Doob - Meyer decoposition for submartingales on time scales Nguyen Thanh Dieua Abstract.. The aim of this paper is to study the Doob - Meyer decomposition for a submartingale on time sca
Trang 1Doob - Meyer decoposition for submartingales on time scales
Nguyen Thanh Dieu(a) Abstract The aim of this paper is to study the Doob - Meyer decomposition for a submartingale on time scales The obtained results can be considered as a generalization of the Doob - Meyer decomposition for submartingale in the discrete and continuous time.
Introduction
The Doob - Meyer decomposition theorem for a submartingale is one of the central topic in the probability theory In [8], P.A Meyer has proved that a submartingale
belonging to the class D admits a unique decomposition into a sum of a uniformly
integrable martingale and a predictable integrable increasing process Later on, this result is considered in the continuous time in [10] by using the increasing natural process instead the concept of prediction
Moreover, in recent years, the theory of dynamic on time scales, which was intro-duced by Stefan Hilger in his PhD thesis [5], has been born in order to unify continu-ous and discrete analysis Since then, this problem has received much attention from many research groups Therefore, it is natural that we need to transfer this theory to
the so-called stochastic calculus on the time scale The first attempt of this topic is to
consider the Doob - Meyer decomposition for submartingales indexed by a time scale and this is the aim of this paper The obtained result can be considered as a common method to present Doob - Meyer decomposition in the discrete and continuous time The organization of this paper is as follows In section 1 we survey some basic notation and properties of the analysis on time scale In section 2 we presents the main result of Doob - Meyer decomposition theorem for a submartingale on time scale
1 Preliminaries on time scales
This section surveys some notations on the theory of the analysis on time scales which was introduced by Stefan Hilger 1988 [5] A time scale is a nonempty closed subset of the real numbers R, and we usually denote it by the symbol T We assume throughout that a time scale T is endowed with the topology inherited from the real
numbers with the standard topology We define the forward jump operator and the
backward jump operator σ, ρ : T → T by σ(t) = inf{s ∈ T : s > t} (supplemented by inf ∅ = sup T ) and ρ(t) = sup{s ∈ T : s < t} (supplemented by sup ∅ = inf T) The
graininess µ : T → R+∪ {0} is given by µ(t) = σ(t) − t A point t ∈ T is said to be
right-dense if σ(t) = t, right-scattered if σ(t) > t, left-dense if ρ(t) = t, left-scattered if ρ(t) < t,
1 NhËn bµi ngµy 22/5/2009 Söa ch÷a xong 17/11/2009.
Trang 2and isolated if t is right-scattered and left-scattered For every a, b ∈ T, by [a, b], we mean the set {t ∈ T : a 6 t 6 b} The set T k is defined to be T if T does not have
a left-scattered maximum; otherwise it is T without this left-scattered maximum
Let f be a function defined on T, valued in R m We say that f is delta differentiable (or simply: differentiable) at t ∈ T k provided there exists a vector f∆(t) ∈ R m, called
the derivative of f, such that for all ² > 0 there is a neighborhood V around t with
kf (σ(t)) − f (s) − f∆(t)(σ(t) − s)k 6 ²|σ(t) − s| for all s ∈V If f is differentiable for every
t ∈ T k , then f is said to be differentiable on T If T = R then delta derivative is f 0
(t) from continuous calculus; if T = Z, the delta derivative is the forward difference, ∆f, from discrete calculus A function f defined on T is rd−continuous if it is continuous at
every right-dense point and if the left-sided limit exists at every left-dense point The
set of all rd−continuous function from T to a Banach space X is denoted by Crd(T, X)
A matrix function f from T to R m × R m is said to be regressive if det(I + µ(t)f(t)) 6= 0 for every t ∈ T.
If f : T → R is a function, then we write f σ : T → R for the function f σ = f ◦ σ;
i.e., f σ
t = f (σ(t)) for all t ∈ T Denote I = {t : right-scattered points of T}.
1.1 Proposition ([7]) The set I of all right-scattered points of T is at most countable.
Let A be an increasing rd−contiuous function defined on T Denote by F1 the family of all left close and right open interval of T:
F1= {[a; b) : a, b ∈ T}.
F1 is semiring of subsets of T Let m1 be the set function defined on F1 by
m1([a, b)) = A(b) − A(a).
It is easy to show that m1 is a countably additive measure on F1 We write µ A
∆for the
Caratheodory extension of the set function m1,associated with the family F1 and call
it the Stieljes - Lebesgue ∆−measure associated with A on T.
