DSpace at VNU: The First Attempt on the Stochastic Calculus on Time Scale

DSpace at VNU: The First Attempt on the Stochastic Calculus on Time Scale tài liệu, giáo án, bài giảng , luận văn, luận...

Stochastic Analysis and Applications, 29: 1057–1080, 2011 Copyright © Taylor & Francis Group, LLC ISSN 0736-2994 print/1532-9356 online DOI: 10.1080/07362994.2011.610169 The First Attempt on the Stochastic Calculus on Time Scale NGUYEN HUU DU1 AND NGUYEN THANH DIEU2 Faculty of Mathematics, Mechanics, and Informatics, University of Science-VNU, Hanoi, Vietnam Department of Mathematics, Vinh University, Nghe An, Vietnam The aim of this article is to study the Doob–Meyer decomposition theorem, -stochastic integration and Ito’s formula for stochastic processes defined on time scale The obtained results can be considered as a first attempt on the stochastic calculus on time scale Keywords Doob–Mayer decomposition; Ito’s formula; Martingale; Natural increasing process; Stochastic integration; Time scale 1991 Mathematics Subject Classification 60H10; 60J60; 34A40; 39A13 Introduction The stochastic calculus for discrete and continuous time had been studying for long times It is used to describe the mathematical stochastic models in economy, physics, biology, medicine and social sciences Some of basic problems show concern for studying are stochastic integration, Doob–Meyer decomposition theorem, stochastic differential equation, Ito’s formula which have been studied carefully for both discrete and continuous time (see for example [9, 11–13]) Moreover, in recent years, the theory of time scale, which was introduced by Hilger in his PhD thesis [6], has been born in order to unify continuous and discrete analysis Since then, this topic has received much attention from many research groups [2, 6, 7] However, almost works for this topic focus only on the deterministic analysis For stochastic analysis, there are not too much in mathematical literature Here, we mention one of the first attempts on this direction, the article of Bhamidi and his research group [1] in which the authors developed the theory of Brownian motion on time scale After, Suman, in his Ph.D dissertation [14], tried to defined “stochastic integral on time scales” but he just deals with discrete time scale Received June 25, 2010; Accepted May 10, 2011 This work was done under the support of the Grand NAFOSTED, no 101.02.63.09 Address correspondence to Nguyen Huu Du, Faculty of Mathematics, Mechanics, and Informatics, University of Science-VNU, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam; E-mail: dunh@vnu.edu.vn 1057 1058 Du and Dieu The aim of this article is to develop the theory of stochastic calculus on the time scale which leads us to a way to unify the presentation of classical problems of stochastic calculus in the discrete and continuous time The first attempt on this topic is to consider the Doob–Meyer decomposition theorem, stochastic integration, Ito’s formula for stochastic processes indexed by a time scale When constructing a stochastic integration on a time scale, the difficulty we are faced here is the forward jump operator t since it can make an adapted process become non-adapted One can avoid this disadvantage by using the -integration on the semi-open intervals of the form ti ti+1 Meanwhile, by the predictable requirement of the integrand, we need to use semi-open intervals ti ti+1 The difference between them makes a wide gap on the stochastic calculus with the -integration In using -integration for stochastic calculus on time scale, we can overcome this difficulty Although it makes some inconveniences when we try to