# Martingale Decomposition and Backward Stochastic Dynamic Equations on Time Scales

Authors: Guofeng Tang; Guangyan Jia
Journal: Journal of Mathematics
 (2022)

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## Abstract

The paper aims to establish the related backward stochastic dynamic equations on time scales, BS∇ Es for short, concerning to ∇-integral on time scales. We present the martingale decomposition theorem on time scales and prove the existence and uniqueness theorem of solutions to BS ∇ Es. This work can be considered as a unification and a generalization of similar results in backward stochastic difference equations and backward stochastic differential equations.


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## Body

## 1. Introduction

In 1988, Hilger [1] introduced the calculus of measure chains to unify continuous and discrete analysis. Since then, this topic, mainly the deterministic analysis, has attracted much attention [2–4]. For the stochastic calculus on time scales, a lot of work [5–8] focused on Δ-integration on the semi-open intervals of the form ti,ti+1. Meanwhile, by the requirement of the predictable integrand (for the martingale property of stochastic integral), it is easy to consider semi-open intervals ti,ti+1. Du and Dieu [9, 10] established the stochastic calculus on time scales for the ∇ case. Some basic problems such as stochastic integration, Doob-Meyer decomposition theorem, Itô’s formula, and stochastic differential equations on time scales have been studied carefully. Zhu [11] studied stochastic optimal control problems on time scales. The corresponding adjoint equations were backward stochastic dynamic equations on time scales. However, there are few studies focused on BS ∇ Es, which stimulates us to discover more in the field.The theory of backward stochastic differential equations on continuous-time (BSDEs) is a mature field. The general nonlinear BSDEs were first studied in the Brownian framework by Pardoux and Peng [12]:(1)Yt=YT+∫tTgu,Yu,Zudu−∫tTZudWu,t∈0,TR;YT=ξ.A solution of equation (1), associated with the terminal value ξ and generator gω,t,y,z, is a couple of adapted stochastic processes Yt,Ztt∈0,TR, which satisfy equation (1). It was followed by a long series of contributions; see, for example, [13] for a survey on BSDEs with jumps and applications to finance.The formal studies of discrete counterpart BSDEs focus on the order of convergence as a numerical scheme, rarely the discrete scheme itself. By switching from the continuous-time Brownian motion to discrete-time, we lose the predictable representation property (PRP). It is well known that we need to include in the formulation of the BSDEs on discrete-time additional orthogonal martingales terms [14]:(2)Yt+1−Yt=−gt,Yt,Zt+ZtWt+1−Wt+NT−Nt,t∈ℕ;YT=ξ,T∈ℕ.By the Galtchouk–Kunita–Watanabe theorem, the solution of (2) is a triple tuple Yt,Zt,Ntt∈ℕ, where Ztt∈ℕ is predictable and Ntt∈ℕ is an orthogonal martingale to the integrals w.r.t the driven process Wtt∈ℕ. Bielecki [15] first studied the existence and uniqueness of the solutions of discrete BSDEs (2) by the Galtchouk–Kunita–Watanabe decomposition. For the discrete BSDEs, based on the driving process, there are mainly two formulations (see [16–22]). One is driving by a finite state process taking values from the basis vectors as in [16–18] and the other is driving by a martingale with independent increments as in [15, 19, 20]. We are more interested in the second case. The theory for the discrete-time counterpart of BSDEs is still a developing field.We point out that finding solutions to the BSDEs (1) and (2) is equivalent to finding the martingale representation or martingale decomposition property of the random variable ξ. Before introducing BS ∇ Es, a very natural and fundamental question in the time scales framework is as follows:What is the form of the Martingale Representation Theorem on time scales?We obtain that every square-integral martingaleM∈ℳT2 with M0=0, can be written as follows:(3)Mt=ItX+Nt,t∈T,where ItX denotes the stochastic integral w.r.t. Brownian motion on time scales and Ntt∈T is a square-integrable martingale with N0=0 satisfying a suitable orthogonality condition that we will make more precise later. As an important consequence, we will see that decomposition (3) allows us to construct a solution to special equations driven by the Brownian motion on time scales, of the following form:(4)Yt=ξ+∫tTg0u∇u−∫tTZu∇Wu−NT−Nt,t∈T,where ξ∈L2Ω,ℱT,P denotes the terminal condition, Ztt∈T is predictable, and Ntt∈T is a square-integrable martingale with N0=0, orthogonal to Wtt∈T in a weak suitable sense.This paper aims to establish the existence and uniqueness of solutions to general BS∇ Es as follows:(5)Yt=YT+∫tTgu,Yu,Zu∇u−∫tTZu∇Wu−NT−Nt,YT=ξ.A triplet of processesYt,Zt,Ntt∈T will satisfy the equations (5), where Ztt∈T is predictable and Ntt∈T is orthogonal to the driving processes Wtt∈T. The BS ∇ Es driven by the Brownian motion on time scales are similar to traditional BSDEs driven by general ca` dla` g martingales beyond the Brownian setting [14, 23, 24].The paper is organized as follows. In Section2, we introduce basic notations of analysis on time scales. In Section 3, we first give some stochastic notations and results on time scales and then prove the martingale decomposition theorem on time scales. Section 4 is devoted to obtain the existence and uniqueness of solutions of BS ∇ Es which is our main result. Also, finally, in Section 5, we apply BS ∇ Es to financial hedging problems.


