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Electr
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Vol. 13 (2008), Paper no. 61, pages 1886–1908.
Journal URL
http://www.math.washington.edu/~ejpecp/
Comparison Results for Reflected Jump-diffusions in the
Orthant with Variable Reflection Directions and Stability
Applications
Francisco J. Piera∗
Ravi R. Mazumdar†
Department of Electrical Engineering
University of Chile
Av. Tupper 2007
Santiago, 8370451, Chile
fpiera@ing.uchile.cl
Department of Electrical
and Computer Engineering
University of Waterloo
Waterloo, ON N2L3G1, Canada
mazum@ece.uwaterloo.ca
Abstract
n
We consider reflected jump-diffusions in the orthant R+
with time- and state-dependent drift, diffusion and jump-amplitude coefficients. Directions of reflection upon hitting boundary faces are
also allow to depend on time and state. Pathwise comparison results for this class of processes
are provided, as well as absolute continuity properties for their associated regulator processes
responsible of keeping the respective diffusions in the orthant. An important role is played by
the boundary property in that regulators do not charge times spent by the reflected diffusion
at the intersection of two or more boundary faces. The comparison results are then applied to
provide an ergodicity condition for the state-dependent reflection directions case.
Key words: Jump-diffusion processes;pathwise comparisons;state-dependent oblique reflection;Skorokhod maps;stability; ergodicity .
∗
Research supported in part by VID, U. of Chile, Chile, DI Project REIN 06/05, by CONICYT, Chile, FONDECYT Project
1070797, and by the Millennium Science Nucleus on Information and Randomness, Dept. of Mathematical Engineering,
U. of Chile, Chile, Program P04-069-F.
†
Research supported in part by NSERC, Canada, through the Discovery Grant Program.
1886
AMS 2000 Subject Classification: Primary 60J60, 60J75; Secondary: 60G17, 60J50, 60K25,
34D20.
Submitted to EJP on September 17, 2007, final version accepted September 30, 2008.
1887
1 Introduction
Reflecting stochastic differential equations have found a wide variety of applications over the last
decades. They play an increasingly important role in several disciplines such as economics and
operations research, where they serve to model portfolio and consumption processes, option pricing
and subsidy phenomena in interdependent economies, among others (see for example [10; 15; 21;
16] and references therein). They also play a central role in several fields of applications in the
electrical engineering context, where they are used to model, in conjunction with weak convergence
methods, from systems such as adaptive antenna arrays to stochastic communication networks (see
for example [12; 13] and references therein).
In the context of stochastic networks, such models appear as heavy-traffic limits of complex network
models, otherwise difficult to analyze, giving rise to corresponding approximations in terms of reflected diffusions. Reflections are taken into account via the Skorokhod map, and are due to the
non-negativeness requirements for the buffer occupation processes in the network (see for example [26; 4] and references therein). Reflected jump-diffusions appear in this queueing application
context when for example network stations are subject to service interruptions (see [11; 25] and
references therein). In the same way, time- and state-dependence in the corresponding drift and
diffusion coefficients, as well as directions of reflection, obey to the corresponding dependence in
network traffic parameters, such as arrival and service rates, and station-to-station routing probabilities (see for example [13; 14] and references therein).
Due to the wide variety of applications of reflected diffusion models, as summarized above, the availability of comparison properties for such class of models have become of practical and theoretical
importance. For example, the pathwise comparison between buffer occupation processes in queueing networks or cumulative subsidies transferred among entities in interdependent economies, both
considered as the respective model parameters change, are of self-explicatory importance. Such
pathwise results in general demand not only the comparison between the respective constrained
jump-diffusion processes (constructed in terms of the Skorokhod map, as detailed further in the
next section), but also the corresponding establishment of absolute continuity properties for the
associated regulator processes (constraining the diffusions to the domain of interest, as for example
the non-negative orthant). In this context, comparison results for reflected jump-diffusions in the
orthant have traditionally been restricted to the case of normal reflection directions upon hitting
boundary faces (see [23]). However a comparison result in the oblique reflection directions case is
available in the context of the deterministic Skorokhod problem in the orthant (see [21]), the framework in which it is established makes its application to the diffusion setting only possible when the
stochastic integral term driving the respective diffusion is state-independent. We will show in the
paper that this requirement is essentially a boundary condition at the faces of the orthant, being able
to include then a controlled state-dependency in that term over the interior of the orthant covering
for example the important case of a product-form setting (see [17; 19]) in queueing network applications. A crucial role is played here by an appropriate boundary behavior characterization, and
in particular by the boundary property in that regulators do not charge times spent by the reflected
diffusion at the intersection of two or more boundary faces (see [18; 19]).
A direct application of the comparison results established in the paper is to provide a simple ergodicity condition for continuous reflected diffusions in the orthant with state-dependent reflection
directions. Stability conditions available in the literature have been established for the constant
reflection directions case (see [1]) and critically depend on the Lipschitz continuity of the corre1888
sponding Skorokhod map, which is not ensured in the state-dependent case.
The organization of the paper is as follows. In Section 2 we specify the setting to be considered
throughout as well as some related notation. In Section 3 we establish the main comparison results
of the paper. In Section 4 we apply those results to derive an ergodicity condition for the continuous
case. Finally, in Section 5 we provide the corresponding proofs of the main results in Section 3.
2 Setting and Notation
Let n ≥ 2 be an integer1 and (Ω, F , (F t ) t≥0 , P) be a stochastic basis satisfying the usual hypotheses,
i.e., F0 contains all the P-null sets of F and the filtration (F t ) t≥0 is right continuous. Throughout
the paper we consider pair of processes (X , Z) satisfying reflecting stochastic differential equations
n
(RSDEs) in the orthant R+
= {x = (x i )ni=1 ∈ Rn : x i ≥ 0 ∀i} = ×ni=1 R+ , with state- and timedependent reflection directions upon hitting boundary faces, of the form
X t = X0 +
Z
t
b(s, X s− )ds +
0
Z
t
γ(s, X s− )dWs
0
+
Z tZ
0
δ(s, X s− , r)N (ds, d r) +
E
Z
t
R(s, X s− )d Zs , t ≥ 0, (2.1)
0
where2
n
• X = (X t ) t≥0 = (X t1 , . . . , X tn ) t≥0 is an (F t ) t≥0 -adapted R+
-valued c àd l à g 3 semi-martingale.
n
n
• b = (bi )ni=1 : R+ × R+
→ Rn and4 γ = (γi j )ni, j=1 : R+ × R+
→ Rn×n are Borel-measurable
.
functions. As usual, we refer to b and a = (ai j )ni, j=1 = γγ T as the drift vector and diffusion
matrix, respectively.
• W = (Wt ) t≥0 = (Wt1 , . . . , Wtn ) t≥0 is an (F t ) t≥0 -standard Brownian motion on Rn .
.
