Stochastic Processes in Vector Lattices
Lectures 5: Doob-Meyer decomposition
JJ Grobler1 and CCA Labuschagne2
18-26 September, 2015
Spring School on Stochastic processes in Functional Analysis,
Potchefstroom
1
North-West University, Potchefstroom Campus, Potchefstroom, South
Africa.
2
University of Johannesburg, South Africa.
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5
Outline
Optional times
Step elements of Stopping times
Doob-Meyer docomposition
Kolmogorov-C̆entsov inequality
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5
Optional times
Proposition
If S is an optional time and T is a stopping time with S T then
PS+ ⊂ PT .
Proposition
If (Sn ) is a sequence of optional times and S = inf Sn , then
∞
\
PS+ =
PSn +.
n=1
Moreover, if each Sn is a stopping time, and S Sn , then
PS+ =
∞
\
PSn .
n=1
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5
Proposition
If (Sn ) is a sequence of optional times, then, for S = sup Sn (which
is optional), we have
[
PSn + ⊂ PS+ .
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5
Step elements of Stopping times
Proposition
Let (Ft ) be a filtration in the Riesz space E and let
{Pk : k = 1, . . . , n} in P be a partition of I. Let S ∈ Orth(E) be a
step element of the form
S :=
n
X
tk Pk ,
k=1
with the tk arranged in strict increasing order, i.e., 0 < tk < tk+1 .
Then, S is an optional time if and only if Pk ∈ Pt + for k = 1, . . . , n
k
and a stopping time if and only if Pk ∈ Ptk for k = 1, . . . , n.
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5
What is XS ?
Classical case: For a stopping time t = S(ω) and a stochastic
process Xt (ω) = X (t, ω) one defines effortless
XS (ω) := X (S(ω), ω).
In the classical case we have
Pn no pointset Ω so this is not so
simple. However, if S := k=1 tk Pk ∈ Orth(E) the element
XS is defined as
n
X
XS =
Pk Xtk
k=1
The process (XS∧t )t∈T is called the stopped process with
reference to the stopping time (or optional time) S.
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5
The Doob-Meyer decomposition
In order to prove this famous theorem in vector lattices, we need to
define a number of notions.
The Doob-Meyer decomposition plays an important part in the
construction of Itô integrals.
Definition
Let (Ft , Ft )t∈T be a filtration on E and let A = (At ) be an
adapted stochastic process. Let Π be the set of all partitions of
the interval [0, t]. We call (At ) tractable on [0, t] whenever
IA (φt − ) := lim
π∈Π
n
X
hφti−1 , Ati − Ati−1 i for
i=1
π = {0 = t0 < t1 < · · · < tn = t},
exists for every adapted bounded dual martingale φ = (φt ). The
process (At ) is called tractable if it is tractable on every finite
interval [0, t].
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5
Every martingale M = (Mt ) is tractable since each of the
sums in the definition equals zero; hence we have IM (φt − ) = 0
for all φ and for all t.
The Poisson process N = (Nt , Ft ) with intensity λ is tractable
and if (Bt ) is a Brownian motion, then the submartingale
(Bt2 ) is tractable; in fact, in the classical case every increasing
process (in the sense of Karatzas and Shreve) is tractable.
A simple calculation shows
Theorem
If X = (Xt ) and A = (At ) are adapted processes such that
M = (Xt − At ) is a martingale, then (Xt ) is tractable if and only if
(At ) is tractable and for each t, IX (φt − ) = IA (φt − ).
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5
Definition
The adapted increasing process A = (At ) is called predictable on
the interval [0, t] if it is tractable and IA (φt − ) = hφt , At i. It is
called predictable if it is predictable on every interval [0, t].