## 2. Preliminaries

Atime scaleT is a nonempty closed subset of the real numbers R. The distance between the points t,s∈T is defined as the normal distance on R: t−s. In this paper, we always suppose T is bounded with 0=infT,T=supT>0. The forward jump operator σ and backward jump operator ρ are, respectively, defined by the following:(6)σt=infs∈T:s>t,(7)ρt=sups∈T:s<t.We say thatt is right-scattered (left-scattered, right-dense, left-dense), if σt>t (ρt<t, σt=t, ρt=t) holds. μt=σt−t is called graininess, νt=t−ρt is called backward graininess. Consider the following:(8)I=t:tis left−scattered.The set of left-scattered points of a time scale is at most countable.Fora,b∈T with a≤b, define the closed interval in T by a,bT=t∈T:a≤t≤b. Other types of intervals are defined similarly. We introduce the set Tk if T has a right-scattered minimum t2, then Tk=T−t2, otherwise Tk=T. If t∈Tk, the ∇-derivative of f at the point t is defined to be the number f∇t (provided it exists) with the property that for each ε>0, there is a neighborhood U (in T) of t such as follows:(9)fρt−fs−f∇tρt−s≤ερt−s,∀s∈U.Now, suppose thatf:T⟶R. Continuity of f is defined in the usual manner. A function f is called right-dense continuous (rd-continuous) on T if and only if it is continuous at every right-dense point and the left-sided limit exists at every left-dense point. Denote limσs↑tfs by ft− and limρs↓tfs by ft+, respectively, if limits exist. It is to be noted that on the right-scattered points, ft≠ft+ for continuous functions on time scales. If t is left-scattered, then ft−=fρt, right-scattered, then ft+=fσt.LetA be an increasing right-continuous function of finite variation defined on T. We denote μ∇A as the Lebesgue ∇-measure associated with A. For any μ∇A-measurable function f:T⟶R, we write ∫0tfs∇As for the integral of f w.r.t. the measure μ∇A on 0,t. It is seen that the function t↦∫0tfs∇As is c a` dla` g. For details, please refer to [9].For any continuous function on time scalesp:T⟶R, with 1+ptμt≠0 for all t∈Tk, the Δ-exponential function ept,t0, defined by ([2], Definition 1.38), is the solution y⋅ of the initial value problem:(10)y∇t=pt−yt,∀t>t0,t∈T,yt0=1,and epσt,t0=1+μtptept,t0, ept,t0=1+νtpt−ept−,t0. Denote e⊖pt,s=1/ept,s.


### 2.1. Stochastic Calculus on Time Scales

LetRk be the k-dimensional Euclidean space, equipped with the standard inner product ⋅,⋅, and the Euclidean norm ⋅, Rk×d be the collection of all k×d real matrices, and z=zijk×d be the matrix with zi≔zi1,…,zid⊤ and z≔trzz⊤, where z⊤ represents the transpose of z. EXt|ℱs or Eℱs is denoted as the conditional expectation w.r.t. the filtration ℱs. Assume that we are working on a probability space Ω,ℱ,F=ℱtt∈T,P with the filtration ℱtt∈T satisfying the usual conditions.Denoted byℳT2, the space of square integrable martingales with EMT2<∞ and consider M∈ℳT2. Since M2 is a submartingale, following the Doob–Meryer decomposition theorem on time scales [9], there exists uniquely a natural increasing process M=Mtt∈T, such that Mt2−Mt is an F-martingale. The natural increasing process Mt is called characteristic of the martingale M.Define the quadratic co-variationM,Nt of two processes similar to ( [9], Definition 3.13). If N=M we write Mt for M,Mt and call it the quadratic variation of M. For partitions s=t0n<t1n<⋯<tin<⋯<tnn=t of s,tT with maxiρti−ti−1≤2−n,(11)EMt−Ms2|ℱs=E∑iMti−Mti−12|ℱs→n→∞EMt−Ms|ℱs.It meansM2−M is an F-martingale, which implies that M−M is also an F-martingale.The one-dimensional Brownian motion is given in [25]. In view of the fact that the multidimensional Brownian motion can be constructed through the classical product space [26], we give a result about multidimensional Brownian motion on time scales.Lemma 1.
LetWt=Wt1,Wt2,…,Wtd,ℱt;t∈T be a d-dimensional Brownian motion. The processes are as follows:(12)Mti=Wti−W0i,ℱt;t∈T,1≤i≤d,are continuous, square-integrable martingales, with the following:(13)Mi,Mj=tδij;1≤i,j≤d,Mi,Mj=δijλt0,t∩T+∑t0≤an<bn≤tWbn−Wan2,∀t∈T,where λ is the classical Lebesgue measure and ∪n=1∞an,bn=t0,∞∖T is the expression for the open subset t0,∞∖T of R as the countable of disjoint open intervals [27]. Furthermore, the vector of martingales M=M1,…,Md is independent of ℱ0.Remark 1.
The Lévy martingale characterization of Brownian motion fails on time scales, that’s a continuous martingale withXi,Xj=tδij, cannot be a Brownian motion.The stochastic integral on time scales in this paper is based on [9], in which the authors established the stochastic ∇-integral w.r.t. square integral martingales and extended to special semimartingales [10]. Consider the integral w.r.t Brownian motion W on time scales: let ℒT20,t;W be the space of all real-valued, predictable processes ϕ=ϕtt∈T satisfying the following:(14)ϕt,W2=E∫0tϕs2∇Ws=E∫0tϕs2∇s<∞.Based on [[9], Definition 3.6], define the integral as follows:(15)ItX=∫0tXs∇Ws,X∈ℒT20,t;W.The spaceℒT20,t;W is actually the L2 space under the measure given by ∇W×dP. Now, let us define the multidimensional Stochastic Integral:Definition 1.
LetWt,t∈T be a d-dimensional Brownian motion on time scales, Xt,t∈T is k×d matrix process, ∀1≤i≤k,1≤j≤d, Xi,j∈ℒT20,t;Wj. Define the following:(16)It,kX=∫0tXs∇Ws=∫0tX11⋯X1d⋮⋮Xn1⋯Xnd∇Ws1⋮∇Wsd,to be the multidimensional stochastic integral for X, also ItX for short. The i-th component of ItX is as follows:(17)ItiX=∑j=1d∫0tXsij∇Wsj,1≤i≤k,t∈T.Clearly it can be seen thatItX belongs to ℳT,k2, the Rk-valued square-integral martingale on time scales. For more properties about the stochastic integral, readers could see [9, 10]. Now, we list the Itô’s formula on time scales [10].Theorem 1.
Letf∈C1,2T×Rd and X=X1,…,Xd be a d-dimensional semimartingales defined by the following:(18)Xti=X0i+∫0tφsi∇s+∫0tϕsi∇Wsi,where ∫0tφsi∇s<∞, a.s. and ϕi∈ℒT20,t;W. For all t∈T, then ft,Xt is a semimartingale and the following formula holds:(19)ft,Xt=ft0,X0+∫0,tf∇s,Xs−∇s+∑i=1d∫0,t∂f∂xis,Xs−∇Xis+12∑i,j∫0,t∂2f∂xixjs,Xs−∇Xi,Xj+∑s∈0,tfs,Xs−fs,Xs−−∑s∈0,t∑i=1d∂f∂xis,Xs−∇∗Xis−12∑s∈0,t∑i,j∂2f∂xixjs,Xs−∇∗Xis∇∗Xjs=ft0,X0+∫0,tf∇s,Xs−∇s+∑i=1d∫0,t∂f∂xis,Xs−1−1Isφsi∇s+∑i=1d∫0,t∂f∂xis,Xs−ϕsi∇Wsi+12∑i,j∫0,t∂2f∂xixjs,Xs−ϕsiϕsj∇Wi,Wj+∑s∈0,tfs,Xs−fs,Xs−−∑s∈0,t∑i=1d∂f∂xis,Xs−ϕsi∇∗Wsi−12∑s∈0,t∑i,j∂2f∂xixjs,Xs−ϕsiϕsj∇∗Wsi∇∗Wsj,where ∇∗Xsi=Xsi−Xs−i=φsiνt+ϕsiWsi−Ws−i for all s∈0,tT.


