficiently large number l (k) so as to have
0 < b(k), c(k) , d (k) ≤ (a (k) )2 .
(34)
Consider the polynomial
⎤
b(k)
c(k)
G0 − γ a (k)
⎢ a (k) G0 − γ d (k)
b(k) ⎥
⎥.
p(k) (γ ) = det((0, S (k) ) − γ I ) = det ⎢
(k)
⎣ b(k)
d
G0 − γ a (k) ⎦
c(k)
b(k)
a (k) G0 − γ
(35)
⎡
(k)
The roots γi
follows:
of this polynomial are related to the critical values βci (S (k) ) as
βci (S (k) ) =
1
γi(k)
,
i = 0, 1, 2, 3;
hence, to prove (32) it suffices to show that
(k)
γ3
(k)
≤ γ2
(k)
< G0 < γ 1
(k)
< γ0 .
(36)
We represent the polynomial p(k) (γ ) in the form
p(k) (γ ) = (G0 − γ )4 + A(k)(G0 − γ )3 + B (k) (G0 − γ )2 + C (k) (G0 − γ ) + D (k) .
Influence of the Configuration of Particle Generation Sources on the Behavior of BRWs
403
The values of the coefficients A(k), B (k) , C (k) are immaterial for us, and the
coefficient D (k) can be easily explicitly evaluated:
D (k) = (a (k))4 − 2(a (k)b(k))2 + (b(k))4 − 2(a (k))2 c(k)d (k) − 2(b(k))2 c(k)d (k)
+ (c(k)d (k) )2 .
Hence by (34) we have
D (k) = (a (k) )4 + o (a (k))4 ,
and therefore,
D (k) > 0,
∀k.
(37)
But by (37) we have p(k) (G0 ) = D (k) > 0, and besides, p(k) (γ ) → ∞ as γ → ∞.
Moreover, by definition γ0(k) > G0 is the maximal eigenvalue of the positive matrix
(0, S (k) ), and hence, is a root of the characteristic polynomial p(k) (γ ). By the
Perron–Frobenius theorem, this eigenvalue (the root of the polynomial p(k) (γ )) is
simple. Hence the derivative of the polynomial p(k) (γ ) is strictly positive at the
(k)
(k)
point γ0 . This means that the polynomial p(k) (γ ) has a root γ1 which is strictly
greater than G0 and is strictly smaller than γ0(k) . A similar analysis shows that the
remaining two roots of the polynomial p(k) (γ ) (which can coalesce, because we do
(k)
not know whether the root γ3 < G0 of the polynomial p(k) (γ ) is simple or not)
lie to the left of G0 (Fig. 1).
Remark 4 Theorem 1 does make clear the behavior of the eigenvalues λi (β, S (k) ),
(k)
i = 1, . . . , N, of the operators Hβ for β ≤ G−1
0 and k → ∞.
(k)
Let β < G−1
0 . By definition of the critical values βci (S ), i = 0, . . . , N − 1,
(k)
each of the eigenvalues λi (β, S ), i = 0, . . . , N −1, qua a function of β, is defined
Fig. 1 Possible graphs of the polynomial p(k) (γ )
404
E. B. Yarovaya
on the interval [βci (S (k) ), ∞). In this case, by assertion (19) of Theorem 1, for the
chosen β < G−1
0 there exists k̃ = k̃(β) such that, for k ≥ k̃(β),
βci (S (k) ) > β,
i = 0, . . . , N − 1.
(k)
Therefore, for k ≥ k̃(β) and the given β < G−1
0 , none of the quantities λi (β, S ),
i = 1, . . . , N, is defined!
The situation is more involved in the case β = G−1
0 < ∞. By assertion (17), in
(k)
(k)
this case the quantities λ0 (β, S ) = λi (0, S ) are defined for all k, whereas the
quantities λN−1 (β, S (k) ) = λN−1 (0, S (k) ) are undefined for any k. As regards the
quantities λi (β, S (k) ) for 1 ≤ i ≤ N − 2, the analysis of BRW’s with simplicial
configurations [18, 21] and Example 1 show that in this case some of the quantities
λi (β, S (k) ), 1 ≤ i ≤ N −2, can be defined for β = G−1
0 , while some of them cannot
be defined. A further analysis is required for a more definite answer.