• E = E1 × · · · × En with each Ei being an arbitrary Polish space (e.g., R with the usual Eun
clidian distance), δ = (δi j )ni, j=1 : R+ × R+
× E → Rn×n is a Borel-measurable function and
.
N (ds, d r) = (N1 (ds, d r1 ), · · · , Nn (ds, d rn )) with each Ni (ds, d ri ) being an independent Poisson random measure over [0, ∞) × Ei with intensity measure λi ds ⊗ Gi (d ri ), where λi ≥ 0
and Gi is a probability distribution on (Ei , B(Ei )). Note since the Poisson measures are independent, any number of them do not “jump” simultaneously at any time (a. s.), and therefore
to ensure that a jump does not take X outside the orthant we ask for each δi j to be such that
n
δi j (t, x, r j ) ≥ −x i for all t ≥ 0, x = (x l )nl=1 ∈ R+
and r j ∈ E j .
1
n
-valued process with each
• Z = (Z t ) t≥0 = (Z t1 , . . . , Z tn ) t≥0 is a continuous (F t ) t≥0 -adapted R+
R∞
i
i
i
i
Z being non-decreasing and such that Z0 = 0 and 0 X s d Zs = 0.
All results in the paper apply to the one-dimensional setting as well (n = 1). However, in order not to include trivial
cases in our proofs, we simply consider n ≥ 2.
2
Throughout, (in)equalities involving vectors and matrices are understood to hold componentwise and elementwise
respectively, and vectors are envisioned as column vectors.
3
Acronym in French standing for continuous from the right with limits from the left.
4
We denote as Rn×n the collection of n × n real matrices.
1889
n
• R = (R i j )ni, j=1 : R+ × R+
→ Rn×n is a Borel-measurable function. As usual, and since the j-th
.
column of R gives the reflection direction upon hitting the interior of the j-th face F j = {x =
n
: x j = 0}, we refer to R as the reflection matrix5 and assume, without loss of
(x l )nl=1 ∈ R+
generality, the normalization R ii (·, ·) ≡ 1 for each i.
We now identify different sets of conditions that will be alternatively considered as assumptions in
the results given in the paper.
Condition 2.1. b and R are continuous. Also, b, γ, δ and R satisfy a linear growth condition and are
n
Lipschitz continuous, both in the state variable x ∈ R+
and uniformly in all the other corresponding
n
variables, i.e., there exists K ∈ (0, ∞) such that, for all x, y ∈ R+
, t ≥ 0 and r ∈ E,
kb(t, x)k2 + kγ(t, x)k2 + kδ(t, x, r)k2 + kR(t, x)k2 ≤ K 2 (1 + kxk2 )
and
kb(t, x) − b(t, y)k + kγ(t, x) − γ(t, y)k
+ kδ(t, x, r) − δ(t, y, r)k + kR(t, x) − R(t, y)k ≤ Kkx − yk,
with the usual Euclidian and Frobenius norms in Rn and Rn×n , respectively. Moreover6 ,
Z
X
sup
λj
δ2i j (t, x, r j )G j (d r j ) < ∞
n
x∈R+
, t≥0
and
X
j
(2.2)
Ej
j
λj
Z
δi j (·, ·, r j )G j (d r j )
Ej
is continuous, for each i.
Condition 2.2. For each i, j, i 6= j, there exists mi j ≥ 0 such that
n
|R i j (t, x)| ≤ mi j , x ∈ R+
, t ≥ 0,
.
.
and, with mii = 0 for each i and M = (mi j )ni, j=1 ∈ Rn×n , we have σ(M ) < 1 with σ(M ) denoting the
spectral radius of M .
Conditions 2.1 and 2.2 in particular guarantee that, given an F0 -measurable initial condition X 0 ∈
n
and b, γ, W , δ, N and R as above, the pair (X , Z) is the pathwise unique strong solution of
R+
RSDE (2.1). Indeed, this follows from [6] in the continuous case (i.e., in absence of jumps), jumps
being taken then into account via standard piecewise construction arguments (see for example [13,
Section 3.7, pp. 134]). Whenever that is the case, we write
(X , Z) = RSDE(X 0 , b, R),
omitting γ, W , δ and N in the notation since in all comparison results between pairs
e ) = RSDE(Xe0 , eb, R
e)
(X , Z) = RSDE(X 0 , b, R) and (Xe , Z
they will be the same for both7 .
.
One in general may assume each R i j as given on F j and extended to the whole orthant by setting R i j (·, x) =
n
π F j (R i j (·, x)), x ∈ R+ \ F j , with π F j the orthogonal projector onto F j .
P
.
6
i
Note from (2.2) in particular follows that 0<s≤t |∆X si | < ∞ a. s., each i and t > 0, with ∆X si = X si − X s−
and
.
i
X s−
= limuցs X ui , s > 0.
7
In some of the corresponding proofs we will find useful to emphasize the common γ though, including it explicitly in
the notation then.
5
1890
Conditions 2.1 and 2.2 in fact guarantee the well posedness of the Skorokhod problem (SP) in
the orthant with state-dependent reflection directions (see [24] for a detailed treatment of the
(modified) SP in the orthant with state-dependent reflection directions), and one may then write
X = Φ(U)
with
.
U· = X 0 +
Z
·
b(s, X s− )ds +
0
Z
·
γ(s, X s− )dWs +
0
Z ·Z
0
δ(s, X s− , r)N (ds, d r)
E
n
and Φ : D([0, ∞) : Rn ) → D([0, ∞) : R+
) the Skorokhod map8 , i.e., with (X , Z) solving the SP for
9
U and R on an a. s. pathwise basis . D([0, ∞) : G) denotes here, as usual, the space of c àd l àg
functions mapping [0, ∞) into G ⊆ Rn .
Condition 2.2 is standard in the context of RSDEs and SPs in non-smooth domains (see for example
n
[6; 5; 21; 8]) as it guarantees that R(t, x) is completely-S, for each x ∈ R+
and t ≥ 0, in that for each
principal sub-matrix R(t, x) extracted from R(t, x) there always exists a non-negative vector v, of
the corresponding proper dimension, such that R(t, x)v > 0. Each such R(t, x) is also non-singular.
This structure, along with Condition 2.1 above and Condition 2.5 below, in particular guarantee that
(see [18; 19]) for each A ⊆ {1, . . . , n} with cardinality |A | ≥ 2
Z∞
1{X j =0, ∀ j∈A } d Zsi = 0 a. s.
0
s
(2.3)
for each i ∈ A , with 1{·} denoting as usual the corresponding indicator function.
As it will be seen in Section 5, the boundary property in equation (2.3) plays an important role in
the establishment of the results in the paper.
Finally, we identify the following conditions on the coefficients of δ and γ and on the diffusion
matrix a.
n
n
Condition 2.3. Each δi j is such that, whenever x = (x l )nl=1 ∈ R+
and y = ( yl )nl=1 ∈ R+
with x i ≤ yi ,
x i + δi j (t, x, r j ) ≤ yi + δi j (t, y, r j ), r j ∈ E j , t ≥ 0.