If F is a conditional expectation on E and F = R(F) we let I be
∼
the ideal generated by F∼ in E∼
00 and note that F maps I into
itself. By Nakano’s theorem we have that the canonical image of E
∼
in I∼
00 is an order dense ideal in I00 . With the notation introduced
here, we have
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5
Theorem
∼
If E is perfect, i.e., E = (E∼
00 )00 , and F-universally complete, then,
in the duality hE, Ii, we have E = I∼
00 .
If hE, τ i is a locally convex-solid Riesz space, a subset A ⊂ E0
is called order-equicontinuous on E if (Un ) is a ρA -Cauchy
sequence whenever 0 ≤ Un ↑≤ U holds in E with
ρA (U) := sup{|φ(U)| : φ ∈ A}, U ∈ E.
Denote by Sa the set of all bounded step elements S of
stopping times with respect to the filtration (Ft , Ft ).
∼
Taking the dual pair hI∼
00 , Ii = hE, Ii, with the σ(E00 , E)
topology, the right continuous stochastic process (Xt , Ft ) is
said to be of class DL if the family {XS }S∈Sa is
order-equicontinuous on I.
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5
We have now defined all the necessary concepts in order to state
the celebrated Doob-Meyer result in Riesz spaces.
Theorem (Doob-Meyer-decomposition)
Let (Ft , Ft ) be a filtration on the perfect Riesz space E and
assume that E is F0 -universally complete. Let X = (Xt ) be a
tractable class DL submartingale, then
Xt = Mt + At , 0 ≤ t < ∞,
with M := {Mt , Ft , t ≥ 0} a right continuous martingale and
A := {At , Ft , t ≥ 0} an increasing predictable process. The
decomposition is unique.
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5
We have defined the element XS for a stochastic process and
a stopping time S which is a step element.
It is possible to define the element XS for a submartingale
that has a Doob-Meyer decomposition.
It may be mentioned at this point that it is still a problem to
define this in a more general setting. However in the case one
has such a decomposition we have a proof of Doob’s optional
sampling theorem:
Theorem
Let (Xt , Ft ) be a right continuous submartingale with the
Doob-Meyer decomposition property and let S ≤ T be two optional
times of the filtration (Ft , Ft ). Then, if either
1
T is bounded or
2
(Xt ) has a last element X∞ ,
we have
FS+ XT ≥ XS .
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5
Stochastic Processes in Vector Lattices
Lectures 6: Kolmogorov-C̆entsov inequality
JJ Grobler1 and CCA Labuschagne2
18-26 September, 2015
Spring School on Stochastic processes in Functional Analysis,
Potchefstroom
1
North-West University, Potchefstroom Campus, Potchefstroom, South
Africa.
2
University of Johannesburg, South Africa.
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5 and 6
Outline
Optional Skipping Theorem
Doob’s Lp -inequality
Kolmogorov-C̆entsov inequality
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5 and 6
Theorem (Optional Skipping Theorem)
Let (Xn , Fn ) be a countable submartingale and let Sn ∈ Orth(Fn )
be such that 0 ≤ Sn ≤ I for all n. Set
Y1 := X1 , Yn := X1 +
n−1
X
Sk (Xk+1 − Xk )
k=1
Then
1
(Yn , Fn ) is a submartingale and F1 (Yn ) ≤ F1 (Xn ) for all n.
2
If (Xn , Fn ) is a martingale, so is (Yn , Fn ) and
F1 (Yn ) = F1 (Xn ) for all n.
3
If Zn = Yn − X1 then (Zn , Fn ) is a submartingale and
F1 (Zn ) ≥ F1 (Zn−1 ) ≥ · · · ≥ F1 (Z1 ) = 0.
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5 and 6
For any two elements X and Y in E we define B(X < Y ) to be
the band generated by (Y − X )+ and B(Y ≤ X ) = B(X < Y )d .
For the intersection of several of these bands, we write for instance
B(X < Y , Z ≤ Y , X > Z ), etc. The corresponding band
projection for the last band will be denoted by
P(X < Y , Z ≤ Y , X > Z ).