### 2.2. Summary of Notations

For readers’ convenience, we collect some spaces and notations used in the paper. For any integerk,Lk2ℱT, short for L2Ω,ℱt,P;Rk: the space of Rk-valued random vectors X that are ℱt-measurable and satisfy EX2<∞,ℒT,k×d2, short for ℒT20,t;Rk×d: the set of ℱtt∈T-predictable, Rk×d-valued integrands processes satisfying E∫0TXs2∇t<∞,ℳT,k2,c: the continuous Rk-valued square-integral martingales on time scales with M0=0P−a.s,ℳT,k2,∗: the subset of ℳT,k2,c, such that for each M∈ℳT,k2,∗, there exists X∈ℒT,k×d2, and Mt=ItX,ST2, short for ST20,T;Rk: the set of Rk valued, adapted and continuous processes ϕtt∈0,TT such that Esupt∈0,TTϕt2<∞,ℳT,k2,⊥: subset of ℳT,k2,c, the set of all the k-dimensional martingales, such that each martingale is orthogonal to that in ℳT,k2,∗.


## 2.1. Stochastic Calculus on Time Scales

LetRk be the k-dimensional Euclidean space, equipped with the standard inner product ⋅,⋅, and the Euclidean norm ⋅, Rk×d be the collection of all k×d real matrices, and z=zijk×d be the matrix with zi≔zi1,…,zid⊤ and z≔trzz⊤, where z⊤ represents the transpose of z. EXt|ℱs or Eℱs is denoted as the conditional expectation w.r.t. the filtration ℱs. Assume that we are working on a probability space Ω,ℱ,F=ℱtt∈T,P with the filtration ℱtt∈T satisfying the usual conditions.Denoted byℳT2, the space of square integrable martingales with EMT2<∞ and consider M∈ℳT2. Since M2 is a submartingale, following the Doob–Meryer decomposition theorem on time scales [9], there exists uniquely a natural increasing process M=Mtt∈T, such that Mt2−Mt is an F-martingale. The natural increasing process Mt is called characteristic of the martingale M.Define the quadratic co-variationM,Nt of two processes similar to ( [9], Definition 3.13). If N=M we write Mt for M,Mt and call it the quadratic variation of M. For partitions s=t0n<t1n<⋯<tin<⋯<tnn=t of s,tT with maxiρti−ti−1≤2−n,(11)EMt−Ms2|ℱs=E∑iMti−Mti−12|ℱs→n→∞EMt−Ms|ℱs.It meansM2−M is an F-martingale, which implies that M−M is also an F-martingale.The one-dimensional Brownian motion is given in [25]. In view of the fact that the multidimensional Brownian motion can be constructed through the classical product space [26], we give a result about multidimensional Brownian motion on time scales.Lemma 1.
LetWt=Wt1,Wt2,…,Wtd,ℱt;t∈T be a d-dimensional Brownian motion. The processes are as follows:(12)Mti=Wti−W0i,ℱt;t∈T,1≤i≤d,are continuous, square-integrable martingales, with the following:(13)Mi,Mj=tδij;1≤i,j≤d,Mi,Mj=δijλt0,t∩T+∑t0≤an<bn≤tWbn−Wan2,∀t∈T,where λ is the classical Lebesgue measure and ∪n=1∞an,bn=t0,∞∖T is the expression for the open subset t0,∞∖T of R as the countable of disjoint open intervals [27]. Furthermore, the vector of martingales M=M1,…,Md is independent of ℱ0.Remark 1.
The Lévy martingale characterization of Brownian motion fails on time scales, that’s a continuous martingale withXi,Xj=tδij, cannot be a Brownian motion.The stochastic integral on time scales in this paper is based on [9], in which the authors established the stochastic ∇-integral w.r.t. square integral martingales and extended to special semimartingales [10]. Consider the integral w.r.t Brownian motion W on time scales: let ℒT20,t;W be the space of all real-valued, predictable processes ϕ=ϕtt∈T satisfying the following:(14)ϕt,W2=E∫0tϕs2∇Ws=E∫0tϕs2∇s<∞.Based on [[9], Definition 3.6], define the integral as follows:(15)ItX=∫0tXs∇Ws,X∈ℒT20,t;W.The spaceℒT20,t;W is actually the L2 space under the measure given by ∇W×dP. Now, let us define the multidimensional Stochastic Integral:Definition 1.
LetWt,t∈T be a d-dimensional Brownian motion on time scales, Xt,t∈T is k×d matrix process, ∀1≤i≤k,1≤j≤d, Xi,j∈ℒT20,t;Wj. Define the following:(16)It,kX=∫0tXs∇Ws=∫0tX11⋯X1d⋮⋮Xn1⋯Xnd∇Ws1⋮∇Wsd,to be the multidimensional stochastic integral for X, also ItX for short. The i-th component of ItX is as follows:(17)ItiX=∑j=1d∫0tXsij∇Wsj,1≤i≤k,t∈T.Clearly it can be seen thatItX belongs to ℳT,k2, the Rk-valued square-integral martingale on time scales. For more properties about the stochastic integral, readers could see [9, 10]. Now, we list the Itô’s formula on time scales [10].Theorem 1.
Letf∈C1,2T×Rd and X=X1,…,Xd be a d-dimensional semimartingales defined by the following:(18)Xti=X0i+∫0tφsi∇s+∫0tϕsi∇Wsi,where ∫0tφsi∇s<∞, a.s. and ϕi∈ℒT20,t;W. For all t∈T, then ft,Xt is a semimartingale and the following formula holds:(19)ft,Xt=ft0,X0+∫0,tf∇s,Xs−∇s+∑i=1d∫0,t∂f∂xis,Xs−∇Xis+12∑i,j∫0,t∂2f∂xixjs,Xs−∇Xi,Xj+∑s∈0,tfs,Xs−fs,Xs−−∑s∈0,t∑i=1d∂f∂xis,Xs−∇∗Xis−12∑s∈0,t∑i,j∂2f∂xixjs,Xs−∇∗Xis∇∗Xjs=ft0,X0+∫0,tf∇s,Xs−∇s+∑i=1d∫0,t∂f∂xis,Xs−1−1Isφsi∇s+∑i=1d∫0,t∂f∂xis,Xs−ϕsi∇Wsi+12∑i,j∫0,t∂2f∂xixjs,Xs−ϕsiϕsj∇Wi,Wj+∑s∈0,tfs,Xs−fs,Xs−−∑s∈0,t∑i=1d∂f∂xis,Xs−ϕsi∇∗Wsi−12∑s∈0,t∑i,j∂2f∂xixjs,Xs−ϕsiϕsj∇∗Wsi∇∗Wsj,where ∇∗Xsi=Xsi−Xs−i=φsiνt+ϕsiWsi−Ws−i for all s∈0,tT.


