Martingale Decomposition and Backward Stochastic Dynamic Equations on Time Scales

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.


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][3][4]. For the stochastic calculus on time scales, a lot of work [5][6][7][8] focused on Δ-integration on the semi-open intervals of the form [t i , t i+1 ). Meanwhile, by the requirement of the predictable integrand (for the martingale property of stochastic integral), it is easy to consider semi-open intervals (t i , t i+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 di erential equations on time scales have been studied carefully. Zhu [11] studied stochastic optimal control problems on time scales. e 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 eld. e theory of backward stochastic di erential equations on continuous-time (BSDEs) is a mature eld. e general nonlinear BSDEs were rst studied in the Brownian framework by Pardoux and Peng [12]: A solution of equation (1), associated with the terminal value ξ and generator g(ω, t, y, z), is a couple of adapted stochastic processes (Y t , Z t ) t∈[0,T] R , 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 nance. e 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 continuoustime 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]: predictable and (N t ) t∈N is an orthogonal martingale to the integrals w.r.t the driven process (W t ) t∈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][17][18][19][20][21][22]). One is driving by a finite state process taking values from the basis vectors as in [16][17][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. e 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 eorem on time scales?
We obtain that every square-integral martingale M ∈ M 2 T with M 0 � 0, can be written as follows: where I t (X) denotes the stochastic integral w.r.t. Brownian motion on time scales and (N t ) t∈T is a square-integrable martingale with N 0 � 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: where ξ ∈ L 2 (Ω, F T , P) denotes the terminal condition, (Z t ) t∈T is predictable, and (N t ) t∈T is a square-integrable martingale with N 0 � 0, orthogonal to (W t ) t∈T in a weak suitable sense. is paper aims to establish the existence and uniqueness of solutions to general BS ∇ Es as follows: A triplet of processes (Y t , Z t , N t ) t∈T will satisfy the equations (5), where (Z t ) t∈T is predictable and (N t ) t∈T is orthogonal to the driving processes (W t ) t∈T . e 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]. e paper is organized as follows. In Section 2, 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.

Preliminaries
A time scale T is a nonempty closed subset of the real numbers R.
e 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. e forward jump operator σ and backward jump operator ρ are, respectively, defined by the following: We say that t is right-scattered (left-scattered, right- is called backward graininess. Consider the following: e set of left-scattered points of a time scale is at most countable. For Other types of intervals are defined similarly. We introduce the set T k if T has a rightscattered minimum t 2 , then T k � T − t 2 , otherwise T k � T. If t ∈ T k , 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: Now, suppose that f: 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 lim σ(s)↑t f(s) by f(t− ) and lim ρ(s)↓t f(s) by f(t+), respectively, if limits exist. It is to be noted that on the right-scattered points, Let A 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 t 0 f s ∇A s for the integral of f w.r.t. the measure μ A ∇ on (0, t]. It is seen that the function t↦ t 0 f s ∇A s is c a ' dla ' g. For details, please refer to [9]. For any continuous function on time scales p: T ⟶ R, with 1 + p(t)μ(t) ≠ 0 for all t ∈ T k , the Δ-exponential function e p (t, t 0 ), defined by ( [2], Definition 1.38), is the solution y(·) of the initial value problem: and e p (σ(t), t 0 ) � (1 + μ(t)p(t))e p (t, t 0 ), e p (t, t 0 ) � (1 + ](t) p(t− ))e p (t− , t 0 ). Denote e ⊖p (t, s) � 1/(e p (t, s)).