Condition 2.4. There exist measurable functions {ηi j }ni, j=1 , mapping R2+ into R, such that for each i, j
n
γi j (t, x) = ηi j (t, x i ), x = (x l )nl=1 ∈ R+
, t ≥ 0.
n
Condition 2.5. The diffusion matrix a is positive definite for each x ∈ R+
and t ≥ 0, i.e.,
X
ai j (t, x)ξi ξ j > 0, ξ = (ξl )nl=1 ∈ Rn , ξ 6= 0.
i, j
8
The map U → (X , Z) is generally known as the solution mapping of the SP (for a given R) or, in queueing theory
jargon, as the reflection map (see [25; 26]). In this same jargon, the Z component is usually referred to as the regulator
process.
9
n
Note the continuity of Z is ensured from the facts that X 0 ∈ R+
and that jumps cannot take X outside the orthant.
1891
As mentioned before, since the Poisson jump measures {Ni (ds, d ri )}ni=1 are independent, any number of them do not “jump” simultaneously at any time (a. s.). Therefore, Condition 2.3 is useful
when comparing processes X and Xe , coming from
e ) = RSDE(Xe0 , eb, R
e),
(X , Z) = RSDE(X 0 , b, R) and (Xe , Z
as it ensures that for each i
i
(X − Xe )si (X − Xe )s−
≥0
at any jump instant (i.e., that jumps cannot alter the order between corresponding components).
In this same comparison context, Condition 2.4 will guarantee for the semi-martingale local time at
level zero associated with each difference (X − Xe )i to be null. It also makes each γi j independent
of the position over the corresponding i-th face Fi , and hence each diagonal diffusion coefficient aii
too. Condition 2.4 is required for comparisons even in the case of normally reflected jump-diffusions
in the orthant (see [23]), and it encompasses the important case of a product-form setting (see
[17; 19]) in queueing network applications.
In addition to play a role in the establishment of relationship (2.3) above, Condition 2.5 also guarantees for reflections from the boundary to be instantaneous (see [18; 19]).
3 Main Results: Comparison Properties
We establish in this section the main results of the paper, regarding comparison properties between
different pairs
e ) = RSDE(Xe0 , eb, R
e).
(X , Z) = RSDE(X 0 , b, R) and (Xe , Z
The following additional notation will be used from now on in the paper, with I denoting the
identity matrix in Rn×n .
Notation. Consider
e ) = RSDE(Xe0 , eb, R
e)
(X , Z) = RSDE(X 0 , b, R) and (Xe , Z
with relationship (2.2) in Condition 2.1 holding, respectively, and define the mapping ψ =
n
e by setting each ψi (resp., ψ
ei ) as the net-drift includ→ Rn (resp., ψ)
(ψ1 , . . . , ψn ) : [0, ∞) × R+
ing jumps in the i-th coordinate, i.e.,
Z
X
.
n
ψi (t, x) = bi (t, x) +
λj
δi j (t, x, r j )G j (d r j ), x ∈ R+
, t ≥ 0,
j
Ej
e with eb in place of b. We write
and similarly for ψ
whenever10
e R
e)
(X 0 , ψ, R) ¹ ( Xe0 , ψ,
e y) and R(t, x) ≤ R
e(t, y)
X 0 ≤ Xe0 a. s., ψ(t, x) ≤ ψ(t,
10
Recall that inequalities among vectors and matrices are understood to hold componentwise and elementwise, respectively. Since we take any reflection matrix as to have normalized to one diagonal elements, inequalities among them
involve then off-diagonal elements only.
1892
n
e(·, ·), then we write
with x ≤ y and t ≥ 0. If in addition R(·, ·) ≤ I ≤ R
as x, y ∈ R+
e R
e).
(X 0 , ψ, R) ¹ I (Xe0 , ψ,
Finally, we write
e << d Z
dZ
e i induces in [0, ∞) is absolutely continuous with respect to the correwhen the random measure each Z
·
sponding one associated to Z·i , denote by
ei
dZ
dZi
the related Radon-Nikodym derivatives, and write
e
dZ
dZ
≤ 1 a. s.
when each Radon-Nikodym derivative above is less than or equal to 1 a. s.
In order not to opaque the continuity in the exposition of the results, we postpone their corresponde(·, ·) ≡ I.
ing proofs to Section 5. We begin with the case when R
Theorem 3.1. Let
e ) = RSDE(Xe0 , eb, I),
(X , Z) = RSDE(X 0 , b, R) and (Xe , Z
both under Conditions 2.1 to 2.4, respectively. Assume that
Then we have
e I).
(X 0 , ψ, R) ¹ ( Xe0 , ψ,
e
¦
©
dZ
e << d Z and
P X t ≤ Xet , t ≥ 0 = 1, d Z
≤ 1 a. s.
dZ
e << d Z with
Remark 3.2. Note d Z
is equivalent to
which in particular implies
e
dZ
dZ
≤ 1 a. s.
¦
©
et − Z
es , t ≥ s ≥ 0 = 1,
P Z t − Zs ≥ Z
¦
©
et , t ≥ 0 = 1.
P Zt ≥ Z
The next result considers the case when R(·, ·) ≡ I. In that case, a full comparison between the tuples
e ) is possible when R
e(·, ·) is constant, a partial comparison being possible otherwise.
(X , Z) and (Xe , Z
As it will become clear in Section 5, the main difficulty in getting a full comparison for non-constant
e(·, ·) relies on the fact that the usual alternative characterization of ( Z
et ) t≥0 ( Z
e0 = 0) when R
e(·, ·)
R
e
e
is constant (say R(·, ·) ≡ R), as being the (unique) pathwise-minimum non-decreasing continuous
process satisfying (see for example [25; 26])
et + R
eZ
et ≥ 0, t ≥ 0,
U
1893
with
. e
e· =
U
X0 +
Z
·
0
eb(s, Xes− )ds +
Z
·
0
γ(s, Xes− )dWs +
Z ·Z
0
E
δ(s, Xes− , r)N (ds, d r),
e(·, ·) (see for example [21]).
is in general not guaranteed for non-constant R
Theorem 3.3. Let
e ) = RSDE(Xe0 , eb, R
e),
(X , Z) = RSDE(X 0 , b, I) and (Xe , Z
both under Conditions 2.1 to 2.4, respectively. Assume that
Then we have
e R
e).
(X 0 , ψ, I) ¹ ( Xe0 , ψ,
¦
©
P X t ≤ Xet , t ≥ 0 = 1.
e(·, ·) is constant, then we also have
If moreover R
e << d Z and
dZ
e
dZ
dZ
≤ 1 a. s.
The following corollary is a direct consequence of Theorems 3.1 and 3.3.