## 2.2. Summary of Notations

For readers’ convenience, we collect some spaces and notations used in the paper. For any integerk,Lk2ℱT, short for L2Ω,ℱt,P;Rk: the space of Rk-valued random vectors X that are ℱt-measurable and satisfy EX2<∞,ℒT,k×d2, short for ℒT20,t;Rk×d: the set of ℱtt∈T-predictable, Rk×d-valued integrands processes satisfying E∫0TXs2∇t<∞,ℳT,k2,c: the continuous Rk-valued square-integral martingales on time scales with M0=0P−a.s,ℳT,k2,∗: the subset of ℳT,k2,c, such that for each M∈ℳT,k2,∗, there exists X∈ℒT,k×d2, and Mt=ItX,ST2, short for ST20,T;Rk: the set of Rk valued, adapted and continuous processes ϕtt∈0,TT such that Esupt∈0,TTϕt2<∞,ℳT,k2,⊥: subset of ℳT,k2,c, the set of all the k-dimensional martingales, such that each martingale is orthogonal to that in ℳT,k2,∗.


## 3. Martingale Decomposition Theorem on Time Scales

In this section, we come back to the martingale decomposition problem on time scales. The fundamental tool on time scales analysis is the countable dense subset. The countable dense subset will play the same role as the dyadic rational numbers that played in the classical analysis from discrete to continuous time. For anyδ>0, consider a partition of 0,TT inductively by letting t0=0 and for i=1,2,⋯, set as follows:(20)ti=supBiifBi≠∅,σti−1ifBi=∅,whereBi=ti−1,ti−1+δT∩0,TT.The partition is given in [3, 9]. On time scales, the size of the interval ti−1,tiT will not converge to zero ti−ti−1↛0 if σti−1=ti. A more specific result was given by David Grow [25]. Now, we provide the optional sampling theorem on time scales to show the basic analysis method on general time scales.Lemma 2 (Optional Sampling Theorem on Time Scales).
IfX is a right-continuous martingale (submartingale) on bounded time scales T with a last element XT and S1,S2 are two bounded stopping times with S1≤S2 on T, then(21)EXS2|ℱS1=XS1≥XS1P−a.s..Proof.
Let0,TT be a time scale, and let Πn=t0,t1,…,tn⊆0,TT be a partition of T, where 0=t0<t1<t2<⋯<tn=T (20). Consider the following sequence of random times:(22)S1nω;Pn=ρti+1ifti≤S1ω<ti+1,and the similarly defined sequences S2n. These are stopping times. For every fixed integer n≥1, both S1n and S2n take on a countable number of values and we also have S1n≤S2n. Therefore, by the discrete optional sampling theorem, we have ∫AXS1ndP≤∫AXS2ndP for every A∈ℱS1n. S1≤S1n implies ℱS1⊂ℱS1n, the preceding inequality also holds for every A∈ℱS1.
The discrete martingale results show that the sequence of random variablesXS1n is uniformly integrable, and the same is of course true for XS2n. XS2=limn⟶∞XS2nω and XS1=limn⟶∞XS1nω hold for a.e. ω∈Ω. It follows from uniform integrability that XS1, XS2 are integrable and that ∫AXS1dP≤∫AXS2dP holds for every A∈ℱS1.Via the identityMt=EMT|ℱt, each M∈ℳT2 can be identified with its terminal value MT∈L2Ω,ℱT,P (in general the terminal variables can be extended to M∞ if exists). ℳT2 becomes a Hilbert space isomorphic to L2Ω,ℱT,P, if endowed with the following inner product:(23)M,NℳT2≔EMTNT,MℳT2=MTL2.Indeed, ifMn is a Cauchy sequence for ⋅ℳT2, then the sequence MTn is Cauchy in L2Ω,ℱT,P and so goes to a limit MT in this space; then if M is the martingale with terminal variable MT, it belongs to ℳT2 and Mn−M⟶0. The set of all continuous elements of ℳT2 denoted by ℳT2,c, is a closed subspace of the Hilbert space ℳT2. Now, define a measure μp on T×Ω,ℬT⊗ℱ by the following:(24)μpA=E∫T1As,ω∇Ws.Similar to Lemma 2.2 chapter 3 in [26], we have the following:Lemma 3.
TheℒT,k×d2 space is a closed space with the norm ⋅T,W.Proof.
We define a Hilbert spaceℋT=L2T×Ω,ℬT⊗ℱT,μp. Obviously, ℒT,k×d2 is a subspace of ℋT. Now, we prove that it is closed. Suppose that Xmm=1∞ is a convergent sequence in ℒT,k×d2 with limit X∈ℋT−. Thus, the sequence has a convergent subsequence which converges almost surely under μp, also denoted by Xmm=1∞. Therefore μpt,ω:limn⟶∞Xtmω≠Xtω;t∈T=0, X∈ℋT−; thus, X is ℬT⊗ℱ-measurable.
