Báo cáo nghiên cứu khoa học: "Biểu diễn Doob - Mayer đối với Martingale trên thang thời gian"

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Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của trường đại học vinh năm 2009 tác giả: 2. Nguyễn Thanh Diệu, Biểu diễn Doob - Mayer đối với Martingale trên thang thời gian.

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Doob - 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, NhËn bµi ngµy 22/5/2009. Söa ch÷a xong 17/11/2009. 1
and isolated if t is right-scattered and left-scattered. For every a, b ∈ T, by [a, b], we mean the set {t ∈ T : a t b}. The set Tk 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 Rm . We say that f is delta diﬀerentiable (or simply: diﬀerentiable) at t ∈ Tk provided there exists a vector f ∆ (t) ∈ Rm , called the derivative of f , such that for all > 0 there is a neighborhood V around t with |σ (t) − s| for all s ∈V. If f is differentiable for every f (σ (t)) − f (s) − f ∆ (t)(σ (t) − s) t ∈ Tk , then f is said to be diﬀerentiable on T. If T = R then delta derivative is f (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 Rm × Rm is said to be regressive if det(I + µ(t)f (t)) = 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., ftσ = 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}. is semiring of subsets of T. Let m1 be the set function defined on F1 by F1 m1 ([a, b)) = A(b) − A(a). It is easy to show that m1 is a countably additive measure on F1 . We write µA 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 µA on E is called ∆ Lebesgue - Stieljes ∆− integral and it is denoted by f (s)∆A(s). E 1.2. Example. If A(t) = t for all t ∈ T we have µA is Lebesgue ∆− measure on T and ∆ f (s)∆A(s) is Lesbesgue ∆− integral. E 1.3. Remark. By the deﬁnition of µA we see that ∆ (1) For each t0 ∈ Tk , the single- point set {t0 } is ∆A − measurable, and µA ({t0 }) = A(σ (t0 )) − A(t0 ) (1.1) ∆
(2) If a, b ∈ T and a b, then µA ((a, b)) = A(b) − A(σ (a)) ∆ µA ((a, b]) = A(σ (b)) − A(σ (a)) ∆ µA ([a, b]) = A(σ (b)) − A(a) ∆ 2. Doob - Meyer decomposition Let a ∈ Tk . We denote Ta = {x ∈ T : x a}. Let (Ω, F, {Ft }t∈T , P) be a space probability with filtration {Ft }t∈T satisfing the usual conditions. The notions of con- a tinuous process, rd−continuous process, cadlag process, martingale, submartingale, a stopping time... for a stochastic processes X = {Xt : t ∈ Ta } on the space probability (Ω, F, {Ft }t∈T , P) are defined as usually. a 2.1. Deﬁnition. A process A = {At }t∈Ta is called increasing if it is Ft −adapted, Aa = 0 and the almost sure sample paths of A are increasing on Ta . 