Acknowledgments This work was supported by the Russian Science Foundation under grant
no. 19-11-00290 and was performed at the Steklov Mathematical Institute of Russian Academy
of Sciences.
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Probab. Appl. 62(3), 413–431 (2018). https://doi.org/10.1137/S0040585X97T98871X
Lanchester Model with the Random
Coefficients
V. G. Zadorozhniy
Abstract A mathematical model of Lanchester’s combat operations is studied
in the form of a system of differential equations, whose coefficients are random
processes. It is assumed that the random coefficients are given by the characteristic
functional. The first and second moment functions of system solutions are found.
The problem is reduced to a deterministic system of differential equations with
ordinary and variational derivatives. Gaussian and uniformly distributed random
coefficients are considered.
Keywords Lanchester model · COVID-19 · Characteristic functional ·
Variational derivative · Moment functions
1 Introduction
Lancaster [1] proposed and investigated a mathematical model of fighting of two
opposing sides
dx
= a1 (t)x + a2 (t)y + f1 (t),
dt
(1)
dy
= a3 (t)x + a4 (t)y + f2 (t),
dt
(2)
where t is time; x(t) and y(t) are the sizes of armies at a given instant of time t; and
a1 , a2 , a3 , and a4 are special functions that characterize the rate of the change of
x(t) and y(t). For example, a1 , a4 characterize the rate of the change of x(t), y(t)
V. G. Zadorozhniy ()
Voronezh State University, Voronezh, Russia
e-mail: zador@amm.vsu.ru
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
A. N. Karapetyants et al. (eds.), Operator Theory and Harmonic Analysis,
Springer Proceedings in Mathematics & Statistics 358,
https://doi.org/10.1007/978-3-030-76829-4_22
407
408
V. G. Zadorozhniy
depending on the diseases and recovery of patients, it characterizes the effectiveness
of the actions of the side “y”, and f1 , f2 the rate of change in the size of armies
associated with disease, reinforcements, etc.
If a1 , a2 , a3 , and a4 are constants, then the solution of the system can be written
out explicitly and the process dynamics can be analyzed [2, 3].
The Lanchester model was generalized and studied for various tasks [4–13]. In
real conditions, random factors play an important role. If the coefficients a1 , a2 , a3 ,
and a4 are random processes, the model becomes more complicated. For random
disturbances generated by white noise, the system is written using the Ito (or
Stratonovich) integral, and implementations for x(t) and y(t) are constructed.
Equations for moment functions can be obtained from the system of equations but
an infinite number of connected differential equations [14, 15]. A different way of
finding the moment functions was suggested by J. Adomian [16]. For linear systems
of differential equations, V. G. Zadorozhniy [17, 18] developed a method for finding
the moment functions of the solution by means of the reduction the problem to a
deterministic problem with ordinary and variational derivatives.
We consider a mathematical model in the form
dx
= ε1 (t)x + ε4 (t)
dt
(3)
dy
= ε2 (t)x + ε3 (t)y
dt
(4)
x(0) = x0 , y(0) = y0 .
(5)
with initial conditions
Here ε1 , ε2 , ε3 , and ε4 are random processes defined by the characteristic functional
[17, p. 30], and x0 and y0 are random variables. Problem (3)–(5) is one of the models
of coronavirus infection (COVID-19), where “x” is the number of viruses and “y” is
the number of people without immunity. The problem of finding the mathematical
expectation and second moment functions of the solution of the problem (3)–(5) is
solved. The problem is reduced to a deterministic differential equation with ordinary
and variational derivatives.
2 Reduction to a Deterministic Problem
Let [t0 , t1 ] = T ⊂ R be the segment on which the problem is studied; L$1 (T ) be the
space of summable functions on the segment T with the norm v = T |v(t)|dt;
u(t) be a vector function with components u1 (t), u2 (t), u3 (t), and u4 (t); and the
Lanchester Model with the Random Coefficients
409
random processes ε1 , ε2 , ε3 , and ε4 be defined by the characteristic functional [17,
p. 30]:
⎡
⎛
ψ(u) = E ⎣exp ⎝i
4
⎞⎤
εj (s)uj (s)ds ⎠⎦ , uj ∈ L1 (T ), j = 1, 2, 3, 4.