Restricted on0,tT for 0<t≤T, repeating the above procedure, by the uniqueness of convergence, we can get that X is ℬ0,tT⊗ℱt−-measurable. Therefore, X is predictable and belongs to ℒT,k×d2. The proof is complete.Now, define an inner product onℒT,k×d2 by X,Y1=E∫0tXs,Ys∇s. Recall the inner product on Lk2ℱT, I1,I22=EI1I2. Consider the mapping X↦ITX from ℒT,k×d2 to Lk2ℱT. From the definition of the stochastic integral on time scales, the mapping is injective. This mapping preserves following inner products:(25)X,Y1=E∫0TXt,Yt∇t=EITXITY=ITX,ITY2.DenoteℛkℱT≜ITX;X∈ℒT,k×d2. Since any convergent sequence in ℛkℱT is also Cauchy, its preimage sequence in ℒT,k×d2 must have a limit in ℒT,k×d2. It follows that ℛkℱT is closed in Lk2ℱT. Let us denote by ℳT2,∗ the subset of ℳT,k2,c which consists of stochastic integrals ItX=∫0tXs∇Ws;0≤t≤T,t∈T, of processes X∈ℒT,k×d2:(26)ℳT,k2,∗≜IX;X∈ℒT,k×d2⊆ℳT,k2,c⊆ℳT,k2.The following result is the “fundamental decomposition theorem” for the martingales w.r.t Brownian motion on time scales.Theorem 2.
For everyM∈ℳT,k2, with M0=0,P−a.s., we have the following decomposition:(27)Mt=ItX+Nt,∀t∈T,where X∈ℒT,k×d2, ItX∈ℳT,k2,∗,N∈ℳT,k2 with N0=0 and N is orthogonal to every element of ℳT,k2,∗.Proof.
We have to show the existence of a processY∈ℒT,k×d2 such that Mt=ItY+Nt, where N∈ℳT,k2 has the property(28)IX,Nt=0,∀X∈ℒT,k×d2.
Such a decomposition is unique (up to indistinguishability); indeed, if we haveM=IY′+N′=IY″+N″ with Y′,Y″∈ℒT,k×d2 and both N′ and N″ satisfy the property, then(29)Z≜N″−N′=IY′−Y″,is in ℳT,k2 with Z0=0 and Z=Z,IY′−Y″=0. Then PZt=0,∀t∈T=1 from [Lemma 2]. The decomposition is unique up to indistinguishability.
SinceℛkℱT is a closed subspace of Lk2ℱT, we can denote its orthogonal complement by ℛk⊥ℱT. The random variable MT is in Lk2ℱT, so it admits the following decomposition:(30)MT=ITY+NT,where Y∈ℒT,k×d2 and NT∈Lk2ℱT satisfies ENTITX=0;∀X∈ℒT,k×d2. We construct a martingale through NT by Nt=ENT|ℱt. Obviously N∈ℳT,k2. Taking conditional expectation under ℱt on MT, we obtain the following:(31)Mt=ItY+Nt,t∈T.
It remains to show thatN is orthogonal to every square-integrable martingale of the form IX;X∈ℒT,k×d2, or equivalently, that NtItX,ℱtt∈T is a martingale. It is to be noted that each martingale has a right continuous modification. So, now, we suppose that N is right continuous.
According to Lemma2, we only need to prove the following:(32)ENSISX=EN0I0X=0.holds for every stopping time S of the filtration ℱtt∈T, with S<T (since I0X=0). The integral has ISX=ITX˜, where X˜tω=Xtω1t≤Sω is a process in ℒT,k×d2. Therefore, by Lemma 2,(33)ENSISX=EENT|ℱSISX=ENTITX˜=0.
The proof is complete.Remark 2.
(1)
Noteft≠ft− at left-scattered points for continuous function on time scales, N∈ℳT,k2,c is continuous, but ∇Nt≠0 at left-scattered points.(2)
LetT=R,k=1, (Karatzas [26], Proposition 4.14 one-dimension decomposition), ℳR2,c and MR2,∗ actually coincide, the component Nt in the decomposition is actually ∇Nt=0. The predictable process space is isomorphic to the adapted process space [28].(3)
For the Brownian motion on general time scales, even on the augmentation filtration of the filtration generated byW, ℳT,k2,c and ℳT,k2,∗ do not coincide, see the following example.(4)
LetT=ℕ (Follmer, Hans, [29], Theorem 10.18), it is the discrete time version of the Kunita–Watanabe decomposition w.r.t a sequence of normal distribution random variables.(5)
The orthogonality condition given in Theorem2 is the weak orthogonality condition in continuous-time, but we call it strong orthogonality condition on time scales. That is, for scattered points t: EXσt−XtYσt−Yt|ℱt=0.Example 1.
For a martingaleWt2−t on time scales T=0,1∪3,4,5, we have the following:(34)Wt2−t=Wt−Ws2−t−s+2WsWt−Ws+Ws2−s.
Fort=3, M3=I3+N3,I3=∫032Ws∇Ws=∫012Ws∇Ws+2W1W3−W1=W12−1+2W1W3−W1,N3=W3−W12−3−1,
withEI3−I1N3−N1|ℱ1=E2W1W3−W1W3−W12−2|ℱ1=0.