Corollary 3.4. Let
e ) = RSDE(Xe0 , eb, R
e),
(X , Z) = RSDE(X 0 , b, R) and (Xe , Z
both under Conditions 2.1 to 2.4, respectively. Assume that
Then we have
e R
e).
(X 0 , ψ, R) ¹ I (Xe0 , ψ,
¦
©
P X t ≤ Xet , t ≥ 0 = 1.
e(·, ·) is constant, then we also have
If moreover R
e << d Z and
dZ
e
dZ
dZ
≤ 1 a. s.
Finally, the next result shows that a full comparison is possible for non-constant oblique reflection
directions, provided reflections upon hitting each boundary tend to bring the remaining coordinates
closer to the origin. In order to compare X and Xe we generally require in this case an ordering at
the boundaries on the drift vectors b and eb, as it will become clear in Section 5, ensuring in turn a
e . It is in this context where
pathwise comparison between the increments of the processes Z and Z
the boundary property in equation (2.3) plays an important role. The result is the following.
Theorem 3.5. Let
e ) = RSDE(Xe0 , eb, R
e),
(X , Z) = RSDE(X 0 , b, R) and (Xe , Z
1894
both under Conditions 2.1 to 2.5, respectively. Assume that
e R
e)
(X 0 , ψ, R) ¹ ( Xe0 , ψ,
e(·, ·) ≤ I, and that further each pair of drift coefficients bi and ebi satisfies the same ordering as
with R
ei but only on the corresponding i-th face Fi , i.e., that11
ψi and ψ
for each i. Then we have
bi (t, x) ≤ ebi (t, y), x, y ∈ Fi , x ≤ y, t ≥ 0,
e
¦
©
dZ
e << d Z and
P X t ≤ Xet , t ≥ 0 = 1, d Z
≤ 1 a. s.
dZ
4 Stability Applications: An Ergodic Result
In this section we use the comparison results of Section 3 to establish an ergodicity criterium related
to solutions of RSDEs as in equation (2.1), the main idea being to exploit those comparison properties, and the stability results in [1] for the constant reflection directions case, to provide a simple
but useful ergodicity condition in the context of state-dependent directions of reflection.
For simplicity we consider the continuous case, i.e., in absence of jumps (δ(·, ·, ·) ≡ 0). Therefore,
we consider RSDEs of the form
Z t
Z t
Z t
X t = X0 +
b(X s )ds +
0
γ(X s )dWs +
0
R(X s )d Zs , t ≥ 0,
(4.1)
0
where of course coefficients are assumed to be time-independent. Note when b and γ are constant,
X in equation (4.1) reduces to a Semi-martinagle Reflecting Brownian Motion (SRBM) in the orthant
with state-dependent reflection directions (see [24]).
n
, whose existence
We denote by X x the process X in equation (4.1) when starting from X 0 = x ∈ R+
and uniqueness is ensured under Conditions 2.1 and 2.2, and introduce accordingly and as usual
n
the family of distributions {P x : x ∈ R+
} on the path space of continuous functions mapping [0, ∞)
n
n
and t ≥ 0, we write
into R+ , denoting as E x expectation with respect to P x . Also, for each x ∈ R+
x
n
Px (t, ·) for the law of X t in R+ , i.e.,
©
. ¦
n
Px (t, A) = P X tx ∈ A = P x X t ∈ A , A ∈ B(R+
),
n
abusing notation in the last equality12 . (Note Px (0, ·) = δ x (·), unit mass at x ∈ R+
.)
We will consider in this section a boundedness condition on b and γ, and a uniform non-degeneracy
condition on the corresponding diffusion matrix a (= γγ T ).
e when jump-amplitude coefficients (δ) do not
Note this condition is in particular implied by the ordering on ψ and ψ
depend on the state.
12
Though P x is a probability measure on the path space, the identification X 0 = x under P x is standard (through the
canonical representation).
11
1895
Condition 4.1. b and γ are bounded, i.e.,
sup kb(x)k + sup kγ(x)k < ∞.
n
x∈R+
n
x∈R+
Moreover, the diffusion matrix a is uniformly elliptic, i.e., there exists ς ∈ (0, ∞) such that
X
ai j (x)ξi ξ j ≥ ςkξk2
i, j
n
and ξ = (ξl )nl=1 ∈ Rn .
for all x ∈ R+
Conditions 2.1 and 2.2, along with the boundedness requirement in Condition 4.1, guarantee the
n
family {X x : x ∈ R+
} satisfies the Feller property13 . Indeed, from [21, Proposition 3.2, pp. 515] and
using the Burkholder-Davis-Gundy inequalities [9, Theorem 26.12, pp. 524], it is easy to see that
n
,
there exists a constant 0 < C < ∞ such that, for all t ≥ 0 and all x, y ∈ R+
E
–
sup
0≤s≤t
kX sx
− X sy k2
™
¦
©
≤ C kx − yk2 + t + t 2 .
(4.2)
Also, from [6, Theorem 5.1, pp. 572] we know for each 0 < T < ∞ there exists a constant
0 < C T < ∞ such that, for all 0 ≤ t ≤ T ,
™ «
¨
™
–
Z t –
E
sup kX sx − X sy k2
kx − yk2 +
≤ CT
0≤s≤t
E
sup
0≤s≤t
kX sx
0≤u≤s
0
Gronwall’s lemma then shows that
™
–
− X sy k2
2
¨
sup kX ux − X uy k2 ds .
E
≤ C T kx − yk exp C T
Z
t
E
0
–
sup
0≤u≤s
kX ux
− X uy k2
™
«
ds ,
and therefore, on invoking again (4.2), and the arbitrariness of 0 < T < ∞, we conclude
y
E[kX tx − X t k2 ] → 0 as kx − yk ց 0,
n
n
} then follows from
. The Feller property of the family {X x : x ∈ R+
for each t > 0 and all x, y ∈ R+
standard arguments (see for example [15], proof of Lemma 8.1.4, pp. 133-134.).
On the other hand, the uniform ellipticity requirement in Condition 4.1, along with Conditions 2.1
and 2.2, in particular guarantee irreducibility in that the probability measure Px (t, ·) and Lebesgue
n
n
measure in R+
are, for each x ∈ R+
and t > 0, mutually absolutely continuous.
We are now in position to state and prove the advertised ergodic result.
n
Theorem 4.2. Consider the family {X x : x ∈ R+
} as above, under Conditions 2.1, 2.2, 2.4 and 4.1.
Set
.
ebi =
sup bi (x) for each i,
n
x∈R+
13
Note then, since each X x is (F t ) t≥0 -adapted with (F t ) t≥0 satisfying the usual hypotheses, the Markov family {X x :
n
x ∈ R+
} is indeed strong-Markov.
1896
n×n
e = (R
e i j )n
e ≤ I, R
e ii = 1 for each i and σ(R
e − I) < 1,
with R
and assume that there exists R
i, j=1 ∈ R
. e n
e and14 , with eb =
and such that R(·) ≤ R
( bi )i=1 ∈ Rn ,
e−1eb < 0.