2.2. Proposition. If M is a right continuous bounded martingale, A is increasing then for any t ∈ Ta , then σ EMt At = E Ms ∆As . (2.1) [a,t) (n) (n) Proof. Fix a t ∈ Ta . For any n ∈ N, consider a partion π (n) = {a = t0 < t1 < ··· < (n) tkn = t} of [a, t]. Denote δπ(n) = maxti ∈π(n) |ti+1 − σ (ti )|. Let  Mσ(a) if s=a   (n) (n) (n) π = Mσ(t(n) ) if s ∈ (ti , ti+1 ] ∀i = 0, ..., kn − 2 Ns  i+1  M (n) if s ∈ (tkn −1 , t). t σ Since M is right continuous, Ms is also right continuous. Therefore, (n) σ π Ms = lim Ns ∀s ∈ [a, t). δπ(n) →0 Hence, by the bounded convergence theorem we have (n) σ π E Ms ∆A(s) = E lim Ns ∆A(s) δπ(n) →0 [a,t) [a,t) (n) π = lim E Ns ∆A(s) = lim E Mσ(a) Aσ(a) δπ(n) →0 δπ(n) →0 [a,t) kn −1 + Mσ(t(n) ) (Aσ(t(n) ) − Aσ(t(n) ) ) + Mt (At − Aσ(t(n) ) kn −1 ) i i i−1 i=1
kn −1 = lim E Mt At + Aσ(t(n) ) (Mσ(t(n) ) δπ(n) →0 i−1 i i=1 − Mσ(t(n) ) ) + Aσ(t(n) (Mt − Mσ(t(n) ) = EMt At . kn −1 ) kn −1 ) i−1 The proof is complete. For any cadlag function f : Ta → R, we define the function ft− by ft− = lims↑t f (s) for each t ∈ T \ {min T}. By convention we put fa− = f (a). 2.3. Deﬁnition. An increasing process A = (At )t∈Ta is said to be natural if for every bounded cadlag martingale M = (Mt )t∈Ta we have EMt At = E Ms− ∆As . (2.2) [a,t) 2.4. Proposition. The rd−continuous, increasing process (At )t∈Ta is natural iﬀ Aσ is t Ft −measurable for t ∈ I ∩ Ta . Proof. Suﬃcient condition. Suppose that Aσ is Ft −measurable for t ∈ I ∩ Ta . Let Mt be a t Ft − cadlag martingale and t ∈ Ta arbitrary. For any n ∈ N, we consider a partition (n) (n) (n) (n) (n) π (n) = {a = t0 < t1 < · · · < tkn = t} of [a, t] such that δπ(n) = max |ti+1 − σ (ti )| 2−n . Let  Ma if s=a   (n) (n) (n) = Mσ(t(n) ) if s ∈ (ti ; ti+1 ] ∀i = 0, .., kn − 2. π (n) Ms  i  (n) M (n) if s ∈ (tkn −1 ; t) σ (t ) kn −1 Since M is a cadlag process, (n) π Ms− = lim Ms ∀s ∈ [a, t). δπ(n) →0 Therefore, by the bounded convergence theorem we have (n) π E Ms− ∆A(s) = E lim Ms ∆A(s) . δπ(n) →0 [a,t) [a,t) We have (n) (n) σ π π E (Ms − Ms− )∆As = E lim (Ns − Ms )∆As δπ(n) →0 [a,t) [a,t) = lim E (Mσ(a) − Ma )(Aσ(a) − Aa ) δπ(n) →0 (n) kn − 2 + (Mσ(t(n) ) − Mσ(t(n) ) (Aσ(t(n) ) − Aσ(t(n) ) ) + (Mt − Mσ(t(n) )(At − Aσ(t(n) ). kn −1 ) kn −1 ) i+1 i i+1 i i=0
(n) (n) (n) Because σ (ti ) ti+1 σ (ti+1 ) , At is rd−continuous and Mt is right continuous we see that the above limit converges to (Ms − Ms )(Aσ − As ) . σ E s s∈I∩[a,t) On the other hand, Aσ is Fs − measurable for s ∈ I ∩ [a, t) then s E [(Ms − Ms )(Aσ − As )] = E [E(Ms − Ms )(Aσ − As )|Fs ] σ σ s s = E [(Aσ − As )E(Ms − Ms |Fs )] = 0. σ s Thus, σ E (Ms − Ms− )∆As = 0. [a,t) By using the proposition 2.2 we get σ E Ms ∆As = E Ms− ∆As = EMt tAt , [a,t) [a,t) i.e., (At ) is natural increasing processes Necessary condition. Let A = (At ) be a natural increasing process. We need drive that Aσ t is Ft −measurable for t ∈ I∩Ta . Let t ∈ I∩Ta . It is easy to see the process As = As −At , s t is also natural on Tt . Therefore, by (2.2), for any cadlad, bounded martingale Mt we have EMσ(t) (Aσ(t) − At ) = E Mτ − ∆Aτ = EMt (Aσ(t) − At ). [t,σ (t)) Or, E(Mσ(t) − Mt )(Aσ(t) − At ) = 0 =⇒ E(Mσ(t) − Mt )Aσ(t) = 0. Since EMt (Aσ(t) − E[Aσ(t) | Ft ]) = 0, E(Mσ(t) − Mt )(Aσ(t) − E[Aσ(t) | Ft ]) = 0. It is easy to see that Mτ = (Aσ(s) − E[Aσ(s) | Fs ]) a s