## 4. BS∇ Es Driven by Brownian Motion on Time Scales

In this section, we denoteΩ,ℱ,F,P to be a probability space equipped with a complete filtration F=ℱtt∈T generated by a d-dimensional Brownian motion Wt on time scales, and augmented by all the P-null sets in ℱ.For simplicity, we consider the following general BS∇ Es on time scales:(35)Yt=YT+∫tTgu,Yu,Zu∇u−∫tTZu∇Wu−NT−Nt,YT=ξ.We callg the driver of the BS ∇ Es and the pair ξ,g the data of the BS ∇ Es.A solution to BS∇ E is a triple of process Y,Z,N satisfying (35), such that Y is a Rk-valued, continuous and adapted process, Z is a Rk×d-valued and predictable process and N is a martingale orthogonal to W. For terminal condition ξ and generator g, we make the following assumptions:Assumption 1.
We assume that for any interk,(H1)g is defined as Ω×T×Rk×Rk×d⟶Rk, such that ∀y,z∈Rk×Rk×d, g⋅,y,z is ℱt progressively measurable,(H2)(36)∫0Tg⋅,0,02∇t∈Lk2ℱT,(H3)g satisfies Lipschitz condition w.r.t. y,z: there exists a constant L>0, ∀y,y′∈Rk, z,z′∈Rk×d, s.t.(37)gt,y,z−gt,y′,z′≤Ly−y′+z−z′,(H4)ξ∈Lk2ℱT.Theorem 3.
(Main Result) If Assumption1 is satisfied, BS ∇ E(35) admits a unique triple of solution Y,Z,N∈ST2×ℒT,k×d2×ℳT,k2,⊥.Remark 3.
IfT=R, Theorem 3 degenerates to the classical BSDEs. The deterministic integral part could also be gω,t,Yt−,Zt or gω,t−,Yt−,Zt−. For simplicity, we only consider the case of g⋅,Yt,Zt, which corresponding to the explicit discrete backward difference equations [22].To prove the main theorem, first considerg real-valued and not depend on y,z.Lemma 4.
Assume thatξ∈Lk2ℱT and g0⋅:Ω×T⟶Rk is a progressively measurable and ℱt-predictable process which satisfies E∫0Tg0s2∇s<∞. Then, for BS ∇ E(4) with parameters ξ and g0, there exists a unique triple solution Y,Z,N∈ST2×ℒT,k×d2×ℳT,k2,⊥. Moreover,(38)Yt2+Eℱt∫tTβ2Ys−2+Zs2eβs,t∇s+∫tTeβs,t∇Ns≤Eℱtξ2eβT,t+2βEℱt∫tTg0s2eβs,t∇s,where β is a positive constant with 1+μtβ>0 for all t∈T, so eβt,0>0.Proof.
Existence: We denote Mt=Eℱtξ+∫0Tg0s∇s. Apparently, Mt is a square integrable martingale. Following the martingale decomposition theorem on time scales, there exists a unique predictable process Zt∈ℒT,k×d2 and N∈ℳT,k2,⊥, s.t. Mt=M0+∫0tZs∇Ws+Nt, MT=M0+∫0,TZs∇Ws+NT. Then Mt=MT−∫tTZs∇Ws−NT−Nt. As also MT=ξ+∫0Tg0s∇s, denote Yt=Mt−∫0tg0s∇s=MT+∫tTg0s∇s−∫tTZs∇Ws−NT−Nt. It can be seen that Y∈ST2, we obtain the existence.
Uniqueness: Applying Itô’s formula (Theorem 1) to Ys2eβs,t on t,TT, where Yt=ξ+∫tTg0s∇s−∫tTZs∇Ws−NT+Nt, or the differential form: ∇Yt=−g0t∇t+Zt∇Wt+∇Nt. We have the following:(39)YT2eβT,t=Yt2+∫tTβeβs,tYs−2∇s+2∫tTeβs,tYs−∇Ys+∫tTeβs,t∇Ys+S=Yt2+∫tTβeβs,tYs−2∇s−2∫tTeβs,tYs−g0s∇s+2∫tTeβs,tYs−Zs∇Ws+2∫tTeβs,tYs−∇Ns+∫tTeβs,tZs2∇Ws+∫tTeβs,t∇Ns+S,where(40)S=∑s∈t,Teβs,tYs2−Ys−2−2Ys−Ys−Ys−−Ys−Ys−2=0.
Thus, we have(41)Yt2+∫tTβeβs,tYs−2∇s+∫tTeβs,tZs2∇Ws+∫tTeβs,t∇Ns=ξ2eβT,t+2∫tTeβs,tYs−g0s∇s−2∫tTeβs,tYs−Zs∇Ws−2∫tTeβs,tYs−∇Ns.
Sinceg0s is ℱs-predictable, the above integrals with respect to ∇W and ∇N all belong to ℳT,k2. By taking the expectation with respect to ℱt, we obtain the following:(42)Yt2+Eℱt∫tTβYs−2+Zs2eβs,t∇s+Eℱt∫tTeβs,t∇Ns≤Eℱtξ2eβT,t+Eℱt∫tTβ2Ys−2+2βg0s2eβs,t∇s.
So, we obtain the inequality.
Now, we suppose there are two solutionsY1,Z1,N1 and Y2,Z2,N2. Define δYt=Yt1−Yt2, δZt=Zt1−Zt2, δNt=Nt1−Nt2, then,(43)δYt=−∫tTδZs∇Ws−δNT+δNt,δYT=0.
Then,δY,δZ,δN satisfy BSDE(35) with ξ=0 and g=0. By the inequality,(44)δY02+E∫0Tβ2δYs−2+δZs2eβs,0∇s+∫0Teβs,0∇δNs≤0.
By the continuity ofδYs,δYs=0,δNs=C−δN0=0,P-a.s.. The uniqueness is proved. □Introduce the new norms on spaces:(45)ST2:Yβ,T=E∫0TYs−2eβs,0∇s1/2,ℒT,k2:Zβ,T=E∫0TZs2eβs,0∇s1/2,ℳT,k2,⊥:Nβ,T=E∫0Teβs,0∇Ns1/2,for any positive integer k. Apparently, for each β>0, ⋅β,T is equivalent to ⋅0,T which is the original norm on the corresponding space. Now, we start to prove Theorem 3:Proof.
For any fixedy⋅,z⋅,n⋅∈ST2×ℒT,k×d2×ℳT,k2,⊥, it follows from Lemma 4 that it admits a unique triple solution Y⋅,Z⋅,N⋅∈ST2×ℒT,k×d2×ℳT,k2,⊥ satisfying(46)Yt=ξ+∫tTgs,ys,zs∇s−∫tTZs∇Ws−NT+Nt,∀t∈0,TT.
Hence, we can define an operator as follows:(47)Y.,Z.,N.=Iy.,z.,n.:ST2×ℒT,k×d2×ℳT,k2,⊥⟶ST2×ℒT,k×d2×ℳT,k2,⊥.
We can prove thatI forms a contraction mapping on the Banach space ST2×ℒT,k×d2×ℳT,k2,⊥. Take any y1⋅,z1⋅,n1⋅,y2⋅,z2⋅,n2⋅∈ST2×ℒT,k×d2×ℳT,k2,⊥, we denote the following:(48)Y1,Z1,N1=Iy1,z1,n1,Y2,Z2,N2=Iy2,z2,n2.and δy=y1−y2, δz=z1−z2, δn=n1−n2. By equation (38), we obtain the following:(49)EδY02+E∫0Tβ2δYs−2+δZs2eβs,0∇s+∫0Teβs,0∇δNs≤2βE∫0Teβs,0gs,ys−1,zs1−gs,ys−2,zs22∇s≤2βE∫0Teβs,0Lδys−+Lδzs2∇s≤2βE∫0Teβs,02L2δys−2+2L2δzs2∇s.
We obtain as follows:(50)E∫0Tβ2δYs−2+δZs2eβs,0∇s+∫0Teβs,0∇δNs≤4L2βE∫0Teβs,0δys−2+δzs2∇s.
Letβ=81+L2,(51)δYβ,T2+δZβ,T2+δNβ,T2≤12δyβ,T2+δzβ,T2.
We can obtain that the operatorI is contractive. Hence, in this case, there exists a unique fixed point Y,Z,N∈ST2×ℒT,k×d2×ℳT,k2,⊥ for the map I. The proof is complete.