R
(4.3)
n
Then there exists a unique invariant distribution for the family {X x : x ∈ R+
}, in that there exists a
n
n
unique probability measure π on (R+ , B(R+ )) such that
Z
Z
f (x)π(d x)
E x f (X t ) π(d x) =
n
R+
n
R+
n
n
n
for all15 f ∈ C b (R+
). Moreover, for each initial distribution π0 on (R+
, B(R+
)) we have
Z
Px (t, A)π0 (d x) = π(A)
lim
tր∞
n
R+
n
), and therefore in particular we have that the measure
for each A ∈ B(R+
weakly to π as t increases to infinity, i.e.,
Z
Z
lim
E x f (X t ) π0 (d x) =
tր∞
n
R+
R
n
R+
Px (t, ·)π0 (d x) converges
f (x)π(d x)
n
R+
n
for each f ∈ C b (R+
).
Before giving the proof of the theorem we make the following remark.
Remark 4.3. Note the key ergodicity condition in Theorem 4.2, equation (4.3), does not coincide in the
e, with the corresponding one in [1], which reads in
case of constant directions of reflection, say R(·) ≡ R
this case
sup θi (x) < 0 for each i,
n
x∈R+
e−1 b(·), i.e., with the supremun being pulled out in equation (4.3).
with θi (·) the i-th component of R
This is a consequence of supporting our result in an auxiliary comparison, as it will be done in the
proof below. However, as mentioned at the beginning of the section, equation (4.3) provides a useful
ergodicity condition for the case of applications with state-dependent reflection directions.
n
Proof. Consider for each x ∈ R+
the auxiliary RSDE given by
Z t
x
eZ
e x , t ≥ 0,
Xet = x + ebt +
γ(Xesx )dWs + R
t
0
ex (t, ·), t ≥ 0, for the corresponding
whose well-posedness is straightforwardly ensured, and write P
laws. From the theorem’s assumptions, the Lipschitz continuity of the Skorokhod map in this constant reflection directions case (see [26]) and [1, Theorem 2.16, pp. 8], we conclude the tightness,
for each M ∈ (0, ∞), of the family of probability measures
©
¦
ex (t, ·) : kxk ≤ M , t ≥ 0 ,
P
14
e not only guarantees for R
e−1 to exist, but also to be (elementwise) non-negative (see for
Note the structure of R
example [2]).
15
n
n
As usual, C b (R+
) denotes the space of bounded continuous functions from R+
into R.
1897
which in turn ensures, since by Theorem 3.5 in Section 3 we have
X x ≤ Xe x a. s.,
the corresponding tightness of the family
Px (t, ·) : kxk ≤ M , t ≥ 0 .
The above tightness, along with the theorem’s assumptions and the Feller structure of the family
n
{X x : x ∈ R+
}, then give the corresponding existence of an invariant distribution π (see [7]). The
convergence
Z
Px (t, A)π0 (d x) = π(A)
lim
tր∞
n
R+
n
for each initial distribution π0 and A ∈ B(R+
), and therefore the uniqueness of π, follow then in
turn from [13, Theorems 1.1. and 1.3, pp. 142 and 144, resp.]. That in particular the measure
Z
Px (t, ·)π0 (d x)
n
R+
converges weakly to π as t increases to infinity, is a direct consequence of Portmanteau’s theorem
(see for example [3]). „
5 Proofs of the Main Results
In this section we give the proofs of the main results of the paper in Section 3. To that aim we first
establish the following lemma.
Lemma 5.1. Let
e ) = RSDE(Xe0 , eb, R
e),
(X , Z) = RSDE(X 0 , b, R) and (Xe , Z
both under Conditions 2.1 to 2.4, respectively, and with X 0 ≤ Xe0 a. s. Then for each constant N ≥ 0,
index i and t ≥ 0 we have both16
Z

t∧TN ∧T
X
i
+
j
j
e ]
E[(φ t∧T ∧T ) ] ≤
E
1{φ i >0} R i j (s, X s )[d Zs − d Z
(5.1)
s
N
and
i
E[(φ t∧T
N ∧T
)+ ] ≤
X
j6=i
where
s
0
j6=i
Z
E
t∧TN ∧T
0
e i j (s, Xes )[d Z j − d Z
e j ] ,
1{φ i >0} R
s
s
s
.
φ ti = X ti − Xeti , t ≥ 0,
and where the (F t ) t≥0 -stopping times TN and T are defined as17
¦
©
.
et k1 > N
TN = inf t > 0 : kX t k1 + kXet k1 + kZ t k1 + k Z
16
17

.
.
.
(x)+ = max{x,
0}, x ∧ y = min{x, y} and x ∧ ∞ = x, x, y ∈ R.
. P
.
n
n
kxk1 = l |x l |, x = (x l )l=1 ∈ R , and inf ; = ∞.
1898
(5.2)
and
.
T = T ⋄ ∧ T ∗ ∧ T †,
with T ⋄ any (F t ) t≥0 -stopping time18 ,
¦
©
.
el (t, Xet ) f or some l
T ∗ = inf t > 0 : ψl (t, X t ) > ψ
and
¦
©
.
e l m (t, Xet ) f or some l, m .
T † = inf t > 0 : R l m (t, X t ) > R
Proof. Consider an index i, fixed throughout the proof. Note since φ i is clearly a semi-martingale
with
X
X
|∆φsi | ≤
[|∆X si | + |∆Xesi |] < ∞ a. s., t > 0,
0<s≤t
0<s≤t
the (jointly) right-continuous in y (∈ R) and continuous in t (∈ [0, ∞)) version of the local time
associated to φ i , with y indicating the corresponding level, exists (see [20]). We denote it by
Lφ i = (Lφ i (t, y)) t≥0, y∈R . In order to prove the lemma, we first verify that
Lφ i (·, 0) ≡ 0 a. s.
Indeed, denote by ([φ i , φ i ]ct ) t≥0 the path-by-path continuous part of the quadratic variation process
.
([φ i , φ i ] t ) t≥0 with [φ i , φ i ]0c = 0, and note that
Z
X ·
i
i c
[γi j (s, X s ) − γi j (s, Xes )]2 ds a. s.
[φ , φ ] =
·
0
j
and that, from Conditions 2.1 and 2.4,
X
X
[γi j (s, X s ) − γi j (s, Xes )]2 =
[ηi j (s, X si ) − ηi j (s, Xesi )]2 ≤ K 2 [φsi ]2 , s ≥ 0.
j
j
Define the mapping ρ : (0, ∞) → (0, ∞) by setting
.
ρ(u) = K 2 u2 , u ∈ (0, ∞).
Let ε > 0 and note that
Z
Now, with
I ti
.
=
Z
[ρ(u)]−1 du = ∞.