Proof. i) If T = N, then any point t ∈ T right- scattered and σ (t) = t + 1. Therefore, by the proposition 2.4, At is natural if and only if At+1 is Ft −measurable, i.e., (An ) is a previsible process. ii) When T = R we have σ (t) = t for all t ∈ T. Since At is increaing, At is Ft −measurable. We recall the Dunford - Pettis theorem in [1]. 2.6. Theorem (Dunford - Pettis [1]). If (Yn )n∈N is uniformly integrable sequence of random variables, there exists an integrable random variable Y and a subsequence (Ynk )k∈N such that weak-limk→∞ Ynk = Y , i.e., for all bounded random variables ξ we have lim E(ξYnk ) = E (ξY ) k→∞ A process X is said to be: of class (D) if the set - {Xτ : τ is a stopping time satisfying a τ < ∞} is uniformly integrable. of class (DL) if for every t ∈ Ta the set - {Xτ : τ is a stopping time satisfying a τ 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 Xt = Mt + At ∀t ∈ Ta 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 and two right continuous natural increasing processes A, A such that Xt = Mt + At = Mt + At ∀t ∈ Ta a.s. This relation implies that Bt = At − At = Mt − Mt is right continuous martingale.
Let ξt be an arbitrary right continuous bounded martingale. For each partition π (n) : a = (n) (n) (n) t0 < t1 < · · · < tn = t of [a, t], we set  ξa if s=a   (n) (n) (n) ξs = ξσ(t(n) ) if s ∈ (ti ; ti+1 ] ∀i = 0, 1, ..., n − 2 . π  i  (n) ξ (n) if s ∈ (t ; t) n−1 σ (tn−1 ) we have (n) π ξs− = lim ξs ∀s ∈ [a, t) δπ(n) →0 By the bounded convergence theorem we have (n) π E ξs− ∆A(s) = E lim ξs ∆A(s) δπ(n) →0 [a,t) [a,t) n−2 = E lim ξ0 (Aσ(0) − A0 ) + ξσ(ti ) (Aσ(ti+1 ) δπ(n) →0 i=0 − Aσ(ti ) ) + ξσ(tn−1 ) (At − Aσ(tn−1 ) ) Therefore, Eξt (At − At ) = E ξs− ∆A(s) − E ξs− ∆A (s) [a,t) [a,t) n−2 = E lim ξ0 (Bσ(0) − B0 ) + ξσ(ti ) (Bσ(ti+1 ) − Bσ(ti ) ) δπ(n) →0 i=0 + ξσ(tn−1 ) (Bt − Bσ(tn−1 ) ) = lim E ξ0 (Bσ(0) − B0 ) δπ(n) →0 n−2 + ξσ(ti ) (Bσ(ti+1 ) − Bσ(ti ) ) + ξσ(tn−1 ) (Bt − Bσ(tn−1 ) ) i=0 Thus Eξt (At − At ) = 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 |Ft )t∈Ta . From the above, E(X (At − At )) = E E(X (At − At )|Ft ) = E (At − At )E(X |Ft ) = Eξt (At − At ) = 0. Since the choice of X was arbitrary, it follows that At = At almost surely for a ﬁxed t > 0. By virtue of the right continuity of A and A , we conclude that A and A are indistinguishable. Hence, M = X − A and M = X − A are indistinguishable as well.