T j =1
Here, E denotes the mathematical expectation by the distribution function random
processes ε1 , ε2 , ε3 , and ε4 and i = (−1)0,5 .
Recall the definition of the variational derivative. Let I : L1 → C be a functional
and h ∈ L1 (T ), v ∈ L1 (T ). Let
ϕ(s, v)h(s)ds + o(h),
I (v + h) − I (v) =
T
where the integral (Lebesgue) is interpreted as a bounded linear functional on L1 (T )
and o(h) denotes an infinitesimal of higher order with respect to h. In this case
ϕ : T ×L1 (T ) → C is called the variational (functional) derivative of the functional
(v)
I and is denoted by δI
δv(s) . The technique of variational differentiation is similar to
conventional differentiation that can be found in the [17].
Let
⎞
⎛
4
εj (s)uj (s)ds ⎠ , uj ∈ L1 (T ),
w(u) = exp ⎝i
T j =1
z1 (t, u) = E[x(t)w(u)], z2 (t, u) = E[y(t)w(u)].
Multiply Eqs. (3) and (4) by the w(u) and take the mathematical expectations. Then,
we obtain
dx
w(u) = E[ε1 (t)xw(u)] + E[ε4 (t)w(u)],
E
dt
dy
w(u) = −E[ε2 (t)xw(u)] + E[ε3 (t)yw(u)].
E
dt
This system of equations can be (formally) written as
∂z1
δz1
δψ
= −i
−i
∂t
δu1 (t)
δu4 (t)
(6)
δz1
δz2
∂z2
=i
−i
.
∂t
δu2 (t)
δu3 (t)
(7)
410
V. G. Zadorozhniy
We assume that random variables x0 andy0 do not depend on the ε1 , ε2 , ε3 , and ε4 .
Multiply equality (5) by w(u) and take the mathematical expectations. Then, we
obtain E[x(0)w(u)] = E[x0 w(u)], E[y(0)w(u)] = E[y0w(u)]. These equalities
can be written as
z1 (0, u) = E[x0 ]ψ(u), z2 (0, u) = E[y0 ]ψ(u).
(8)
Thus, we arrive at the linear deterministic problem (6)–(8) in the form of a system
of differential equations with ordinary and variational derivatives. Since z1 (t, 0) =
E[x(t)], z2 (t, 0) = E[y(t)], it is natural to give the following definition:
Definition 1 The mathematical expectation of the solution of the Cauchy problem
(3)–(5) is called E[x(t)] = z1 (t, 0), E[y(t)] = z2 (t, 0), where z1 andz2 are the
solutions of problem (6)–(8) in some neighborhood of the point u = 0.
3 Solving the Equation with Ordinary and Variational
Derivatives
Let the function χ of the variable s be defined by the rule χ(s) = χ(t0 , t, s) =
sign(s − t0 ), where s ∈ [min{t0 , t}, max{t0 , t}] and χ(s) = 0 for s ∈
/
[min{t0 , t}, max{t0 , t}], and let g : L1 (T ) → C.
Theorem 1 Let a be a continuous function, and there exist variational derivative
δg(v)
dg(v+aχ(t0,t ))
and
δv(t ) and then the derivative
dt
δg(v + aχ(t0 , t))
dg(v + aχ(t0 , t))
= a(t)
.
dt
δv(t)
Proof Let Δt be the increment of the variable t; then,
g(v + aχ(t0 , t + Δt)) − g(v + aχ(t0 , t))
Δt
=
1
Δt
δg(v + aχ(t0 , t))
a(s)(χ(t0 , t + Δt, s) − χ(t0 , t, s))ds
δv(s)
T
+
=
1
Δt
o(χ(t0 , t + Δt, s) − χ(t0 , t, s))
Δt
o(χ(t, t + Δt, s))
δg(v + aχ(t0 , t))
a(s)(χ(t, t + Δt, s))ds +
.