(5.3)
(0,ε]
t
1{0<φ i ≤ε} [ρ(φsi )]−1 d[φ i , φ i ]sc , t ≥ 0,
s
0
we have a. s.
I ti
=
Z
t
1{0<φ i ≤ε} [ρ(φsi )]−1
0
≤ K2
18
s
Z
X
j
t
[γi j (s, X s ) − γi j (s, Xes )]2 ds
1{0<φ i ≤ε} [ρ(φsi )]−1 [φsi ]2 ds ≤ t < ∞.
0
s
For convenience T ⋄ is left here unfixed, being chosen appropriately when applying the lemma.
1899
(5.4)
On the other hand, by the occupation times formula of semi-martingale local times (see [20])
Z
[ρ(u)]−1 Lφ i (t, u)du a. s., t ≥ 0,
I ti =
(0,ε]
and therefore, in light of equations (5.3) and (5.4) and since
Lφ i (t, u) → Lφ i (t, 0) as u ց 0 a. s., t ≥ 0,
we conclude, by the same arguments as in [22, Lemma 3.3, pp. 389], that
Lφ i (t, 0) = 0 a. s., t ≥ 0.
The claim then follows by invoking the sample path continuity of Lφ i (·, 0). We now turn into proving
the lemma. From Meyer-Itô’s formula (see [20]) we have19
i
(φ·∧T
)+
N ∧T
− (φ0i )+
=
Z
·∧TN ∧T
dφsi
s− >0}
1{φ i
0+
+
+
X
1
2
Lφ i (· ∧ TN ∧ T, 0)
1{φ i
s−
0<s≤·∧TN ∧T
But, from Condition 2.3 we have
X
1{φ i
s−
0<s≤·∧TN ∧T
X
i −
>0} (φs ) =
X
i −
>0} (φs ) +
1{φ i
0<s≤·∧TN ∧T
1{φ i
s− ≤0}
0<s≤·∧TN ∧T
s− ≤0}
(φsi )+ a. s.
(φsi )+ ≡ 0 a. s.
and, since also (φ0i )+ = 0 and Lφ i (·, 0) ≡ 0 a. s., it is therefore easy to see that
i
E[(φ t∧T
)+ ]
N ∧T
≤
X
j
Z
E

t∧TN ∧T
0
1{φ i >0} [R i j (s, X s )d Zsj
s
e i j (s, Xes )d Z
e j ]
−R
s
for each t ≥ 0, where we have replaced X s− by X s since X is c àd l àg and Z is continuous, and
e . Thus, by writing
similarly for Xe and Z
e i j (s, Xes )d Z
ej
R i j (s, X s )d Zsj − R
s
and using that
=
e j ] + [R i j (s, X s ) − R
e i j (s, Xes )]d Z
ej
R i j (s, X s )[d Zsj − d Z
s
s
e i ] = 1{φ i >0} [d Z i − d Z
ei ]
1{φ i >0} R ii (s, X s )[d Zsi − d Z
s
s
s
s
s
e we have
and that from the definition of Z and Z
i
e i ] = 1 i ei
ei
1{φ i >0} [d Zsi − d Z
{X −X >0} d Zs − 1{X i −Xe i >0} d Zs
s
s
s
s
s
ei
= − 1{X i −Xe i >0} d Z
s
s
s
≤ 0
19
Rt . R
R0
P
.
.
(x)− = − min{x, 0}, x ∈ R, 0+ = (0,t] , t > 0, and 0+ = 0<s≤0 = 0.
1900
s
and also, on {T † > 0},
e i j (s, Xes )]d Z
e j ≤ 0, s ∈ (0, T † ) ∩ (0, ∞),
[R i j (s, X s ) − R
s
we obtain equation (5.1). In the same way, by writing
e i j (s, Xes )d Z
ej
R i j (s, X s )d Zsj − R
s
=
e i j (s, Xes )]d Z j + R
e i j (s, Xes )[d Z j − d Z
e j]
[R i j (s, X s ) − R
s
s
s
we find, by similar arguments than before, equation (5.2). The lemma is then proved. „
.
Having established the lemma, we now give the proofs of the results in Section 3. Set N∗ = {1, 2, . . .}
n
and, for each k ∈ N∗ , index i and x = (x l )nl=1 ∈ R+
,
.
x (k,i) = (x 1 , . . . , x i−1 , x i − k−1 , x i+1 , . . . , x n ).
In addition,
(k)
Fi
©
. ¦
n
= x = (x l )nl=1 ∈ R+
: x i ≤ k−1 ,
for each k ∈ N∗ and index i as well.
Proof of Theorem 3.1. We first consider the comparison between20
.
(X (k) , Z (k) ) = RSDE(X 0 , b, γ(k) , R)
and
(5.5)
.
e (k) ) =
(Xe (k) , Z
RSDE(Xe0 , eb(k) , γ(k) , I),
(5.6)
n
and t ≥ 0,
where for each k ∈ N∗ , x ∈ R+
for each i, and
(k)
γi j (t, x)
.
=
¨
. e
eb(k) (t, x) =
bi (t, x) + k−1
i
(k)
ηi j (t, 0)
if x ∈ Fi
(k,i)
−1
γi j (t, x
) = ηi j (t, x i − k ) elsewhere
for each i, j, with {ηi j }ni, j=1 taken from Condition 2.4. Note (5.5) and (5.6) are clearly well defined
since the linear growth and Lipschitz continuity conditions are correspondingly inherited. As in
e(k) to (5.5) and (5.6), respectively, and note that ψ(k) ≡ ψ and
Section 3, we associate ψ(k) and ψ
that, since
e y)
ψ(t, x) ≤ ψ(t,
n
as x, y ∈ R+
with x ≤ y and t ≥ 0, we have
e y) + k−1 = ψ
e(k) (t, y)
ψ(k) (t, x) < ψ(t,
n
with x ≤ y and t ≥ 0 as well. Fix now a k ∈ N∗ . From Lemma 5.1,
for each k ∈ N∗ , as x, y ∈ R+
equation (5.2), applied to
(X (k) , Z (k) ) = ((X (k),l )nl=1 , (Z (k),l )nl=1 )
20
Recall from Section 2 when we want to emphasize the γ-term we explicitly add it in the notation.
1901
and
.
and with T ⋄ = ∞, we have21
e (k) ) = ((Xe (k),l )n , ( Z
e (k),l )n ),
(Xe (k) , Z
l=1
l=1
(k),i
E[(X t∧T
N ∧T
(k),i
∗
− Xet∧T
N ∧T
∗
)+ ] ≤ 0, t ≥ 0,
for each i. By letting N ր ∞, Fatou’s lemma then shows that
(k)
and therefore right-continuity gives
(k)
X t∧T ∗ ≤ Xet∧T ∗ a. s., t ≥ 0,
X ·(k) ≤ Xe·(k) on [0, T ∗ ] ∩ [0, ∞) a. s.