Consider the finite submartingale (Xk , Fk )nk=1 and two elements
F , G ∈ F1 , F < G . We will construct disjoint upcrossing bands in
E as follows:
E = B(X1 > F ) + B(X1 ≤ F );
E = B(X1 > F , X2 > F ) + B(X1 > F , X2 ≤ F )+
+ B(X1 ≤ F , X2 < G ) + B(X1 ≤ F , X2 ≥ G );
..
.
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5 and 6
We continue the process by splitting each band in the following
way in two disjoint bands in each consecutive step:
B(. . . , Xk−1 > F ) = B(. . . , Xk−1 > F , Xk > F )+
B(. . . , Xk−1 > F , Xk ≤ F ),
B(. . . , Xk−1 ≤ F ) = B(. . . , Xk−1 ≤ F , Xk < G )+
B(. . . , Xk−1 ≤ F , Xk ≥ G ),
B(. . . , Xk−1 < G ) = B(. . . , Xk−1 < G , Xk < G )+
B(. . . , Xk−1 < G , Xk ≥ G ),
B(. . . , Xk−1 ≥ G ) = B(. . . , Xk−1 ≥ G , Xk > F )+
B(. . . , Xk−1 ≥ G , Xk ≤ F ),
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5 and 6
In this manner we obtain 2n disjoint bands, which we shall call the
upcrossing bands.
We shall say that the process is in an upward phase at k on the
bands B(. . . , Xk ≤ F ) and B(. . . , Xk < G ) and in a downward
phase at k on the bands B(. . . , Xk > F ) and B(. . . , Xk ≥ G ).
The number of upcrossings u(B) on the upcrossing band B is the
number of times that the process changes from an upward phase
to a downward phase on B.
We define the upcrossing orthomorphism
X
UFG =
u(B)PB .
B
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5 and 6
With these preparations the proof of the inequality is easy
Theorem (Upcrossing inequality)
Let (Xk , Fk )nk=1 be a finite submartingale and let F , G be two
elements in F1 = F1 (E) such that F < G . Let UFG be the
upcrossing orthomorphism of the process. Then
F1 (UFG (G − F )) ≤ F1 [(Xn − F )+ ].
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5 and 6
We now consider a continuous time submartingale (Xt , Ft )t∈T , an
arbitrary closed interval [σ, τ ] and elements F , G ∈ F0 with
F < G . Let π = {σ = t1 < t2 < . . . < tn = τ } be a partition of
[σ, τ ]. Then (Xtk )tk ∈π is a finite submartingale with upcrossing
orthomorphism UFG ,π . We note that if π2 is finer than π1 , then the
number of upcrossings increase, i.e., we have UFG ,π1 ≤ UFG ,π2 ; it
follows that (UFG ,π )π is an upwards directed system of
orthomorphisms. For any Z ≥ 0 we then define the upcrossing
orthomorphism of the submartingale (Xt , Ft )t∈T over [F , G ] on the
interval [σ, τ ] ⊂ T as
UFG ,[σ,τ ] Z = sup UFG ,π Z ∈ Es .
π
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5 and 6
We then have
Theorem (Upcrossing inequality: the continuous case)
Let
(Xt , Ft )t∈T be a submartingale in the Dedekind complete Riesz
space E. Let F , G be two elements in F1 = F1 (E) such that
F < G and let UFG ,[σ,τ ] be the upcrossing orthomorphism of the
process over [F , G ] on the interval [σ, τ ] ⊂ R. Then
F1 (UFG ,[σ,τ ] (G − F )) ∈ E and
F1 (UFG ,[σ,τ ] (G − F )) ≤ F1 [(Xτ − F )+ ].
Moreover, if E is F1 universally complete, then
UFG ,[σ,τ ] (G − F ) ∈ E.