Let E be an A∆− measurable set of T\{max T, min T} and f : T → R, be an A∆−
measurable function The integral of f associated with the measures µ A
∆on E is called
Lebesgue - Stieljes ∆− integral and it is denoted by
Z
E
f (s)∆A(s).
1.2 Example If A(t) = t for all t ∈ T we have µ A
∆ is Lebesgue ∆− measure on T and
R
E f (s)∆A(s) is Lesbesgue ∆− integral.
1.3 Remark By the definition of µ A
∆ we see that
(1) For each t0∈ T k , the single- point set {t0} is ∆ A − measurable, and
µ A∆({t0}) = A(σ(t0)) − A(t0) (1.1)
Trang 3(2) If a, b ∈ T and a 6 b, then
µ A∆((a, b)) = A(b) − A(σ(a))
µ A∆((a, b]) = A(σ(b)) − A(σ(a))
µ A∆([a, b]) = A(σ(b)) − A(a)
2 Doob - Meyer decomposition
Let a ∈ T k We denote Ta = {x ∈ T : x > a} Let (Ω, F, {F t } t∈T a , P) be a space
probability with filtration {F t } t∈T a satisfing the usual conditions The notions of
con-tinuous process, rd−concon-tinuous process, cadlag process, martingale, submartingale, stopping time for a stochastic processes X = {X t : t ∈ T a } on the space probability
(Ω, F, {F t } t∈T a , P)are defined as usually
2.1 Definition A process A = {A t } t∈T a is called increasing if it is Ft −adapted, A a = 0
and the almost sure sample paths of A are increasing on T a
2.2 Proposition If M is a right continuous bounded martingale, A is increasing then for
any t ∈ T a , then
EM t A t= E
Z
[a,t)
Proof Fix a t ∈ T a For any n ∈ N, consider a partion π (n) = {a = t (n)0 < t (n)1 < · · · <
t (n) k n = t} of [a, t] Denote δ π (n) = maxt i ∈π (n) |t i+1 − σ(t i )| Let
N s π (n) =
M σ(a) if s = a
M σ(t (n)
i+1) if s ∈ (t (n) i , t (n) i+1] ∀i = 0, , k n − 2
M t if s ∈ (t (n) k n −1 , t).
Since M is right continuous, M σ
s is also right continuous Therefore,
M s σ = lim
δ π(n) →0 N s π (n) ∀s ∈ [a, t).
Hence, by the bounded convergence theorem we have
E
Z
[a,t)
M s σ ∆A(s) = E
h lim
δ π(n) →0
Z
[a,t)
N s π (n) ∆A(s)
i
δ π(n) →0E
h Z
[a,t)
N s π (n) ∆A(s)
i
δ π(n) →0E
h
M σ(a) A σ(a)
+
kXn −1 i=1
M σ(t (n)
i )(A σ(t (n)
i )− A σ(t (n)
i−1)) + M t (A t − A σ(t (n)
kn−1)) i
Trang 4= lim
δ π(n) →0E
h
M t A t+
kXn −1 i=1
A
σ(t (n) i−1)(M
σ(t (n) i )
− M σ(t (n)
i−1)) + A σ(t (n)
kn−1)(M t − M σ(t (n)
kn−1))
i
= EM t A t
For any cadlag function f : T a → R , we define the function f t− by f t− = lims↑t f (s)
for each t ∈ T \ {min T} By convention we put f a− = f (a)
2.3 Definition An increasing process A = (A t)t∈T a is said to be natural if for every
bounded cadlag martingale M = (M t)t∈T a we have
EM t A t= E
Z
[a,t)
2.4 Proposition The rd−continuous, increasing process (A t)t∈T a is natural iff A σ
t is
Ft −measurable for t ∈ I ∩ T a
Proof.
Sufficient condition Suppose that A σ
t is Ft −measurable for t ∈ I ∩ T a Let M t be a
Ft − cadlag martingale and t ∈ T a arbitrary For any n ∈ N, we consider a partition
π (n) = {a = t (n)0 < t (n)1 < · · · < t (n) k n = t} of [a, t] such that δ π (n) = max |t (n) i+1 −σ(t (n) i )| 6 2 −n
Let
M s π (n)=
M σ(t (n)
i ) if s ∈ (t (n) i ; t (n) i+1] ∀i = 0, , k n (n) − 2.
M σ(t (n)
kn−1) if s ∈ (t (n) k n −1 ; t) Since M is a cadlag process,
M s−= lim
δ π(n) →0 M s π (n) ∀s ∈ [a, t).
Therefore, by the bounded convergence theorem we have
E Z
[a,t)
M s− ∆A(s) = E
à lim
δ π(n) →0
Z
[a,t)
M s π (n) ∆A(s)
!
.