define a stochastic dynamic equations on time scale because the -dynamic equations are more popular in references, -dynamic equations are also interesting in both theory and practice The organization of this article is as follows In Section we survey some basic notions and properties of the analysis on time scale Section presents Doob–Meyer decomposition theorem for a submartingale indexed by a time scale In Section 3, we give a definition of the -stochastic integration, and deals with some basic properties Preliminaries on Time Scale This section surveys some basic notions on the theory of the analysis on time scale which was introduced by Hilger [6] A time scale is a nonempty closed subset of the real numbers , and we usually denote it by the symbol We assume throughout that a time scale 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 → by t = inf s ∈ s > t (supplemented by inf ∅ = sup ) and t = sup s ∈ s < t (supplemented by sup ∅ = inf ) The forward (respectively backward) graininess is given by t = t − t (respectively t = t − t ) A point t ∈ is said to be right-dense if t = t, right-scattered if t > t, left-dense if t = t, left-scattered if t < t and isolated if t is right-scattered and left-scattered For every a b ∈ , by a b , we mean the set t ∈ a≤t≤b The set k is defined to be if does not have a right-scattered minimum; otherwise it is without this right-scattered minimum The set k is defined to be if does not have a left-scattered maximum; otherwise it is without this left-scattered maximum Let f be a function defined on , valued in m We say that f is -differentiable (or simply: differentiable) at t ∈ k provided there exists a vector f t ∈ m , called the derivative of f , such that for all > there is a neighborhood V around t with f t − f s − f t t −s ≤ t − s for all s ∈ V If f is differentiable for every t ∈ k , then f is said to be differentiable on If = then delta derivative is f t from continuous calculus; if = , the delta derivative is the backward difference, f , from discrete calculus A function f defined on is ld-continuous if it is continuous at every left-dense point and if the right-sided limit exists at every right-dense point The set of all ld-continuous function from to a Banach space X is denoted by Cld X It is similar to notation of rd-continuous If f → is a function, then we write f → Stochastic Calculus on Time Scale 1059 for the function f = f ; i.e., ft = f t for all t ∈ k Denote lim s ↑t f s by f t− or ft− if this limit exists It is easy to see that t is left-scatted then ft− = ft Let = t t is left-scattered Proposition 1.1 ([2]) The set most countable = t t is right-scattered = ∪ of all left-scattered or right-scattered points of is at We give an outline of constructing a Lebesgue–Stieltjes measure on time scale Let A be an increasing right continuous function defined on Denote by = a b a b∈ the family of all left open and right closed interval of It is seen that is semi-ring of subsets of Let m1 be the set function defined on by = Ab − Aa m1 a b (1.1) It is easy to show that m1 is a countably additive measure on We write A for the Caratheodory extension of the set function m1 associated with the family and call it the Stieltjes–Lebesgue -measure associated with A on Let E be a A -measurable subset of k and f → be an A -measurable function The integral of f associated with the measures A on E, denoted by f A , is called Lebesgue–Stieltjes -integral If A t = t for all t ∈ we have A E is Lebesgue -measure on and E f is Lesbesgue -integral By the definition of A we see that • For each t ∈ k , the single-point set t is A • For a b ∈ A a b t A -measurable, and = At − At− and a ≤ b, = Ab− − Aa A a b = Ab− − Aa− A a b = Ab − Aa− For the details, we can refer to [4] Doob–Meyer Decomposition For a ∈ k , denote a = x ∈ x ≥ a Let be a probability t t∈ a space with filtration t t∈ a satisfying the usual conditions (see [10]) The notions of continuous process, rd-continuous process, ld-continuous process, cadlag process, martingale, submartingale, semimartingale, stopping time for a stochastic process X = Xt t ∈ a on probability space are defined as usually t t∈ a A right continuous process A = At t∈ a is said to be increasing if it is t adapted, Aa = and the sample paths of A are increasing functions on a for almost sure ∈ The increasing process A = At t∈ a is called integrable if At < for all t ∈ a By convention, we write fa− = fa for all function defined on a 1060 Du and Dieu Proposition 2.1 If M is a bounded martingale, A is increasing, integrable, then for any t ∈ a , we have M t At = n M A at n n n Proof Let a = t0 < t1 < · · · < tkn = t n n maxi ti+1 − ti ≤ 2−n of a t and Ns n = B a partition satisfying kn Mti n ti−1 ti i=1 where be s n is the indicator function of the set B Since M is right continuous, n Ms = lim Ns ∀s ∈ a t n→ Hence, by the bounded convergence theorem, we obtain at M A = lim n→ = lim n→ = A at kn = lim n→ n N Mt n A t n − A t n i i=1 i M t At + kn i−1 At n M t n − M t n i=1 i i−1 i−1 Mt At The proof is complete Definition 2.2 An increasing process A = At holds Mt At = for every bounded martingale M = Mt Proposition 2.3 Let At t∈ a at t∈ a t∈ a is said to be natural if there (2.1) M− A be an increasing process These statements hold 1) If At is rd-continuous and At is t− -measurable for t ∈ Where t− stands for s ti − measurable by Proposition 2.3 Therefore, i ti+1 −ti > i ti − i =0 i Stochastic Calculus on Time Scale 1063 Thus, S − T = − i i ti ti+1 −ti ≤ ≤ i ≤ T sup i i ti+1 −ti ≤ ti+1 − ti ≤ We see that T supi ≤ T2 and Further, the continuity i ti+1 − ti ≤ T < of t on the compact set a T implies that supi → as → i ti+1 − ti ≤ with probability Hence, by using Lebesgue’s dominated convergence theorem, we obtain S − T −→ as →0 We now proceed to prove the general case Set n t n Processes t and for any n; limn→ n t n t n ∧n t = t − n t are continuous and monotonically nondecreasing and = a.s Moreover, n t T where S = n T −S ≤ −S n is defined by (2.4) for the process 0≤ S −S n n T = n T + n t + S −S n t ≤n n From the relation ≤ T T + n T >n it follows that lim sup →0 T →0 = Letting n → n T − S ≤ lim sup n T + T lim T −S T n + T T >n >n we get →0 −S =0 The proof is complete Remark 2.6 T i) If < then relationship (2.5) holds in the sense of convergence in ii) = the above theorem is the content of Theorem 21 in [5] iii) The conclusion of Theorem 2.5 is still valid if in the expression(2.4), a = t0 < t1 < · · · < tk = T is an arbitrary partition of T , provided maxi ti+1 − ti ≤ for i = k − 1064 Du and Dieu We recall the Dunford–Pettis theorem in [10] Theorem 2.7 (Dunford–Pettis [10]) If Yn n∈ is uniformly integrable sequence of random variables, there exists an integrable random variable Y and a subsequence Ynk k∈ such that weak − limk→ Ynk = Y , that is, for all bounded random variables we have Ynk = lim k→ Y Lemma 2.8 If Yn n∈ is integrable sequence of random variables on probability space , which converges weakly in to an integrable random variable Y , then for each -field ⊂ the sequence Yn converges weakly to Y in Proof For an arbitrary bounded random variable = Yn = on we have = Yn Yn = Yn Yn On the other hand, lim = Yn n→ = Y Y Hence, lim n→ Yn = Y The proof is completed Definition 2.9 A process X is said belong to class DL if for any t ∈ a the