### 4.1. Linear BS∇ Es

As a particular Lipschitz case, we give a linear case result. First, we cite the exponential martingale on time scales [9].Example 2.
LetM be a bounded martingale satisfying ∇∗Ms=Ms−Ms−≠−1 for any s∈0,TT,s∈T. Set(52)M˜t=Mt−∑s≤t,s,t∈TMs−Ms−,Et=∏s0,t1+∇∗MsexpM˜t−12M˜t.
Then,Et satisfies the Doleans exponential equation(53)Et=1+∫0,tEs−∇Ms.Lemma 5.
Leta,b,c be predictable bounded processes, let E be the exponential martingale of the martingale Mt=∫0tbs∇Ws, define γt=eat,0, Γt=γtE. Suppose thatE is a positive uniformly integrable martingale;EsuptγtE2<∞.
If the linear backward equation(54)−∇Yt=atYt+btZt+ct∇t−Zt∇Wt−∇Nt,YT=ξ,has a solution Y,Z,N, then Y is given by(55)Yt=EξΓTΓt+∫tTΓsΓtcs∇s|ℱt.Proof.
By Girsanov’s theorem on time scales [30], W˜t=Wt−∫0tbs∇s is a Q-martingale, the probability measure Q which has density E with respect to P on ℱT. Under the measure Q, the equation becomes −∇Yt=atYt+ct∇t−Zt∇W˜t−∇Nt,YT=ξ. Applying Itô’s formula to the process γY, we obtain the following:(56)γtYt=γ0Y0+∫0tγsZs∇W˜s+∫0tγs∇Ns−∫0tcsγs∇W˜s+S,where S=∑s∈0,tγsYs−γsYs−−γsYs−Ys−=0. Since ∫0tγsZs∇W˜s and ∫0tγs∇Ns are martingales,(57)γtYt=EQξγT+∫tTcsγs∇s|ℱt,and consequently,(58)ΓtYt=EξΓT+∫tTcsΓs∇s|ℱt.
This concludes the proof.


### 4.2. Comparison Theorem

The linear case allows us to obtain a comparison theorem on time scales.Theorem 4.
LetY1,Z1,N1, Y2,Z2,N2 be the corresponding solutions to the data g1t,Yt1,Zt1,ξ1, g2t,Yt2,Zt2,ξ2 and suppose g1,g2 satisfy conditions (H2) (H3). Assume that ξ1≥ξ2 and satisfy (H4), g1t,Yt2,Zt2≥g2t,Yt2,Zt2a.s.−P. Then, for any t, we have the following:(59)Yt1≥Yt2a.s.−P.Proof.
We defineδYt=Yt1−Yt2, δZt=Zt1−Zt2, δNt=Nt1−Nt2. Then, δY solves the following linear BS ∇ E(60)−∇δYt=atδYt+btδZt+ct∇t−δZt∇Wt−∇δNt,δYT=ξ1−ξ2,where(61)at=g1t,Yt1,Zt1−g1t,Yt2,Zt1δYt1δYt≠0,(62)bt=g1t,Yt2,Zt1−g1t,Yt2,Zt2δYt1δZt≠0,(63)ct=g1t,Yt2,Zt2−g2t,Yt2,Zt2.
According to the linear BS∇ Es, if the coefficients at,bt,ct satisfy the suitable conditions, we have the following:(64)ΓtδYt=EΓTξ1−ξ2+∫tTδcsΓs∇s|ℱt.
Assume now thatξ1≥ξ2, and for any t, ct≥0a.s.−P. Then, for any t, Yt1≥Yt2a.s.−P.Example 3.
ConsiderT=0,1∪2,3, the BS ∇ E:(65)Yt=YT+∫tTgu,Yu,Zu∇u−∫tTZu∇Wu−NT−Nt,YT=ξ.
The solutionYt,Zt,Nt on 2,3 equals to the classical solution on continuous-time with terminal condition Y3=ξ and NT−Nt=0. For right-scattered point t=1,(66)Y1=Y2+∫12gu,Yu,Zu∇u−∫12Zu∇Wu−N2−N1=Y2+g2,Y2,Z2−Z2W2−W1−N2−N2.
In the conditional expectation form:(67)Y1=EY2+g2,Y2,Z2|ℱ1.
Recall (Example1), in a particular case, when Y2+g2,Y2,Z2=W22−2,(68)W22−2=W12−1+2W1W2−W1+W22−W12−1.(69)Y1=W12−1,Z2=2W1,NT−N2=W22−W12−1.


## 4.1. Linear BS∇ Es

As a particular Lipschitz case, we give a linear case result. First, we cite the exponential martingale on time scales [9].Example 2.
LetM be a bounded martingale satisfying ∇∗Ms=Ms−Ms−≠−1 for any s∈0,TT,s∈T. Set(52)M˜t=Mt−∑s≤t,s,t∈TMs−Ms−,Et=∏s0,t1+∇∗MsexpM˜t−12M˜t.
Then,Et satisfies the Doleans exponential equation(53)Et=1+∫0,tEs−∇Ms.Lemma 5.
Leta,b,c be predictable bounded processes, let E be the exponential martingale of the martingale Mt=∫0tbs∇Ws, define γt=eat,0, Γt=γtE. Suppose thatE is a positive uniformly integrable martingale;EsuptγtE2<∞.
If the linear backward equation(54)−∇Yt=atYt+btZt+ct∇t−Zt∇Wt−∇Nt,YT=ξ,has a solution Y,Z,N, then Y is given by(55)Yt=EξΓTΓt+∫tTΓsΓtcs∇s|ℱt.Proof.
By Girsanov’s theorem on time scales [30], W˜t=Wt−∫0tbs∇s is a Q-martingale, the probability measure Q which has density E with respect to P on ℱT. Under the measure Q, the equation becomes −∇Yt=atYt+ct∇t−Zt∇W˜t−∇Nt,YT=ξ. Applying Itô’s formula to the process γY, we obtain the following:(56)γtYt=γ0Y0+∫0tγsZs∇W˜s+∫0tγs∇Ns−∫0tcsγs∇W˜s+S,where S=∑s∈0,tγsYs−γsYs−−γsYs−Ys−=0. Since ∫0tγsZs∇W˜s and ∫0tγs∇Ns are martingales,(57)γtYt=EQξγT+∫tTcsγs∇s|ℱt,and consequently,(58)ΓtYt=EξΓT+∫tTcsΓs∇s|ℱt.
This concludes the proof.