We have
E
Z
[a,t)
(M s σ − M s− )∆A s= E lim
δ π(n) →0
Z
[a,t)
(N s π (n) − M s π (n) )∆A s
δ π(n) →0E
·
(M σ(a) − M a )(A σ(a) − A a)
+
k n (n)X−2
i=0
(M σ(t (n)
i+1)− M σ(t (n)
i )(A σ(t (n)
i+1)− A σ(t (n)
i )) + (M t − M σ(t (n)
kn−1))(A t − A σ(t (n)
kn−1))
¸
.
Trang 5Because σ(t (n) i ) 6 t (n) i+1 6 σ(t (n) i+1 ) , A t is rd−continuous and M t is right continuous we see that the above limit converges to
s∈I∩[a,t)
(M s σ − M s )(A σ s − A s)
i
.
On the other hand, A σ
s is Fs − measurable for s ∈ I ∩ [a, t) then
E [(M s σ − M s )(A σ s − A s )] = E [E(M s σ − M s )(A σ s − A s )|F s]
= E [(A σ s − A s )E(M s σ − M s |F s )] = 0.
Thus,
E Z
[a,t)
(M s σ − M s− )∆A s = 0.
By using the proposition 2.2 we get
E Z
[a,t)
M s σ ∆A s = E
Z
[a,t)
M s− ∆A s = EM t tA t ,
i.e., (A t) is natural increasing processes
Necessary condition Let A = (A t ) be a natural increasing process We need drive that A σ
t
is Ft −measurable for t ∈ I∩T a Let t ∈ I∩T a It is easy to see the process eA s = A s −A t , s > t
is also natural on Tt Therefore, by (2.2), for any cadlad, bounded martingale M t we have
EM σ(t) (A σ(t) − A t) = E
Z
[t,σ(t))
M τ − ∆A τ = EM t (A σ(t) − A t ).
Or,
E(M σ(t) − M t )(A σ(t) − A t ) = 0 =⇒ E(M σ(t) − M t )A σ(t) = 0.
Since EM t (A σ(t) − E[A σ(t) | F t]) = 0,
E(M σ(t) − M t )(A σ(t) − E[A σ(t) | F t ]) = 0.
It is easy to see that
M τ = X
a6s<τ
(A σ(s) − E[A σ(s) | F s])
is a Fτ −martingale Therefore,
E(M σ(t) − M t )(A σ(t) − E[A σ(t) | F t])
= E(A σ(t) − E[A σ(t) | F t ])(A σ(t) − E[A σ(t) | F t ]) = 0, which implies that A σ(t) − E[A σ(t) | F t] = 0 a.s The proof is complete ¤
2.5 Corollary (A t ) is increasing process on time scale T
i) T = N then (A t ) is natural iff it is previsible.
ii) T = R then evry increasing process (A t ) is natural if it is continuous.
Trang 6i) If T = N, then any point t ∈ T right- scattered and σ(t) = t + 1 Therefore, by the proposition 2.4, A t is natural if and only if A t+1is Ft −measurable, i.e., (A n) is a previsible process
ii) When T = R we have σ(t) = t for all t ∈ T Since A t is increaing, A t is Ft −measurable.
¤
We recall the Dunford - Pettis theorem in [1]
2.6 Theorem (Dunford - Pettis [1]) If (Y n)n∈N is uniformly integrable sequence of random variables, there exists an integrable random variable Y and a subsequence (Y n k)k∈N such that weak-lim k→∞ Y n k = Y , i.e., for all bounded random variables ξ we have
lim
k→∞ E(ξY n k ) = E(ξY )
A process X is said to be:
- of class (D) if the set
{X τ : τ is a stopping time satisfying a 6 τ < ∞}
is uniformly integrable
- of class (DL) if for every t ∈ T athe set
{X τ : τ is a stopping time satisfying a 6 τ 6 t}
is uniformly integrable
2.7 Theorem (Doob-Meyer decomposition) Let X be a right continuous submartingale
of class (DL) Then, there exist a right continuous martingale and a right continuous increasing process A such that
X t = M t + A t ∀t ∈ T a a.s.
If A is natural then M and A are uniquely determined up to indistinguishability If X is of class (D) then M and A are uniformly integrable.
Proof First we proof uniqueness Suppose there exist two right continuous martingales M ,
M 0 and two right continuous natural increasing processes A, A 0 such that
X t = M t + A t = M t 0 + A 0 t ∀t ∈ T a a.s.
This relation implies that
B t = A t − A 0 t = M t 0 − M t is right continuous martingale.