set is a stopping time satisfying a ≤ ≤ t X is uniformly integrable Theorem 2.10 (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 ∈ a a.s If A is natural then M and A are uniquely determined up to indistinguishability Proof We firstly prove the 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 ∈ a as Stochastic Calculus on Time Scale 1065 This relation implies that Bt = At − At = Mt − Mt is right continuous martingale For each partition n a = t0 < t1 < · · · < tkn = t such that maxi −n ≤ of a t set Bs n = Ba a + ti+1 − ti kn −1 Bti ti ti+1 i=0 Then, Bs− = lim Bs n ∀s ∈ a t n→ By (2.1) and the bounded convergence theorem we have Bt At − At = at B− A − kn = lim n→ at B− A Bti−1 Bti − Bti−1 =0 i=1 Thus, Bt At − At = At − At = which implies At − At = a.s for all t ∈ a These inequalities imply that At = At a.s for any t ∈ a Next, we prove the existence By uniqueness, it suffices to prove the existence of the processes M and A on the interval a b for fixed b ∈ a Without loss of generality n n we may assume that Xa = Consider a sequence of partitions n a = t0 < t1 < n n n −n n n+1 · · · < tkn = b such that maxi ti+1 − ti ≤ of a b and ⊂ Consider the Doob–Meyer decomposition of the submartingale X n = Xtj tj ∈ n n n Xtj = Mtj + Atj n where Atj = ji=1 Xti − Xti−1 n n Mtj = Xtj − Atj Further, n Mtj = is an ti−1 Mb j=0 n tj = (2.6) kn kn tj j=0 –previsible increasing process and n Xb − Ab tj n It is easy to see that if X belong to class DL then Ab n = is uniformly integrable Therefore, by Dunford–Pettis theorem, there is a subsequence n Ab k k∈ weakly converging to an integrable random variable Ab We define the processes M and A by Mt = Xb − A b t At = Xt − Mt ∀t ∈ a b In this definition we take a right continuous version of the martingale Mt and the n n process At Since weak − limk→ Ab k = Ab , it follows that weak − limk→ Mb k = Mb 1066 Du and Dieu Therefore, by Lemma 2.8, weak − lim Mb k→ for any sub -field Let = n∈ n nk = Mb of and a ≤ s ≤ t ≤ b with s t ∈ At − As = Xt − Xs − Mb − t Mb = Xt − Xs − weak − lim Mb k→ be fixed It is easy to see that s nk − t = weak − lim Xt − Xs − Mb nk = weak − lim Xt − Xs − Mt nk − Ms nk k→ k→ = weak − lim At nk − Asnk k→ Mb − t nk Mb s nk s ≥0as Since is countable and dense with respect to a b and A is right continuous, At ≥ As a.s for all t > s in a b This means that A is increasing Next, we check A to be natural Let t be any bounded martingale Put n s kn = ti−1 ti−1 ti s i=1 As is seen s− n = lim ∀s ∈ a b s n→ Further, ab s− n As = lim n→ ab ab s ti−1 n→ For a fixed n, we can find a sequence mk ↑ n kn As = lim s such that kn As = lim mk ti−1 mk → Ati − Ati−1 i=1 m − Ati−1k A ti i=1 By the predictability of A mk , kn mk ti−1 A ti m − Ati−1k = i=1 kn mk − Ati−1k n As = A ti b m = i=1 Thus, ab s− As = lim n→ ab s b Ab mk b Ab Stochastic Calculus on Time Scale 1067 Hence, As = s− ab b Ab i.e., A = At is natural The proof is complete Denote by the set of the square integrable t -martingales and consider M ∈ Since M is a submartingale, there exists uniquely a natural increasing process M = M t t∈ a such that Mt2 − M t is an t -martingale The natural increasing process M t is called characteristic of the martingale M Proposition 2.11 For any T ∈ a , let ti be defined by (2.3) Suppose that the characteristic of a martingale M is continuous Then, M T = lim →0 kn Mti − Mti−1 ti−1 i=1 Proof By assumption, M t is a natural increasing continuous process Therefore, by virtue of Theorem 2.5, we have M T = lim S →0 On the other hand, it is easy to verify that S = kn Mti − Mti−1 ti−1 i=1 The proof is complete The Ito–Stieltjes Integrals Denote by the set of all real-valued