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5 and 6
Similarly, we construct downcrossing bands for a finite
submartingale
(Xk , Fk )nk=1 and two elements F , G ∈ F1 = F1 (E). Counting the
number of downcrossings d(B) on the downcrossing band B we
define the downcrossing orthomorphism
X
DFG =
d(B)PB .
B
We then obtain downcrossing inequalities; the first inequality is
obtained by replacing UFG by DFG on the left and F by G on the
right; the second inequality is obtained by replacing UFG ,[σ,τ ] by
DFG ,[σ,τ ] (defined in the obvious way) on the left and again F by G
on the right.
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5 and 6
Theorem
(i) Let (Xk , Fk )N
k=1 be a finite submartingale. Then, for every
λ > 0 we have
λF1 [P(sup Xk ≥ λE )(E )] ≤ F1 [P(sup Xk ≥ λE )(XN )]
k
k
≤ F1 [P(sup Xk ≥ λE )(|XN |)].
k
(ii) Let (Xk , Fk )N
k=1 be a martingale or a positive submartingale.
Then, for every λ > 0 and p ≥ 1 we have
λp F1 [P(sup |Xk | ≥ λE )(E )] ≤ F1 [|XN |p ].
k
(iii) Let (Xk , Fk )N
k=1 be a finite submartingale. Then, for every
λ > 0 and p > 1 we have
p
p
p
p
F1 (|XN | ) ≤ F1 [(sup |Xk |) ] ≤
F1 (|XN |p ).
p
−
1
k
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5 and 6
Theorem
(i) Let X = (Xt )t∈T be a martingale or a positive submartingale,
[σ, τ ] ⊂ T be a closed interval and let λ > 0 be a real
number. Then, if X ∗ = sup |Xt | ∈ Es we have for p ≥ 1
t∈[σ,τ ]
that λp F1 (P(X ∗ ≥ λE )E) ≤ F1 [|Xτ |p ].
For p > 1, F1 [(X ∗ )p ] ≤
p
p−1
p
F1 (|Xτ |p ).
(ii) (Doob’s Lp -inequality) Let X = (Xt )t∈T be a martingale or a
positive submartingale with T an interval in R and let λ > 0
be a real number. Then, if X ∗ = sup |Xt | ∈ Es we have for
t∈T
∗ ≥ λE )E ) ≤ sup F [|X |p ]. If p > 1 we
p ≥ 1 that λp F1 (P(X
t
t 1
p
p
∗
p
p
have F1 [(X ) ] ≤ p−1 supt F1 (|Xt | ).
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5 and 6
Kolmogorov-C̆entsov inequality.
Definition
Let E be a Dedekind complete Riesz space with weak order unit E .
We say that the stochastic process (Xt ) is locally
Hölder-continuous with exponent γ if there exist a number δ > 0
and a strictly positive orthomorphism S ∈ Orth E such that, for all
s, t ∈ T satisfying |t − s|I ≤ S on a band C, we have
|Xt − Xs | ≤ δ|t − s|γ E on the band C.
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5 and 6
Theorem (Kolmogorov-C̆entsov)
Let E be a Dedekind complete Riesz space with weak order unit E
and let F be a conditional expectation defined on E with FE = E .
Suppose that (Xt ) is a stochastic process in E such that for some
positive constants α, β and C , we have
F(|Xt − Xs |α ) ≤ C |t − s|1+β E , 0 ≤ s, t ∈ T .
Then the process is locally Hölder-continuous with exponent γ for
every γ ∈ (0, β/α); in fact, there exists a maximal disjoint
sequence of bands Ck in E such that for n ≥ k and for s, t ∈ [a, b]
with |s − t| < 2−n we have
PCk (|Xt − Xs |) ≤ δ|t − s|γ E
for some constant δ and for every γ ∈ (0, β/α) and
∞
X
S :=
2−k PCk which is strictly positive.
k=1
JJ Grobler and CCA Labuschagne
Stochastic Processes, Lecture 5 and 6