Trang 7Let ξ t be an arbitrary right continuous bounded martingale For each partition π (n) : a =
t (n)0 < t (n)1 < · · · < t (n) n = t of [a, t], we set
ξ s π (n) =
ξ σ(t (n)
i ) if s ∈ (t (n) i ; t (n) i+1] ∀i = 0, 1, , n − 2
ξ σ(t (n)
n−1) if s ∈ (t (n) n−1 ; t)
.
we have
ξ s−= lim
δ π(n) →0 ξ s π (n) ∀s ∈ [a, t)
By the bounded convergence theorem we have
E
Z
[a,t)
ξ s− ∆A(s) = E lim
δ π(n) →0
Z
[a,t)
ξ s π (n) ∆A(s)
= E lim
δ π(n) →0
h
ξ0(A σ(0) − A0) +
n−2
X
i=0
ξ σ(t i)(A σ(t i+1)
− A σ(t i)) + ξ σ(t n−1)(A t − A σ(t n−1))
i Therefore,
Eξ t (A t − A 0 t) = E
Z
[a,t)
ξ s− ∆A(s) − E
Z
[a,t)
ξ s− ∆A 0 (s)
= E lim
δ π(n) →0
h
ξ0(B σ(0) − B0) +
n−2
X
i=0
ξ σ(t i)(B σ(t i+1)− B σ(t i))
+ ξ σ(t n−1)(B t − B σ(t n−1))
i
δ π(n) →0E
h
ξ0(B σ(0) − B0) +
n−2
X
i=0
ξ σ(t i)(B σ(t i+1)− B σ(t i)) + ξ σ(t n−1)(B t − B σ(t n−1))
i
Thus
Eξ t (A t − A 0 t ) = 0.
Now let X be an arbitrary bounded random variable and let us define the bounded mar-tingale ξ by taking a right continuous version of E(X|F t)t∈T a From the above,
E(X(A t − A 0 t)) = E
·
E(X(A t − A 0 t )|Ft)
¸
= E£(A t − A 0 t )E(X|Ft)¤= Eξ t (A t − A 0 t ) = 0 Since the choice of X was arbitrary, it follows that A t = A 0 t almost surely for a fixed
t > 0 By virtue of the right continuity of A and A 0 , we conclude that A and A 0 are
indistinguishable Hence, M = X − A and M 0 = X − A 0 are indistinguishable as well
Trang 8Next, we prove the existence of the decomposition By uniqueness, it suffices to prove
the existence of the processes M and A on the interval [a; b] for fixed b ∈ T a Without loss
of generality we may assume that X a = 0 Let π (n) : a = t (n)0 < t (n)1 < · · · < t (n) N = b be
a partition of [a, b] such that δ π (n) = maxt i ∈π (n) |t i − σ(t i−1 )| 6 1
2n Consider the Doob
-Meyer decomposition of the finite submartingale X (n) = (X t (n)
j
)t (n)
j ∈π (n)
X t (n)
j
= M (n)
t (n) j + A (n)
t (n) j
Thus, M (n) = {M (n)
t (n) j }
t (n) j ∈π (n) is a martingale satisfying M a (n) = X a and A (n) = {A (n)
t (n) j }
t (n) j ∈π (n)
is a previsible and increasing Therefore,
M (n)
t (n) j = E(M b (n) |Ft(n)
j
) = E(X b − A (n) b |Ft(n)
j
)
2.8 Lemma {A (n) b : n = 1, 2, · · · } is uniformly integrable.
Proof Let λ > 0 be fix and define the random variable T λ (n) by
T λ (n)=
min{t (n) j−1: j = 1, 2, , N and A (n)
t (n) j > λ}
b if {t (n) j−1 : j = 1, 2, , N and A (n)
t (n) j > λ} = ∅ .
Since A (n) is increasing,
{T λ (n) 6 t (n) j−1 } = {A (n)
t (n) j > λ},
and this set belongs to Ft (n)
j−1 by the previsibility of A (n) It is easy to see that T λ (n) is a stopping time By noting that
{T λ (n) < b} = {A (n) b > λ}
and that A (n)
T λ (n) 6 λ on this set, we obtain
0 6 1
2
Z
A (n) b >2λ
A (n) b dP 6
Z
A (n) b >2λ
(A (n) b − λ)dP
6 Z
A (n) b >2λ
(A (n) b − A (n)
T λ (n) )dP 6
Z
Ω
(A (n) b − A (n)
T λ (n) )dP
= Z
Ω
(X b − X T (n)
λ
)dP =
Z
{A (n) b >λ}
(X b − X T (n)
λ
)dP.
By using Chebyshev inequality we have:
P{A (n) b > λ} 6 EA
(n)
b
EX b
λ → 0 where λ → ∞.