stochastic processes = t t∈ a defined on with the left continuous paths on a and a × t -adapted Let be the -algebra of the subsets of a × generated by the stochastic processes in Definition 3.1 Every set in the -algebra is called predictable A process to be predictable if it is measurable with respect to is said It is easily to see that is generated by the family of sets s t × F s t ∈ a s < t F ∈ s We note that in general a left continuous process is not necessarily predictable Remark 3.2 i) If = then the process t is predictable if t is t−1 -measurable ii) If = then the process t is predictable if it is understood by stochastic process, measurable with respect to -algebra generated by adapted left continuous processes 1068 = Du and Dieu Let M ∈ t t∈ a Denote by satisfying M the space of all real-valued, predictable processes 2 tM = M < at for any t > a Fix a b > a and let a b M be the restriction of We endow a b M with the norm bM = 2 M on a b M ab and identify and in a b M if − b M = A process defined on a b is called simple if there exist a partition of a b a = t0 < t1 < · · · < tn = b and bounded random variables fi such that fi is ti−1 measurable for all i = n and n t = fi t∈ a b t ti−1 ti (3.1) i=1 Denote the set of the simple processes by Lemma 3.3 is dense in d = a b M with respect to the metric − bM = − ab M Proof Clearly, ⊂ a b M Take ∈ a b M and set K t = K t Then ∈ a b M and − K b M → as K → + t −K K Thus, it is enough to show that for any bounded ∈ a b M we can find n ∈ n=1 such that − n b M → as n → Let = ∈ n ∈ is bounded and there exist a b M − such that n bM → as n → is linear space and it is easy to see that if n ∈ Let ∈ Set K > and n ↑ then ∈ n t = ti Definition 3.4 For a simple process and call it the < K for some constant if t ∈ ti ti+1 for i = kn − where ti is a partition of a b such that n n ∈ and − b M → as n → From [9, Proposition I-5.1, p 21] we can predictable processes Thus, = a b ab n M = -stochastic integral of maxi ti+1 − ti ≤ 2−n It is clear that by the bounded convergence theorem conclude that contains all bounded M The proof is complete with the form (3.1) in kn fi Mti − Mti−1 , define (3.2) i=1 with respect to the martingale M on a b Stochastic Calculus on Time Scale It is easy to prove that the -stochastic integral and the following relations hold Lemma 3.5 For any ∈ 1069 M is ab b -measurable , there hold M =0 ab (3.3) M ab find = (3.4) M ab Next, we define the integration for ∈ a b M By Lemma 3.3 we can n ⊂ such that − n b M → as n → Since M − n ab m = M ab m − n bM n M is a Cauchy sequence In consequences, it converges in to a random variables with < This limit does not depend n on the choice of the sequence This leads to the following definition ab Definition 3.6 The -stochastic integration of a process ∈ a b M with respect to the square integrable martingale M on a b , denoted by a b M , is defined by M = lim n n→ ab where the limit in the sense of processes satisfying ab (3.5) n is a sequence of simple and − lim n→ M ab n M =0 Example 3.7 i) Let = Further, is in iff n is n−1 measurable ∀n ∈ ∗ , that is, it is previsible b M = ab ii) If = then Mi − Mi−1 a b M consists of all predictable processes and ab where the integral i i=a+1 b a M = b dM a dM is defined in [13] 1070 Du and Dieu The stochastic integration has the following usual properties Proposition 3.8 Let ∈ following relations hold (i) (ii) ab ab a b M and let be two real numbers Then, the M is b -measurable; M = 0; M (iii) ab (iv) a b + (vi) If is a real and 2 = M ; ab M = M + M a.s ab ab -measurable bounded random variable, then a M = ab ∈ and a b M (3.6) M ab Proof Those above properties are true for