Stochastic Calculus on Time Scales.
Let R k be the k-dimensional Euclidean space, equipped with the standard inner product (·, ·), and the Euclidean norm | · |, R k×d be the collection of all k × d real matrices, and z � (z ij ) k×d be the matrix with z i ≔ (z i1 , . . . , z i d ) ⊤ and |z| ≔ ������ � tr(zz ⊤ ), where z ⊤ represents the transpose of z. E[X t |F s ] or E F s is denoted as the conditional expectation w.r.t. the filtration F s . Assume that we are working on a probability space (Ω, F, F � (F t ) t∈T , P) with the filtration F t t∈T satisfying the usual conditions.
Denoted by M 2 T , the space of square integrable martingales with E |M| 2 T < ∞ and consider M ∈ M 2 T . Since M 2 is a submartingale, following the Doob-Meryer decomposition theorem on time scales [9], there exists uniquely a natural increasing process 〈M〉 � (〈M〉 t ) t∈T , such that M 2 t − 〈M〉 t is an F-martingale. e natural increasing process 〈M〉 t is called characteristic of the martingale M.
Define the quadratic co-variation [M, N] t of two processes similar to ( [9], Definition 3.13).
e 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.
Brownian motion. e processes are as follows: are continuous, square-integrable martingales, with the following: where λ is the classical Lebesgue measure and ∪ ∞ n�1 (a n , b n ) � [t 0 , ∞)∖T is the expression for the open subset [t 0 , ∞)∖T of R as the countable of disjoint open intervals [27]. Furthermore, the vector of martingales e Lévy martingale characterization of Brownian motion fails on time scales, that's a continuous martingale with 〈X i , X j 〉 � tδ ij , cannot be a Brownian motion. e 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 L 2 T ((0, t]; W) be the space of all real-valued, predictable processes ϕ � ϕ t t∈T satisfying the following: Based on [ [9], Definition 3.6], define the integral as follows: e space L 2 T ((0, t]; W) is actually the L 2 space under the measure given by ∇〈W〉 × dP. Now, let us define the multidimensional Stochastic Integral: . Define the following: to be the multidimensional stochastic integral for X, also I t (X) for short. e i-th component of I t (X) is as follows:

Journal of Mathematics 3
Clearly it can be seen that I t (X) belongs to M 2 T,k , the R k -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. Let f ∈ C 1,2 (T × R d ) and X � (X 1 , . . . , X d ) be a d-dimensional semimartingales defined by the following: where t 0 φ i s ∇s < ∞, a.s. and ϕ i ∈ L 2 T ((0, t]; W). For all t ∈ T, then f(t, X(t)) is a semimartingale and the following formula holds: where

Summary of Notations.
For readers' convenience, we collect some spaces and notations used in the paper. For any integer k, T,k : the continuous R k -valued square-integral martingales on time scales with M 0 � 0 P − a.s, M 2, * T,k : the subset of M 2,c T,k , such that for each M ∈ M 2, * T,k , there exists X ∈ L 2 T,k×d , and M t � I t (X), T,k : subset of M 2,c T,k , the set of all the k-dimensional martingales, such that each martingale is orthogonal to that in M 2, * T,k .

Martingale Decomposition Theorem on Time Scales
In this section, we come back to the martingale decomposition problem on time scales. e fundamental tool on time scales analysis is the countable dense subset. e 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 any δ > 0, consider a partition of [0, T] T inductively by letting t 0 � 0 and for i � 1, 2, · · ·, set as follows: 4 Journal of Mathematics e partition is given in [3,9]. On time scales, the size of the interval (t i− 1 , t i ) T will not converge to zero 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). If
X is a right-continuous martingale (submartingale) on bounded time scales T with a last element X T and S 1 , S 2 are two bounded stopping times with S 1 ≤ S 2 on T, then Proof.
Let [0, T] T be a time scale, and let Π n � t 0 , t 1 , . . . , t n ⊆[0, T] T be a partition of T, where 0 � t 0 < t 1 < t 2 < · · · < t n � T (20). Consider the following sequence of random times: and the similarly defined sequences S n 2 . ese are stopping times. For every fixed integer n ≥ 1, both S n 1 and S n 2 take on a countable number of values and we also have S n 1 ≤ S n 2 . erefore, by the discrete optional sampling theorem, we have A X S n 1 dP ≤ A X S n 2 dP for every A ∈ F S n 1 . S 1 ≤ S n 1 implies F S 1 ⊂ F S n 1 , the preceding inequality also holds for every A ∈ F S 1 . e discrete martingale results show that the sequence of random variables X S n 1 is uniformly integrable, and the same is of course true for X S n 2 . X S 2 � lim n⟶∞ X S n 2 (ω) and X S 1 � lim n⟶∞ X S n 1 (ω) hold for a.e. ω ∈ Ω. It follows from uniform integrability that X S 1 , X S 2 are integrable and that T can be identified with its terminal value M T ∈ L 2 (Ω, F T , P) (in general the terminal variables can be extended to M ∞ if exists). M 2 T becomes a Hilbert space isomorphic to L 2 (Ω, F T , P), if endowed with the following inner product: Indeed, if M n is a Cauchy sequence for ‖ · ‖ M 2 T , then the sequence M n T is Cauchy in L 2 (Ω, F T , P) and so goes to a limit Similar to Lemma 2.2 chapter 3 in [26], we have the following: e L 2 T,k×d space is a closed space with the norm ‖ · ‖ T,W .