In particular,
(k)
(k)
e(k) (T ∗ , Xe ∗ ) on {T ∗ < ∞}.
ψ(k) (T ∗ , X T ∗ ) < ψ
T
Since
(5.7)
(k)
e(k) (0, Xe0 ) = ψ
e(k) (0, Xe (k) ) a. s.,
ψ(k) (0, X 0 ) = ψ(0, X 0 ) < ψ
0
e(k) then give T ∗ > 0 a. s., and therefore
right-continuity of X (k) and Xe (k) and continuity of ψ(k) and ψ
the consideration, on {T ∗ < ∞}, of
(k)
(k)
(X T ∗ +t ) t≥0 and (XeT ∗ +t ) t≥0
shows, by a direct argument by contradiction based on equation (5.7), that T ∗ = ∞ a. s. Thus, we
conclude that
n
o
(k)
(k)
P X t ≤ Xet , t ≥ 0 = 1.
(5.8)
Consider now an s ≥ 0 and write
(k)
(k)
e (k) − U
e (k) ) + ( Z
e (k) − Z
e (k) ), t ≥ s,
Xet∧T = Xes∧T + (U
t∧T
s∧T
t∧T
s∧T
N
i.e., with
e (k)
U
·
.
=
Xe0 +
Set
Z
N
N
·
0
eb(k) (u, Xe (k) )du +
u−
N
Z
N
·
γ
(k)
0
(k)
(u, Xeu− )dWu +
Z ·Z
0
(k)
E
δ(u, Xeu− , r)N (du, d r).
. (k)
(k)
e (k) − U
e (k) ) + (Z (k) − Z (k) ), t ≥ s,
Yet∧T = Xes∧T + (U
s∧T
t∧T
t∧T
s∧T
N
N
N
N
N
(5.9)
N
N
(5.10)
e by Z in the right-hand-side of equation (5.9). (Note
i.e., with the corresponding replacement of Z
e(·, ·) ≡ I, it has been therefore correspondingly omitted in equations (5.9) and (5.10).) Then,
since R
since X ·(k) ≤ Xe·(k) a. s., by using Meyer-Itô’s formula as in the proof of Lemma 5.1, it is easy to see
that, for each i,
(k),i
(k),i
E[(X t∧T − Yet∧T )+ ] ≤ 0, t ≥ s.
N
21
a. s.
N
e (k) . Obviously, T † = ∞
Note T , T and TN are defined, throughout the proof, with respect to X (k) , Xe (k) , Z (k) and Z
†
∗
1902
By letting N ր ∞ as before, Fatou’s lemma then shows that
i.e., by right-continuity
e t(k) − U
e (k) ) + (Z t(k) − Z (k) ) ≥ 0 a. s., t ≥ s,
Xes(k) + (U
s
s
e (k) − U
e (k) ) + (Z (k) − Z (k) ) ≥ 0 on [s, ∞) a. s.,
Xes(k) + (U
·
s
·
s
(k)
e −Z
e (k) )h≥0 (see Section 3), we conclude
and therefore, by the alternative characterization of ( Z
s
s+h
n
o
(k)
et(k) − Z
e (k) , t ≥ s ≥ 0 = 1.
P Z t − Zs(k) ≥ Z
s
(5.11)
n
Finally, from the uniform convergence (on [0, ∞) × R+
)
eb(k) → eb and γ(k) → γ as k ր ∞,
the regularity requirements from Condition 2.1 and the continuity results for the solution mapping
of the SP in the orthant with state-dependent reflection directions from [24], we have that for each
t ≥ 0 there exists a subsequence {nk }∞
along which
k=1
(nk )
lim X t
kր∞
and
(nk )
= X t and lim Z t
kր∞
= Z t a. s.
(n )
et(nk ) = Z
et a. s.
lim Xet k = Xet and lim Z
kր∞
kր∞
The theorem then follows from equations (5.8) and (5.11) by the right-continuity of X and Xe and
e. „
the continuity of Z and Z
Theorem 3.3 follows by the same corresponding arguments as in the previous proof (save the corresponding consideration of equation (5.1) in Lemma 5.1), and its proof is therefore omitted. The
proof of Corollary 3.4 is also omitted, as it is a direct consequence of Theorems 3.1 and 3.3 by noting
that we can always “sandwich” a function ψ
with b defined by
e z), x, y, z ∈ Rn , x ≤ y ≤ z, t ≥ 0,
ψ(t, x) ≤ ψ(t, y) ≤ ψ(t,
+
X
.
b i (·, ·) = ψi (·, ·) −
λj
j
Z
δi j (·, ·, r j )G j (d r j ), each i,
Ej
satisfying the regularity requirements in Condition 2.1, the same as b and eb. We then proceed to
the proof of Theorem 3.5.
Proof of Theorem 3.5. As in the proof of Theorem 3.1, we first consider the comparison between
.
(X (k) , Z (k) ) = RSDE(X 0 , b, γ(k) , R(k) )
and
.
e (k) ) =
e)
(Xe (k) , Z
RSDE(Xe0 , eb(k) , γ(k) , R
1903
(5.12)
(5.13)
n
and t ≥ 0,
with eb(k) and γ(k) defined the same as before and, for each k ∈ N∗ , x ∈ R+
¨
R i j (t, x) − (k + k0 )−1 if i 6= j
.
(k)
R i j (t, x) =
1
if i = j
with k0 sufficiently large so as for R(k) to satisfy Condition 2.2, inherited from22 R. Also as before,
e(k) to (5.13), and fix in what follows a k ∈ N∗ . From Lemma
we associate ψ(k) (≡ ψ) to (5.12) and ψ
23
5.1, equation (5.2), applied to (5.12) and (5.13) above24 , we obtain for each i and t ≥ 0
Z

t∧TN ∧T
X
(k),i
(k),i
e i j (s, Xe (k) )[d Z (k), j − d Z
e (k), j ] ,
E[(X t∧T ∧T − Xet∧T ∧T )+ ] ≤
E
1{X (k),i −Xe(k),i >0} R
s
s
s
N
N
j6=i
s
0
s
(5.14)
where we set T ⋄ in the corresponding definition of T as
n
o
.
(k),l
(k),l
(k),l
T ⋄ = inf t > 0 : X t
> Xet
and Xet
= 0 f or some l .
We claim that T ⋄ > 0 a. s. Indeed, set
n
o
.
(k),l
I0 = l : Xe0 = 0 a. s. .
(k),l
(Note X 0 = 0 a. s. too for l ∈ I0 .) If I0 = ;, then by right-continuity of Xe (k) we have T ⋄ > 0 a. s.
Assume then I0 6= ; and set the (F t ) t≥0 -stopping times
o
n
.
(k),l
(k),l
T1 = inf t > 0 : X t
> k−1 or Xet
> k−1 f or some l ∈ I0
and
n
o
.