obtain them by taking limits Theorem 3.9 Let M ∈ ∈ in For in a b M we a b M Then M a ab =0 as (3.7) M a ab = M a ab as Proof The formula (3.7) follows directly from the definition of integral and (3.6) Further, A ab -stochastic = M (3.8) ab = M A = M A ab A M ab Thus, A = M ab A ∀A ∈ M ab that is, M ab (3.8) is proved a = ab M a as a Stochastic Calculus on Time Scale ∈ Definition 3.10 Let 1071 a b M For each t ∈ a b we define I a =0 It = ab ∀a < t ≤ b M at and call it the indefinite -stochastic integral of We often write a t M for I t with respect to the martingale M the indefinite -stochastic integral Theorem 3.11 For any ∈ a b M I t t∈ a b is an t -square integrable martingale In particular, sup ≤4 M at a≤t≤b (3.9) M ab Proof Clearly, I t t∈ a b is square-integrable The martingale property of I t follows from (3.7): for a ≤ s < t ≤ b we have It = s Is + s M =I s s st The inequality (3.9) is deduced from Doob’s martingale inequality (see [9, p 28]) Let n t t∈ be an adapted, cadlag process satisfying a be a certain partition of a b We define the process n = t kn ti−1 ti−1 ti t M t t ab n sampled at t < and n to be ∀t ∈ a b i=1 Since limn→ n t = t− a.s for every t ∈ a b , we can define, Consider ab Am = t M = n lim n→ ab ab − (3.10) M is a cadlag process but the assumption the case when t m M t < does not satisfy For any m ∈ , let t = t · t ≤m Put m ∀t ∈ a b Since t is cadlag, Am ↑ Further, if m1 < m2 then t = t t = t m1 = m By virtue of the boundedness of m = ab m − t t ∀t ∈ a b − lim n→ kn m ti−1 i=1 Denote mn = ∈ Am1 , we can define = M m2 kn i=1 m ti−1 Mti − Mti−1 Mti − Mti−1 (3.11) 1072 Du and Dieu Since, mn converges to any > 0, Am ∩ mn − m m ≥ in 2, it converges to ≤ Am mn m − in probability Therefore, for ≥ m → as n → From (3.11) we see that if m1 < m2 then m1 = m2 for all ∈ Am1 , except possibly for a null set Thus, on the set Am , m = limk→ k = and mn = kn i=1 ti−1 Mti − Mti−1 Hence, Am ∩ mn − Paying attention Am ↑ ≥ ≤ Am mn − ≥ m → as n → , we see that kn ti−1 Mti − Mti−1 i=1 converges in probability to Thus, we can define ab − M = lim m m→ = − lim m→ m (3.12) M ab Remark 3.12 To simplify the presentation, we define the integration only for a martingale M ∈ However, it is easy to deal with the definition of stochastic integration for a semimartingale which can be decomposed as the sum of a bounded variation process and a square integrable martingale (square semimartingale for short) We come to the definition of the quadratic co-variation Let X Y be two stochastic processes defined on a For each t ∈ a we consider the partition n n n n n n a = t0 < t1 < · · · < tkn = t satisfying maxi ti+1 − ti ≤ 2−n of a t and put Bn t = kn Xt n − X t n i i=1 i−1 Yt n − Y t n i i−1 Definition 3.13 The quadratic co-variation of X and Y on the interval a t is the limit (as n → ) in the sense of convergence in probability of the sums Bn t , provided this limit exists The quadratic co-variation of the process Xt and Yt on the interval a t is X t is called the quadratic variation of denoted by X Y t If X = Y then X X t X, that is X t = lim n→ kn i=1 Xt n − X t n i Proposition 3.14 If M and N are martingales in always exists and it satisfies the relation Mt Nt = Ma Na + at M− N + i−1 at 2, then the quadratic co-variation N− M + M N t (3.13) Stochastic Calculus on Time Scale 1073 Proof By calculation kn Mt Nt − Ma Na = M t n Nt n − M t n Nt n i i=1 kn = i i−1 Mt n − M t n i i=1 + kn i−1 i−1 Nt n − N t n i i−1 Nt n M t n − M t n i i−1 i=1 kn + i−1 Mt n Nt n − N t n i i−1 i=1 i−1 From (3.12) we get kn − lim n→ Nt n M t n − M t n = Mt n N t n − N t n = i i−1 i=1 i−1 at N− M and kn − lim n→ i i−1 i=1 i−1 at M− N