Trang 9Hence, limλ→∞ P(A (n) > λ) = 0 uniformly Since X is assumed to be of class (DL),
Z
{A (n) b >λ}
(X b − X
T λ (n) )dP → 0 where λ → ∞.
A (n) >2λ
A (n) b dP → 0 where λ → ∞,
Now we return to the proof of Theorem 2.8 By the Dunford - Pettis theorem, there
is a subsequence (A (n k)
b )k∈N converging weakly to an integrable random variable A b We
claim that for any sub σ− algebra G of F,
weakly- lim
k→∞ E(A (n k)
b |G) = E(A b |G).
To prove this, fix an arbitrary bounded random variable η Then,
lim
k→∞ E(ηE(A (n k)
b |G)) = lim
k→∞ E(E(ηE(A (n k)
b |G)|G))
= lim
k→∞ E(E(A (n k)
b |G)E(η|G)) = lim
k→∞ E(E(A (n k)
b E(η|G)|G))
= lim
k→∞ E(A (n k)
b E(η|G)) = E(A b E(η|G)) = E(ηE(A b |G)).
We now define the processes M and A by
M t = E(X b − A b |F t ); A t = X t − M t ; ∀t ∈ [a, b], where A b is the weak limit point of an appropriate subsequence of (A (n) b ) In the first
definition we take a right continuous version of the martingale M t which implies that the
process A is right continuous, A t is integrable for each t We see that
A a = X a − E(X b − A b |F a ) = X a − weak- lim
k→∞ E(X b − A (n k)
b |F a)
= X a − weak- lim
k→∞ E(M (n k)
b |F a ) = X a − weak- lim
k→∞ M (n k)
a
= weak- lim
k→∞ A (n k)
a = 0.
Let Π =Sn∈N π (n) If a 6 s 6 t 6 b with s, t ∈ Π are fixed then
A t − A s = X t − X s − E(X b − A b |F t ) − E(X b − A b |F s)
= X t − X s − weak- lim
k→∞
³
E(X b − A (n k)
b |F t ) − E(X b − A (n k)
b |F s)
´
= X t − X s − weak- lim
k→∞
³
E(M (n k)
b |F t ) − E(M (n k)
b |F s)
´
weak- lim
k→∞
³
EA (n k)
t − EA (n k)
s
´
> 0.
Trang 10Since Π is countable and A is right continuous, it follows that there is a version of A t such
that A t > A s for all t > s in [a; b] almost surely It follows that A is increasing Next
we check that A is natural Let ξ be any right continuous bounded martingale By the predictability of A (n) then
E
·
ξ b A (n) b
¸
= Eξ b (A (n) σ(a) − A (n) a ) + X
t (n) k ∈π (n)
Eξ a (A (n)
σ(t (n) k )− A (n)
σ(t (n) k−1))
= Eξ σ(a) (A (n) σ(a) − A (n) a ) + X
t (n) k ∈π (n)
Eξ σ(t (n)
k−1)(A (n)
σ(t (n) k )− A (n)
σ(t (n) k−1)) Where n enough large the right hand of the obove relation equals to
Eξ σ(a) (A σ(a) − A a) + X
t (n) k ∈π (n)
Eξ σ(t (n)
k−1)(A σ(t (n)
k )− A σ(t (n)
k−1)).
Letting n → ∞ we obtain
Eξ σ(a) (A σ(a) − A a) + X
t (n) k ∈π (n)
Eξ σ(t (n)
k−1)(A σ(t (n)
k )− A σ(t (n)
k−1)) → E
Z
[a,b)
ξ s− ∆A(s),
and
E
h
ξ b A (n) b
i
→ E [ξ b A b ]
So, we have
E [ξ b A b] = E
Z
[a,b)
ξ s− ∆A(s).
Replacing ξ = (ξ s ) by ξ = ξ t∧s for each t ∈ [a, b] it easy to conclude that
E [ξ t A t] = E
Z
[a,t)
ξ s− ∆A(s).
Thus A = (A t) is natural
Finally, if X is of class (D), then X is uniformly integrable and the limit X ∞ = limt→∞ X t exists almost surely and this limit belongs to L1 The Doob - Meyer
decom-positions of the discrete submartingales X (n) along the partitions π (n) = {t (n) j : j ∈ N} of
Ta , are then uniformly integrable as well, and we may define A ∞ as the weak limit of an
appropriate subsequence of (A (n) ∞)n∈N , where A (n) ∞ := limj→∞ A (n)
t (n) j The details carry over
References
1 Lav Kallenberg, Foundations of Modern Probability, Springer Verlag, New York Berlin Heidelberg, 2001