δv(s)
Δt
T
(9)
Lanchester Model with the Random Coefficients
411
Note that
χ(t, t + Δt) =
T
t +Δt
|a(s)χ(t, t + Δt, s)|ds = |a(s)|ds = O(Δt)
t
and
1
Δt
t +Δt
t
δg(v + aχ(t0 , t))
δg(v + aχ(t0 , t))
a(s)ds → a(t)
δv(s)
δv(t)
when Δt → 0. Passing to the limit as Δt → 0, we prove the existence of the
0,t ))
derivative dg(v+aχ(t
and formula (9).
dt
We consider the problem:
δy
∂y
= a(t)
+ b(t, v)
∂t
δv(t)
(10)
y(t0 ) = y0 (v)
(11)
Here, y : T × L1 (T ) → C is to be determined mapping, b : T × L1 (T ) → C is the
given mapping, and y0 : L1 (T ) → C is given.
Theorem 2 Let a be a continuous function, and let the variational derivatives
δy0 (v)
δb(t,v)
δv(t ) and δv(t ) exist; then, the function
y(t, v) = y0 (v + aχ(t0 , t)) +
t
b(s, v + aχ(s, t))ds
(12)
t0
is the solution of problem (10).
Proof Using Theorem 1, we find
∂y
δy0 (v + aχ(t0 , t))
= a(t)
+ b(t, v + χ(t, t)) +
∂t
δv(t)
δy0 (v + aχ(t0 , t))
δy(v)
=
+
δv(t)
δv(t)
t
a(t)
t0
δb(s, v + aχ(s, t))
ds,
δv(t)
t
δb(s, v + aχ(s, t))
ds.
δv(t)
t0
Substituting these expressions to (10) shows that y(t, u) is the solution of equation
(10). Further, y(t0 , v) = y0 (v + aχ(t0 , t0 )) = y0 (v), and therefore, the initial
condition is satisfied.
412
V. G. Zadorozhniy
4 Solving Problem (6)–(8)
Theorem 3 Let the characteristic functional ψ have the second variational derivatives; then,
t
δψ(u1 − iχ(s, t), u2 , u3 , u4 )
ds
δu4 (s)
0
(13)
z2 (t, u) = E[y0 ]ψ(u1 , u2 , u3 − iχ(0, t), u4 )
z1 (t, u) = E[x0 ]ψ(u1 −iχ(0, t), u2 , u3 , u4 )−i
+iE[x0]
+
t s
0
0
t
δψ(u1 − iχ(0, s), u2 , u3 − iχ(s, t), u4 )
ds+
δu2 (s)
0
δ 2 ψ(u1 − iχ(τ, t), u2 , u3 − iχ(s, t), u4 )
dτ ds
δu4 (τ )δu2 (s)
(14)
is the solution of problem (6)–(8).
Proof Obviously, initial conditions (8) are satisfied. Equation (6) looks like equation (11), and using formula (12), we find (13). Substituting (13) in Eq. (7), we
obtain an equation of the form (11). Using formula (12), we obtain (14).
Formulas (13) and (14) are fairly general.
Theorem 4 Let, in problem (3)–(5), the random processes ε1 , ε2 , ε3 , and ε4 be
independent and their characteristic functionals
ψj (uj ) = E exp i
εj (s)uj (s)ds , j = 1, 2, 3, 4,
T
be known. Then, the solution of problem (6)–(8) has the form
z1 (t, u) = E[x0 ]ψ1 (u1 − iχ(0, t))ψ2 (u2 )ψ3 (u3 )ψ4 (u4 )
−i
t
ψ1 (u1 − iχ(s, t))ψ2 (u2 )ψ3 (u3 )
0
δψ4 (u4 )
ds
δu4 (s)
(15)
z2 (t, u) = E[y0 ]ψ1 (u1 )ψ2 (u2 )ψ3 (u3 − iχ(0, t))ψ4 (u4 )
+i
t
ψ1 (u1 − iχ(0, s))
0
+
t s
0
0
ψ1 (u1 − iχ(τ, t))
δψ2 (u2 )
ψ3 (u3 − iχ(s, t))ψ4 (u4 )ds
δu2 (s)
δψ2 (u2 )
δψ4 (u4 )
ψ3 (u3 − iχ(s, t))
dτ ds
δu2 (s)
δu4 (τ )
(16)
Lanchester Model with the Random Coefficients
413
Proof Since random processes ε1 , ε2 , ε3 , and ε4 are independent, we have ψ(u) =
ψ1 (u1 )ψ2 (u2 )ψ3 (u3 )ψ4 (u4 ). Substituting this expression in (13) and (14), we
obtain (15) and (16).