Thus, there exists the limit − lim n→ kn Mt n − M t n i i=1 i−1 Nt n − N t n i i−1 = M N t and the following relation holds Mt Nt = Ma Na + at M− N + N− M + M N at t So, we get (3.13) It is seen from Theorem 3.11 that if M ∈ Yt = is a martingale in X M at and X ∈ ∀t ∈ M then the process (3.14) a Lemma 3.15 (Associativity) Let M ∈ Then, ab H Y = X∈ a b M and let H ∈ HX M a b Y (3.15) ab Proof It is clear that if X H ∈ then (3.15) holds Suppose that the sequence n Xt t∈ a b in converges in the norm · b M to Xt t∈ a b By definition, 1074 Du and Dieu n Yt = a t X n M converges to Yt = since H ∈ , it is easy to verify that ab at H Y = lim n→ = lim n→ = · Let H n converge to H in Therefore, ab bY X M in H Y for each t ∈ a b Further, n ab HXn M ab HX M ab then ab H Y = lim n→ = lim n→ = H n X converge to H n ab H X in · b M Y ab HnX M ab HX M ab The proof is complete Let t ∈ a and G → be a continuous function Consider a partition a = t0 < t1 < · · · < tkn = t such that maxi ti+1 − ti ≤ 2−n of a t Set Sn t = kn G Mti−1 Mti − Mti−1 n i=1 In order to establish Ito’s formula, we need the following lemma Lemma 3.16 For any M ∈ we have lim Sn t = n→ at G M− M where the limit is in probability sense Proof Define stopping times m = inf t Mt ≥ m Then, the stopped martingale Mt∧ m is bounded by m Further, because Ms is pathwise bounded with probability 1, it follows that if Lemma 3.16 is true for Mt∧ m for each m, it is true for M as well by taking the limit as m → Therefore, we assume that M is bounded Hence, M − M is a martingale in From (3.13) and Lemma 3.15, it yields at G M− M2 − M =2 at G M− M− M Stochastic Calculus on Time Scale 1075 By (3.10) this equation is equivalent to at G M− = lim n→ = lim n→ M kn G Mti−1 Mt2i − Mt2i−1 − i=1 kn kn G Mti−1 Mti−1 Mti − Mti−1 i=1 G Mti−1 Mti − Mti−1 i=1 The proof is complete Theorem 3.17 (Ito’s Formula) Let f ∈ C f Mt = f Ma + + at f M− M + and M ∈ f M− at f Ms − f Ms− − f Ms− ∗ s∈ a t Where ∗ For all t ∈ a, then M M s − f M s− ∗ Ms Ms = Ms − Ms− for all s ∈ a t Proof Taylor’s Theorem says that for f ∈ C defined on compact set we have f y −f x =f x y−x + f x y−x 2+R x y where R x y ≤ r y − x y − x such that r function with limu↓0 r u = Define stopping times m = inf t + → + is an increasing Mt ≥ m Then, the stopped martingale Mt∧ m is bounded by m and if Ito’s formula is valid for Mt∧ m for each m, it is valid for M as well Therefore, we assume that M is bounded Let > Since the discontinuous points of the martingale M is at most countable and the series M s − M s− < s∈ a t we can split the set of jumps on a t of M into two classes C1 is a finite set and C2 is the set of jumps for which Ms − Ms− ≤ s∈C2 Consider a partition n a = t0 < t1 < · · · < tkn = t such that maxi 2−n of a t Since C1 is finite and as M is cadlag, lim n→ C1 ∩ ti−1 ti =∅ f Mti − f Mti−1 = f Ms − f Ms− s∈C1 ti+1 − ti ≤ (3.16) 1076 Du and Dieu To simplify notation, we denote C1 ∩ ti−1 ti =∅ by and C1 ∩ ti−1 ti =∅ by Adding up the increments of other intervals and by Taylor’s formula f Mti − f Mti−1 = f Mti−1 Mti − Mti−1 + = kn f Mti−1 Mti − Mti−1 + i=1 2 f Mti−1 Mti − Mti−1 + kn R Mti−1 Mti f Mti−1 Mti − Mti−1 i=1 f Mti−1 Mti − Mti−1 + f Mti−1 Mti − Mti−1 − 2 + R Mti−1 Mti As C1 is finite, the expression 1 f Mti−1 Mti − Mti−1 + f Mti−1 Mti − Mti−1 2 goes to s∈C1 f Ms− Ms − Ms− + f Ms− Ms − Ms− One can estimate the expression R Mti−1 Mti ≤ r 2 (3.17) R Mti−1 Mti as max C1 ∩ ti−1 ti =∅ Mti − Mti−1 Mti − Mti−1 2 Therefore, lim sup n→ R Mti−1 Mti ≤ r + lim sup n→ ≤r + M Mti − Mti−1 ti ∈ n t Letting ↓ we see that this expression goes to zero and the difference of (3.16) and (3.17) goes to f Ms − f Ms− − f Ms− a