Proof.
We define a Hilbert space T,k×d is a subspace of H T . Now, we prove that it is closed. Suppose that X m { } ∞ m�1 is a convergent sequence in L 2 T,k×d with limit X ∈ H T− . us, the sequence has a convergent subsequence which converges almost surely under μ p , also denoted by Restricted on [0, t] T for 0 < t ≤ T, repeating the above procedure, by the uniqueness of convergence, we can get that X is B([0, t] T ) ⊗ F t− -measurable. erefore, X is predictable and belongs to L 2 T,k×d . e proof is complete.
. Consider the mapping X↦I T (X) from L 2 T,k×d to L 2 k (F T ). From the definition of the stochastic integral on time scales, the mapping is injective. is mapping preserves following inner products: Denote R k (F T ) ≜ I T (X); X ∈ L 2 T,k×d . Since any convergent sequence in R k (F T ) is also Cauchy, its preimage sequence in L 2 T,k×d must have a limit in L 2 T,k×d . It e following result is the "fundamental decomposition theorem" for the martingales w.r.t Brownian motion on time scales.

Theorem 2. For every M ∈ M 2
T,k , with M 0 � 0, P − a.s., we have the following decomposition: where X ∈ L 2 T,k×d , I t (X) ∈ M 2, * T,k , N ∈ M 2 T,k with N 0 � 0 and N is orthogonal to every element of M 2, * T,k .

Journal of Mathematics 5
Proof. We have to show the existence of a process Y ∈ L 2 T,k×d such that M t � I t (Y) + N t , where N ∈ M 2 T,k has the property 〈I(X), N〉 t � 0, ∀X ∈ L 2 T,k×d .
Such a decomposition is unique (up to indistinguish- T,k×d and both N ′ and N ″ satisfy the property, then ]. e decomposition is unique up to indistinguishability. Since , so it admits the following decomposition: where T,k . Taking conditional expectation under F t on M T , we obtain the following: It remains to show that N is orthogonal to every squareintegrable martingale of the form I(X); X ∈ L 2 T,k×d , or equivalently, that N t I t (X), F t t∈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 Lemma 2, we only need to prove the following: holds for every stopping time S of the filtration F t t∈T , with S < T (since I 0 (X) � 0). e integral has I S (X) � I T (X), where X t (ω) � X t (ω)1 t≤S(ω) { } is a process in L 2 T,k×d . erefore, by Lemma 2,  (1) Note f(t) ≠ f(t− ) at left-scattered points for continuous function on time scales, N ∈ M 2,c T,k is continuous, but ∇N t ≠ 0 at left-scattered points.
(2) Let T � R, k � 1, (Karatzas [26], Proposition 4.14 one-dimension decomposition), M 2,c R and M 2, * R actually coincide, the component N t in the decomposition is actually ∇N t � 0. e 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 by W, M 2,c T,k and M 2, * T,k do not coincide, see the following example. (4) Let T � N (Follmer, Hans, [29], eorem 10.18), it is the discrete time version of the Kunita-Watanabe decomposition w.r.t a sequence of normal distribution random variables. (5) e orthogonality condition given in eorem 2 is the weak orthogonality condition in continuoustime, but we call it strong orthogonality condition on time scales. at is, for scattered points t: { }, we have the following: For t � 3, M 3 � I 3 + N 3 ,

BS = Es Driven by Brownian Motion on Time Scales
In this section, we denote (Ω, F, F, P) to be a probability space equipped with a complete filtration F � (F t ) t∈T generated by a d-dimensional Brownian motion W t 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: We call g 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 R k -valued, continuous and adapted process, Z is a R k×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 inter k, (H3) g satisfies Lipschitz condition w.r. t. (y, z): there exists a constant L > 0, ∀y, y ′ ∈ R k , z, z ′ ∈ R k×d , s.t.