5 Mathematical Expectation of the Solution of Problem
(3)–(5)
Theorem 5 Let the characteristic functional ψ have the second variational derivatives; then,
E[x(t)] = E[x0]ψ(−iχ(0, t), 0, 0, 0) − i
t
E[y(t)] = E[y0 ]ψ(0, 0, −iχ(0, t), 0) + iE[x0 ]
+
t s
0
0
δψ(−iχ(s, t), 0, 0, 0)
ds
δu4 (s)
0
(17)
t
δψ(−iχ(0, s), 0, −iχ(s, t), 0)
ds
δu2 (s)
0
δ 2 ψ(−iχ(τ, t), 0, −iχ(s, t), 0)
dτ ds
δu4 (τ )δu2 (s)
(18)
is the mathematical expectation of the solution of problem (3)–(5).
Proof According to Definition 1, we have E[x(t)] = z1 (t, 0), E[y(t)] = z2 (t, 0).
Substituting u = 0 in (15) and (16), we obtain (17) and (18).
For independent random processes ε1 , ε2 , ε3 , and ε4 , formulas (17) and (18) can be
simplified.
Theorem 6 Let the conditions of Theorem 4 be fulfilled; then, the mathematical
expectation of the solution of problem (3)–(5) has the form
E[x(t)] = E[x0 ]ψ1 (−iχ(0, t)) +
t
ψ1 (−iχ(s, t))E[ε4 (s)]ds,
(19)
0
E[y(t)] = E[y0 ]ψ3 (−iχ(0, t)) −
t
ψ1 (−iχ(0, s))ψ3 (−iχ(s, t))E[ε2 (s)]ds
0
−
t s
ψ1 (−iχ(τ, t))ψ3 (−iχ(s, t))E[ε2 (s)]E[ε4(τ )]dτ ds
0
(20)
0
Proof From the definition of the characteristic functional, we can easily find
δψ (0)
ψj (0) = 1, δujj (t ) = iE[εj (t)]. Setting in (15) and (16) u1 = u2 = u3 = u4 = 0,
we obtain (19) and (20).
Gaussian random processes are the most common.
414
V. G. Zadorozhniy
Theorem 7 Let, in problem (3)–(5), the processes εj , j = 1, 2, 3, 4, be independent and be Gaussian with the characteristic functionals
1
ψj (uj ) = exp i
aj (s)uj (s)ds −
bj (s1 , s2 )uj (s1 )uj (s2 )ds1 ds2 ,
2 T T
T
where aj (t) = E[εj (t)], bj (s1 , s2 ) = E[εj (s1 )εj (s2 )] − E[εj (s1 )]E[εj (s2 )]; then,
the mathematical expectation of the solution of problem (3)–(5) has the form
E[x(t)] = E[x0 ] exp
+
t
t
exp
0
1
2
s
s
bj (s1 , s2 )ds1 ds2
0
1
2
t
t s
0
t t
b3 (s1 , s2 )ds1 ds2
0
0
s s
b1 (s1 , s2 )ds1 ds2
0
0
t t
b3 (s1 , s2 )ds1 ds2 )E[ε2 (s)]ds
s
s
τ
1
+
2
t
a3 (ξ )dξ +
s
1
2
t t
b1 (s1 , s2 )ds1 ds2
τ
τ
t t
b3 (s1 , s2 )ds1 ds2 E[ε2(s)]E[ε4 (τ )]dτ ds.
s
(21)
s
1
a3 (τ )dτ +
2
0
a1 (ξ )dξ +
exp
0
b1 (s1 , s2 )ds1 ds2 E[ε4(s)]ds
t
+
t t
1
a1 (τ )dτ +
a3 (τ )dτ +
2
0
s
exp
0
t t
t
s
t
0
−
1
a1 (s)ds +
2
0
a1 (τ )dτ +
E[y(t)] = E[y0 ] exp
−
t
(22)
s
Proof Let us make use of formulas (19) and (20). At the same time, by the
properties of the χ function, we have
a1 (s)(−iχ(0, t, s))ds =
i
1
2
a1 (s)ds,
0
T
−
t
b1 (s1 , s2 )(−iχ(0, t, s1 ))(−iχ(0, t, s2 ))ds1 ds2
T
T
1
=
2
t t
b1 (s1 , s2 )ds1 ds2 .