(k)
(k)
(k)
T2 = inf t > 0 : bl (t, X t ) > ebl (t, Xet ) f or some l ∈ I0 .
e(k) , R(k) , R
e, b and eb(k) , and the facts that a. s.
Right-continuity of X (k) and Xe (k) , continuity of ψ, ψ
e(k) (0, Xe0 ),
ψ(0, X 0 ) < ψ
e(0, Xe0 ) − I
R(k) (0, X 0 ) − I < R
and
(k)
bl (0, X 0 ) < ebl (0, Xe0 ), l ∈ I0 ,
show that
T1 T2 T ∗ T † > 0 a. s.
Thus, again by right-continuity of X (k) and Xe (k) , and the fact that Xe (k),l > 0 for l ∈
/ I0 , we conclude
that for t in some vicinity [0, ξ), with ξ > 0 depending on the paths,
Z t
Z
X t
(k)
(k),i
(k)
(k)
Xt
=
Bi (s, ω)ds + Γi (t, ω) +
Θi j (s, ω)d Zs(k), j
0
j∈I0
0
n
Note analogously as for eb(k) and γ(k) , R(k) → R uniformly on [0, ∞) × R+
as k ր ∞.
As the reader will notice, equation (5.1) in Lemma 5.1 serves the same to prove the theorem since off-diagonal
elements of R(k) are strictly negative, in particular non-positive.
24
e (k) .
Note as before, T † , T ∗ and TN are defined, throughout the proof, with respect to X (k) , Xe (k) , Z (k) and Z
22
23
1904
and
(k),i
Xet
=
Z
t
0
X
e(k) (s, ω)ds + Γ
e i (t, ω) +
B
i
j∈I0
for each i ∈ I0 , where for s ∈ [0, ξ) and P-a.e. ω ∈ Ω
Z
t
0
e i j (s, ω)d Z
e (k), j
Θ
s
.
. e(k)
(k)
(k)
Bi (s, ω) = bi (s, X s(k) (ω)) ≤ ebi (s, Xes(k) (ω)) = B
i (s, ω)
and
. (k)
. e (k)
(k)
e i j (s, Xe (k) (ω)) =
Θi j (s, ω) = R i j (s, X s(k) (ω)) ≤ R
Θi j (s, ω),
s
.
(k)
e i (0, ·) =
and where Γi (0, ·) = Γ
0 and, for s, u ∈ [0, ξ), s ≤ u, and P-a.e. ω ∈ Ω as well,
(k)
(k)
Γi (u, ω) − Γi (s, ω)
Zu
Z
X u
. X
(k)
j
(k)
(k), j
ηi j (v, 0)dWv +
=
(ω)
R i j (v, X v )d Z v
j∈I0c
s
j
≤
s
X Z
j
P
u
ηi j (v, 0)dWvj
s
. e
e
(ω) = Γ
i (u, ω) − Γi (s, ω)
.
= 0 (recall off-diagonal elements of R(k) are
e(·, ·) ≤ I, the same arguments leading to
negative, in particular non-positive). Hence, since also R
the proof of [21, Theorem 4.1, pp. 521] show that
with I0c denoting the complement of I0 and
j∈;
X ·(k),l ≤ Xe·(k),l on [0, ξ) ∩ [0, TN ) a. s., each l ∈ I0 ,
(5.15)
and therefore, equation (5.15) holding for each N ≥ 0,
T ⋄ > 0 a. s.,
as claimed. Note then T (= T ⋄ ∧ T ∗ ∧ T † ) > 0 a. s. as well. Assume now S(ω) ∈ [0, T (ω)) is a
e (k),i (some index i). Then Xe (k),i = 0 and, since S < T ⋄ , X (k),i = 0 too. Thus,
point of increase of Z
S
S
by right-continuity of X (k) and Xe (k) we have, for t in some vicinity [S, S + ς) ∩ [0, T ) with ς > 0
depending on the paths,
(k),i
Xt
=
Z
t
bi (u, X u(k) )du
S
+
X
j
and
(k),i
Xet
=
Z
Z
t
ηi j (u, 0)dWuj
S
j6=i
t
S
+
X
eb(k) (u, Xe (k) )du +
u
i
X
j
Z
Z
t
(k)
(k),i
R i j (u, X u(k) )d Zu(k), j + (Z t
(k),i
− ZS
)
S
t
(k),i
S
et
ηi j (u, 0)dWuj + ( Z
(k),i
e
−Z
S
),
(k), j
where for this last expression we have assumed without lost of generality that XeS > 0 for all j 6= i,
in light of the boundary property (see equation (2.3))
Z∞
e (k),i = 0 a. s.
1 e(k),l
dZ
0
{X u
=0, ∀l∈A }
1905
u
for each A ⊆ {1, . . . , n} with |A | ≥ 2 and A ⊃ {i}. Right-continuity of X (k) and Xe (k) , continuity of
b and eb(k) , the ordering
(k)
bi (t, x) < ebi (t, y), x, y ∈ Fi , x ≤ y, t ≥ 0,
and the standard fixed-point-equation characterization of
(k)
(k)
(k)
(k)
e
e )h≥0
(ZS+h − ZS )h≥0 and ( Z
−Z
S
S+h
(see [24]), then show that for t in some vicinity [S, S + ε) ∩ [0, T ), with ε > 0 depending on the
paths as well,
(k),i
Zt
(k),i
− ZS
where25
Vu(k),i
.
=
Z
(k),i
e (k),i )+ = Z
et
= sup (−Vu(k),i )+ ≥ sup (−V
u
S≤u≤t
u
bi (u, X u(k) )du +
S
Thus, since also
e (k),i
V
u
.
=
Z
Z
u
ηi j (u, 0)dWuj
(k)
X
j6=i
eb(k) (u, Xe (k) )du +
u
i
S
+
S
u
Zt
we conclude
X
j
and
S≤u≤t
X
j
Z
Z
(k),i
e
−Z
S
,
(5.16)
u
(k)
R i j (u, X u(k) )d Zu(k), j
(5.17)
S
u
ηi j (u, 0)dWuj .
S
− Zs(k) ≥ 0 for all t ≥ s ≥ 0,
n
o
(k)
et(k) − Z
e (k) , T > t ≥ s ≥ 0 = 1.
P Z t − Zs(k) ≥ Z
s
Equation (5.14) then shows, by the same corresponding arguments as in the proof of Theorem 3.1,
that indeed T = ∞ a. s. and that
n
o
(k)
(k)
P X t ≤ Xet t ≥ 0 = 1.
Therefore, we conclude that for each k ∈ N∗
n
o
n
o
(k)
(k)
(k)
et(k) − Z
e (k) , t ≥ s ≥ 0 = 1,
P X t ≤ Xet t ≥ 0 = P Z t − Zs(k) ≥ Z
s
and a limiting argument as in the proof of Theorem 3.1 gives then the desired result. „
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