## 4.2. Comparison Theorem

The linear case allows us to obtain a comparison theorem on time scales.Theorem 4.
LetY1,Z1,N1, Y2,Z2,N2 be the corresponding solutions to the data g1t,Yt1,Zt1,ξ1, g2t,Yt2,Zt2,ξ2 and suppose g1,g2 satisfy conditions (H2) (H3). Assume that ξ1≥ξ2 and satisfy (H4), g1t,Yt2,Zt2≥g2t,Yt2,Zt2a.s.−P. Then, for any t, we have the following:(59)Yt1≥Yt2a.s.−P.Proof.
We defineδYt=Yt1−Yt2, δZt=Zt1−Zt2, δNt=Nt1−Nt2. Then, δY solves the following linear BS ∇ E(60)−∇δYt=atδYt+btδZt+ct∇t−δZt∇Wt−∇δNt,δYT=ξ1−ξ2,where(61)at=g1t,Yt1,Zt1−g1t,Yt2,Zt1δYt1δYt≠0,(62)bt=g1t,Yt2,Zt1−g1t,Yt2,Zt2δYt1δZt≠0,(63)ct=g1t,Yt2,Zt2−g2t,Yt2,Zt2.
According to the linear BS∇ Es, if the coefficients at,bt,ct satisfy the suitable conditions, we have the following:(64)ΓtδYt=EΓTξ1−ξ2+∫tTδcsΓs∇s|ℱt.
Assume now thatξ1≥ξ2, and for any t, ct≥0a.s.−P. Then, for any t, Yt1≥Yt2a.s.−P.Example 3.
ConsiderT=0,1∪2,3, the BS ∇ E:(65)Yt=YT+∫tTgu,Yu,Zu∇u−∫tTZu∇Wu−NT−Nt,YT=ξ.
The solutionYt,Zt,Nt on 2,3 equals to the classical solution on continuous-time with terminal condition Y3=ξ and NT−Nt=0. For right-scattered point t=1,(66)Y1=Y2+∫12gu,Yu,Zu∇u−∫12Zu∇Wu−N2−N1=Y2+g2,Y2,Z2−Z2W2−W1−N2−N2.
In the conditional expectation form:(67)Y1=EY2+g2,Y2,Z2|ℱ1.
Recall (Example1), in a particular case, when Y2+g2,Y2,Z2=W22−2,(68)W22−2=W12−1+2W1W2−W1+W22−W12−1.(69)Y1=W12−1,Z2=2W1,NT−N2=W22−W12−1.


## 5. Applications to the Financial Market

Consider the financial market on general time scales, such as0,1R∪2,3R, the market is no longer complete with any time gap. Let us consider a riskless bond Pt0 solution to(70)∇P0s=rtP0s∇s,with P00=1 and a risky asset St(71)∇St=btSt−∇t+σtSt−∇Wt,S0=s,where W is a standard Brownian motion on a probability space Ω,ℱ,F=ℱtt∈T,P. The solution to equation (71) is given in [10]. We will denote by Vt=Vt:t∈0,TT the wealth stochastic process representing the total value of the investor’s portfolio at time t, given an initial wealth V0>0. In particular, the investor, at a given time t, holds πt share of the risky stock. The trading strategy V,π is called self-financing if(72)Vt=V0+∫0tVt−πt∇P0P0+πt∇StSt−,or equivalently,(73)Vt=V0+∫0trtVt+bt−rtπt∇t+σtπt∇Wt,where πt is a predictable process.In incomplete markets, Föllmer [31] introduced the broader concept of the mean-self-financing strategy. The cost process C is defined by the difference Ct=Vt−∫0trtVt+bt−rtπt∇t+σtπt∇Wt. The hedging strategy V,π,C against contingent claims H∈Lk2ℱT is called mean-self-financing if the corresponding cost process C is a martingale. That is,(74)Vt=VT−∫tTrtVt+bt−rtπt∇t+σtπt∇Wt−CT−Ct,VT=H,where Ct is a martingale orthogonal to ∫0tσs∇Ws. The process Ct is the tracking error. In particular, at the terminal time, the tracing error measures the spread between the contingent claim and the portfolio value, and Ct corresponds to the cost process introduced by Föllmer and Schweizer [32]. Notice that the tracking error of the self-financing hedging strategy equals to zero.When hedging the contingent claim with terminal payoff given byH=hST, we assume the risk-free rate rt, the trend bt and the volatility σt to be constant. We have the following BS ∇ E:(75)Vt=VT−∫tTrVt+b−rπt∇t+σπt∇Wt−CT−Ct,VT=hST,that is, a linear BS ∇ E. Applying the linear BS∇ E(54), we have that the value V0 of the portfolio at initial time is given by (57) under some measure Q:(76)Y0=EQe⊖rThST.Q is the minimal martingale measure introduced by Föllmer and Schweizer [32], which coincides with variance-optimal signed martingale measure [33].Remark 4.
(1)
On continuous time, the market driven by the Brownian motion is complete. The option price is the well-known.(77)Y0=EQexp−rThST, where the Q turn to be the unique equivalent probability measure. Let θ=b−r/σ,dQ/dP|ℱt=exp−∫0tθdWt−1/2∫0tθ2dt.(2)
On discrete-time, the market driven by the Brownian motion is incomplete. In incomplete market, the equivalent probability measure contains more than one measure according to the so-called Second Fundamental Theorem of Asset Pricing. Upon imposing additional assumptions on the martingale measure, one can distinguish a unique measure and hence, a unique price.The discrete case can be seen in [29].


## 6. Conclusions

In this paper, we provided martingale decomposition on time scales. This allows us to prove the existence and uniqueness of solution for the backward stochastic dynamic equations on time scales.



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