0
0
Other expressions are calculated similarly and get (21) and (22).
Lanchester Model with the Random Coefficients
415
Remark The mathematical expectation of the solution depends on the mathematical
expectation of E[ε2(s)] of a random coefficient ε2 and does not depend on moment
functions of higher orders of this coefficient.
For the “y” side, the coefficient ε2 is important, which characterizes the
effectiveness of the side’s “x” actions. Its values can be uniformly distributed at any
given time. The characteristic functional of a uniformly distributed random process
has the form
$
sin a(s)u2 (s)ds
ψ2 (u2 ) = $ T
E[ε2 (s)]u2 (s)ds , (ψ2 (0) = 1),
(23)
exp i
T
T a(s)u2 (s)ds
where a(t) ≥ 0 is a given function.
Theorem 8 Let the random processes ε1 , ε2 , ε3 , and ε4 be independent; ε1 , ε3 , and
ε4 be Gaussian processes defined by the characteristic functionals (19); and ε2 be a
uniformly distributed random process with characteristic functionals (23); then, the
solution of problem (6)–(8) has the form
z1 (t, u) = E[x0 ]ψ1 (u1 − iχ(0, t))ψ2 (u2 )ψ3 (u3 )ψ4 (u4 )
−i
t
ψ1 (u1 − iχ(s, t))ψ2 (u2 )ψ3 (u3 )ψ4 (u4 )(ia4 (s) −
0
b4 (s, s2 )u4 (s2 )ds2 )ds,
T
z2 (t, u) = E[y0 ]ψ1 (u1 )ψ2 (u2 )ψ3 (u3 − iχ(0, t))ψ4 (u4 )
+i
t
ψ1 (u1 − iχ(0, s)) exp i
E[ε2 (τ )]u2 (τ )dτ
0
T
$
$
a(s)( T a(τ )u2 (τ )dτ cos T a(τ )u2 (τ )dτ − sin T a(τ )u2(τ )dτ )
$
×
( T a(τ )u2 (τ )dτ )2
$
+iψ2 (u2 )E[ε2 (s)]] ψ3 (u3 − iχ(s, t))ψ4 (u4 )ds
+
t s
0
ψ1 (u1 − iχ(τ, t)) exp i
E[ε2 (τ )]u2 (τ )dτ
0
T
$
$
$
a(s)( T a(τ )u2 (τ )dτ cos T a(τ )u2 (τ )dτ − sin T a(τ )u2(τ )dτ )
$
×
( T a(τ )u2 (τ )dτ )2
×iψ2 (u2 )E[ε2 (s)]] ψ3 (u3 − iχ(s, t))ψ4 (u4 )(ia4(τ )
− b4 (τ, s2 )u4 (s2 )ds2 )dτ ds
T
416
V. G. Zadorozhniy
Proof Let us make use of formulas (15) and (16). To do this, we need to calculate
the variational derivatives of the functionals ψ2 and ψ4 :
δψ4 (u4 )
= ia4 (s) −
b4 (s, s2 )u4 (s2 )ds2 ,
δu4 (s)
T
$
$
$
a(s)( T a(τ )u2 (τ )dτ cos T a(τ )u2 (τ )dτ − sin T a(τ )u2(τ )dτ )
δψ2 (u2 )
$
=
×
δu2 (s)
( T a(τ )u2 (τ )dτ )2
× exp i
E[ε2 (τ )]u2 (τ )dτ + iψ2 (u2 )E[ε2 (s)]
T
Substituting these expressions in formulas (15) and (16), we obtain the desired
expressions for z1 and z2 .
Comment Under the assumptions of Theorem 8, the mathematical expectation for
the solution of problem (3)–(5) has the form (21) and (22).