Theorem 3. (Main Result) If Assumption 1 is satisfied, BS ∇ E(35) admits a unique triple of solution
Remark 3. If T � R, eorem 3 degenerates to the classical BSDEs. e deterministic integral part could also be g(ω, t, Y t− , Z t ) or g(ω, t− , Y t− , Z t− ). For simplicity, we only consider the case of g(·, Y t , Z t ), which corresponding to the explicit discrete backward difference equations [22].
To prove the main theorem, first consider g real-valued and not depend on (y, z).
. Apparently, M t is a square integrable martingale. Following the martingale decomposition theorem on time scales, there exists a unique predictable process Z t ∈ L 2 T,k×d and en, (δY, δZ, δN) satisfy BSDE(35) with ξ � 0 and g � 0. By the inequality, By the continuity of δY s , δY s � 0, δN s � C − δN 0 � 0, P-a.s.. e uniqueness is proved. □ □ Introduce the new norms on spaces: 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 eorem 3: Proof. For any fixed (y(·), z(·), n(·)) ∈ S 2 T × L 2 T,k×d × M 2,⊥ T,k , it follows from Lemma 4 that it admits a unique triple Hence, we can define an operator as follows: (Y., Z., N.) � I[(y., z., n.)]: We can prove that I forms a contraction mapping on the Banach space Take any (y 1 (·), z 1 (·), n 1 (·)), (y 2 (·), z 2 (·), n 2 (·)) ∈ S 2 T × L 2 T,k×d × M 2,⊥ T,k , we denote the following: and δy � y 1 − y 2 , δz � z 1 − z 2 , δn � n 1 − n 2 . By equation (38), we obtain the following: (49) We obtain as follows: (50) We can obtain that the operator I is contractive. Hence, in this case, there exists a unique fixed point en, E t satisfies the Doleans exponential equation Lemma 5. Let a, b, c be predictable bounded processes, let E be the exponential martingale of the martingale Suppose that E is a positive uniformly integrable martingale; If the linear backward equation has a solution (Y, Z, N), then Y is given by e solution (Y t , Z t , N t ) on (2, 3] equals to the classical solution on continuous-time with terminal condition Y 3 � ξ and N T − N t � 0. For right-scattered point t � 1, In the conditional expectation form: Recall (Example 1), in a particular case, when

Applications to the Financial Market
Consider the financial market on general time scales, such as [0, 1] R ∪ [2, 3] R , the market is no longer complete with any time gap. Let us consider a riskless bond P 0 t solution to ∇P 0 (s) � r t P 0 (s)∇s, with P 0 (0) � 1 and a risky asset S t where W is a standard Brownian motion on a probability space (Ω, F, F � F t t∈T , P). e solution to equation (71) is given in [10]. We will denote by V t � V t : t ∈ [0, T] T the wealth stochastic process representing the total value of the investor's portfolio at time t, given an initial wealth V 0 > 0. In particular, the investor, at a given time t, holds π t share of the risky stock. e trading strategy (V, π) is called self-financing if or equivalently, where π t is a predictable process.
In incomplete markets, Föllmer [31] introduced the broader concept of the mean-self-financing strategy. e cost process C is defined by the difference C t � V t − t 0 ([r t V t + (b t − r t )π t ]∇t + σ t π t ∇W t ). e hedging strategy (V, π, C) against contingent claims H ∈ L 2 k (F T ) is called mean-self-financing if the corresponding cost process C is a martingale. at is, where C t is a martingale orthogonal to t 0 σ s ∇W s . e process C t 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 C t 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 by H � h(S T ), we assume the risk-free rate r t , the trend b t and the volatility σ t to be constant. We have the following BS ∇ E: that is, a linear BS ∇ E. Applying the linear BS∇ E(54), we have that the value V 0 of the portfolio at initial time is given by (57) under some measure Q: Q is the minimal martingale measure introduced by Föllmer and Schweizer [32], which coincides with varianceoptimal signed martingale measure [33].

Remark 4.
(1) On continuous time, the market driven by the Brownian motion is complete. e option price is the well-known.
where the Q turn to be the unique equivalent probability measure. Let θ � (b − r/σ), (dQ/dP)| F t � exp(− t 0 θdW t − (1/2) t 0 θ 2 dt). (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 eorem of Asset Pricing. Upon imposing additional assumptions on the martingale measure, one can distinguish a unique measure and hence, a unique price. e discrete case can be seen in [29].

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

Data Availability
e data used to support the findings of this study are included within the article.

Disclosure
is work was preprinted on arxiv named "Martingale Decomposition and BSDE on Time Scales" [34].

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.