Next, 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 ∈ Ta . Without loss (n) (n) (n) of generality we may assume that Xa = 0. Let π (n) : a = t0 < t1 < · · · < tN = b be 1 a partition of [a, b] such that δπ(n) = maxti ∈π(n) |ti − σ (ti−1 )| 2n . Consider the Doob - Meyer decomposition of the ﬁnite submartingale X (n) = (Xt(n) )t(n) ∈π(n) j j (n) (n) Xt(n) = M (n) + A (n) tj tj j (n) (n) (n) Thus, M (n) = {M (n) }t(n) ∈π(n) is a martingale satisfying Ma = Xa and A(n) = {A (n) }t(n) ∈π(n) tj tj j j is a previsible and increasing. Therefore, (n) (n) (n) M (n) = E(Mb |Ft(n) ) = E(Xb − Ab |Ft(n) ) tj j j (n) 2.8. Lemma. {Ab : n = 1, 2, · · · } is uniformly integrable. (n) Proof. Let λ > 0 be ﬁx and deﬁne the random variable Tλ by  min{t(n) : j = 1, 2, ..., N and A((n)) > λ} n  j −1 tj (n) Tλ = . (n) (n) b if {tj −1 : j = 1, 2, ..., N and A (n) > λ} = ∅  t j Since A(n) is increasing, (n) (n) (n) {Tλ tj −1 } = {A (n) > λ}, tj (n) and this set belongs to Ft(n) by the previsibility of A(n) . It is easy to see that Tλ is a j −1 stopping time. By noting that (n) (n) {Tλ < b} = {Ab > λ} (n) and that A λ on this set, we obtain (n) Tλ 1 (n) (n) 0 Ab dP (Ab − λ)dP 2 (n) (n) Ab >2λ Ab >2λ (n) (n) (n) (n) (Ab −A )dP (Ab −A )dP (n) (n) Tλ Tλ (n) Ab >2λ Ω = (Xb − XT (n) )dP = (Xb − XT (n) )dP. (n) λ λ Ω {Ab >λ} By using Chebyshev inequality we have: (n) EAb EXb (n) P{Ab > λ} = → 0 where λ → ∞. λ λ
Hence, limλ→∞ P(A(n) > λ) = 0 uniformly. Since X is assumed to be of class (DL), (Xb − XT (n) )dP → 0 where λ → ∞. (n) λ {Ab >λ} Therefore, (n) Ab dP → 0 where λ → ∞, A(n) >2λ (n) i.e., {Ab } is uniform integrability. Now we return to the proof of Theorem 2.8. By the Dunford - Pettis theorem, there (n ) is a subsequence (Ab k )k∈N converging weakly to an integrable random variable Ab . We claim that for any sub σ − algebra G of F, (nk ) weakly- lim E(Ab |G ) = E(Ab |G ). k→∞ To prove this, ﬁx an arbitrary bounded random variable η . Then, (nk ) (nk ) lim E(η E(Ab |G )) = lim E(E(η E(Ab |G )|G )) k→∞ k→∞ (nk ) (nk ) = lim E(E(Ab |G )E(η |G )) = lim E(E(Ab E(η |G )|G )) k→∞ k→∞ (nk ) = lim E(Ab E(η |G )) = E(Ab E(η |G )) = E(η E(Ab |G )). k→∞ We now deﬁne the processes M and A by Mt = E(Xb − Ab |Ft ); At = Xt − Mt ; ∀t ∈ [a, b], (n) where Ab is the weak limit point of an appropriate subsequence of (Ab ). In the ﬁrst deﬁnition we take a right continuous version of the martingale Mt which implies that the process A is right continuous, At is integrable for each t. We see that (nk ) Aa = Xa − E(Xb − Ab |Fa ) = Xa − weak- lim E(Xb − Ab |Fa ) k→∞ (nk ) = Xa − weak- lim Mank ) ( = Xa − weak- lim E(Mb |Fa ) k→∞ k→∞ = weak- lim A(nk ) = 0. a k→∞ π (n) . Let Π = If a s t b with s, t ∈ Π are ﬁxed then n∈N At − As = Xt − Xs − E(Xb − Ab |Ft ) − E(Xb − Ab |Fs ) (nk ) (nk ) = Xt − Xs − weak- lim E(Xb − Ab |Ft ) − E(Xb − Ab |Fs ) k→∞ (nk ) (nk ) = Xt − Xs − weak- lim E(Mb |Ft ) − E(Mb |Fs ) k→∞ (nk ) − EA(nk ) weak- lim EAt 0. s k→∞