For the proof, it is sufficient u1 = u2 = u3 = u4 = 0 in (15) and (16) to put.
6 Second Mixed Moment Functions
Alongside with the mathematical expectation for the solution of problem (3)–(5),
the second moment functions play an important role. Using the definition of the
mappings z1 and z2 , we find
δw(u)
δz1 (t, u)
|u=0 = E x(t)
|u=0 = E[x(t)w(u)iεj (τ )]|u=0 = iE[x(t)εj (τ )]
δuj (τ )
δuj (τ )
and similarly
δz2 (t, u)
|u=0 = iE[y(t)εj (τ )].
δuj (τ )
Thus, the second mixed moment functions are found by variational differentiation.
Let us write out some second mixed moment functions.
Theorem 9 If the characteristic functional ψ has all the third-order variational
derivatives, then
t
δ 2 ψ(−iχ(s, t), 0, 0, 0)
ds,
δu1 (τ )δu4 (s)
0
(24)
δψ(0, 0, −iχ(0, t), 0)
E[y(t)ε2 (τ )] = −iE[y0]
+
δu2 (τ )
E[x(t)ε1 (τ )] = −iE[x0]
δψ(−iχ(0, t), 0, 0, 0)
−
δu1 (τ )
Lanchester Model with the Random Coefficients
+E[x0]
−i
417
t
δ 2 ψ(−iχ(0, s), 0, −iχ(s, t), 0)
ds−
δu2 (s)δu2 (τ )
0
t s
0
δ 3 ψ(−iχ(ξ, t), 0 − iχ(s, t), 0)
dξ ds
δu4 (ξ )δu2 (s)δu2 (τ )
0
(25)
Proof The proof is carried out by variational differentiation of z1 and z2 .
We consider some special cases. Let all random processes ε1 , ε2 , ε3 , ε4 be
independent and be Gaussian with characteristic functionals (19); then, using
formulas (24) and (25), we obtain
E[x(t)ε1 (τ )] = E[x0] exp
+
t
1
a1 (s)ds +
2
0
t
t
exp
0
1
a1 (ξ )dξ +
2
s
E[y(t)ε2 (τ )] = E[y0] exp
s
t
exp
0
b1 (s1 , s2 )ds1 ds2 E[ε1 (τ )]E[ε4(s)]ds,
s
s
1
2
a3 (s)ds +
a1 (ξ )dξ +
0
1
+
2
+
t s
t
exp(
0
0
1
2
s s
0
t t
0
b3 (s1 , s2 )ds1 ds2 E[ε2 (τ )]
0
b1 (s1 , s2 )ds1 ds2 +
t
0
a3 (ξ )dξ +
s
t t
b3 (s1 , s2 )ds1 ds2 )b2 (s, τ )ds
s
s
a1 (η)dη +
ξ
1
+
2
0
0
+E[x0 ]
b1 (s1 , s2 )ds1 ds2 E[ε4 (τ )]
0
t t
t
t t
1
2
t t
ξ
ξ
b1 (s1 , s2 )ds1 ds2 +
t
a3 (η)dη+
s
t t
b3 (s1 , s2 )ds1 ds2 )b2 (s, τ )a4 (ξ )dξ ds.
s
s
7 Conclusion
When conducting military operations, there are always present random factors. To
reduce the dependence of accidents, they use intelligence or measures in order to
disinform the enemy, etc. Mathematical modeling is used to assess the prospects
of combat operations. The model with random coefficients describes the process
of interaction between the parties more adequately. The considered mathematical
model describes the spread of viral infection (COVID-19). The method proposed in
the article allows us to find explicit expressions for the mathematical expectation
418
V. G. Zadorozhniy
of changes in the number of opposing parties. The results are fairly general, as
long as the characteristic functional of random coefficients is known. In the process
of solving the problem, we first find z1 (t, u) and z2 (t, u). It turns out that they
not only allow us to find the mathematical expectations of solving of the problem
but also help us obtain the second mixed functions of the solutions. If Gaussian or
evenly distributed processes are used in the simulation, then one must first define
two functions, the mathematical expectations and covariance functions, for random
coefficients.
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