Since Π is countable and A is right continuous, it follows that there is a version of At such that At As 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 (n) (n) (n) (n) = Eξb (Aσ(a) − Aan) ) + ( E ξb Ab Eξa (A −A ) (n) (n) σ (tk ) σ (tk−1 ) (n) tk ∈π (n) (n) (n) (n) = Eξσ(a) (Aσ(a) − Aan) ) + ( Eξσ(t(n) ) (A −A ). (n) (n) σ (tk ) σ (tk−1 ) k −1 (n) tk ∈π (n) Where n enough large the right hand of the obove relation equals to Eξσ(a) (Aσ(a) − Aa ) + Eξσ(t(n) ) (Aσ(t(n) ) − Aσ(t(n) ) ). k−1 k k−1 (n) tk ∈π (n) Letting n → ∞ we obtain Eξσ(a) (Aσ(a) − Aa ) + Eξσ(t(n) ) (Aσ(t(n) ) − Aσ(t(n) ) ) → E ξs− ∆A(s), k −1 k k−1 [a,b) (n) tk ∈π (n) and (n) E ξb Ab → E [ξb Ab ] . So, we have E [ξb Ab ] = E ξs− ∆A(s). [a,b) Replacing ξ = (ξs ) by ξ = ξt∧s for each t ∈ [a, b] it easy to conclude that E [ξt At ] = E ξs− ∆A(s). [a,t) Thus A = (At ) is natural. Finally, if X is of class (D), then X is uniformly integrable and the limit X∞ = limt→∞ Xt exists almost surely and this limit belongs to L1 . The Doob - Meyer decom- (n) positions of the discrete submartingales X (n) along the partitions π (n) = {tj : j ∈ N} of Ta , are then uniformly integrable as well, and we may deﬁne A∞ as the weak limit of an (n) (n) (n) appropriate subsequence of (A∞ )n∈N , where A∞ := limj →∞ A (n) . The details carry over tj almost verbatim. References 1. Lav Kallenberg, Foundations of Modern Probability, Springer Verlag, New York Berlin Heidelberg, 2001.
2. Doob. J.L. Stochastic processes, John Wiley and Sons, New York, 1953. 3. N. Ikeda and S. Wantanabe, Stochastic differential equations and diffusion pro- cesses, North Holland, Amstardam, 1981. 4. Peter Medvegyev, Stochastic Integration Theory, Oxford University Press Inc., New York, 2007. 5. S. Hilger, Ein Makettenkalkul mit Anwendung auf Zentrumsmannigfaltigkeiten, a Ph.D. thesis, Universitaat Waurzburg, 1988. 6. S. Hilger, Analysis on measure chains a unified approach to continuous and discrete calculus, Results Math., 18 (1990), No. 1-2, 1856. 7. M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhauser ¨ Boston, Massachusetts, 2001. 8. P.A. Meyer, A decomposition theorem for supermartingales, Illinois J. Math., 6 (1962), 193205. 9. P.A. Meyer, Decomposition of supermartingales: The uniqueness theorem, Illinois J. Math., 7 (1963), 117. 10. Kunita. H and Wantanabe. S, On square integrable martingales, Nagoya Math. J., 30 (1967), 209-245 11. Alberto Cabada, Dolores R. Vivero, Expression of the Lebesgue ∆-Integral on Time Scales as a usual Lebesgue Integral; Application to the calculus of ∆- Antiderivatives, Mathematical and Computer Modelling, 43, (2006)194207. 12. I.I . Gihman, A.V Skorokhod, The Theory of Stochastic processes III, Springer - Verlag New York Inc, 1979. Tãm t¾t BiÓu diÔn doob - Meyer ®èi víi martingale díi trªn thang thêi gian Môc ®Ých cña bµi b¸o nµy lµ nghiªn cøu biÓu diÔn Doob - Mayer ®èi víi Martingale díi trªn thang thêi gian. KÕt qu¶ ®¹t ®îc cã thÓ xem lµ sù tæng qu¸t hãa cña biÓu diÔn Doob - Mayer víi thêi gian rêi r¹c vµ thêi gian liªn tôc. (a) Department of Mathematics,Vinh University, Nghe An, Vietnam.