Martingale Decomposition and Backward Stochastic Dynamic Equations on Time Scales

Tang, Guofeng; Jia, Guangyan

doi:https://doi.org/10.1155/2022/6217582

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusions Data Availability Disclosure Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 6217582 | https://doi.org/10.1155/2022/6217582

Martingale Decomposition and Backward Stochastic Dynamic Equations on Time Scales

Guofeng Tang¹and Guangyan Jia¹

Academic Editor: Barbara Martinucci

Received28 Dec 2021

Accepted09 Mar 2022

Published24 May 2022

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.

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 -integration on the semi-open intervals of the form . Meanwhile, by the requirement of the predictable integrand (for the martingale property of stochastic integral), it is easy to consider semi-open intervals . 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]:

A solution of equation (1), associated with the terminal value and generator , is a couple of adapted stochastic processes , 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]:

By the Galtchouk–Kunita–Watanabe theorem, the solution of (2) is a triple tuple , where is predictable and is an orthogonal martingale to the integrals w.r.t the driven process . 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 . 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 martingale with , can be written as follows:where denotes the stochastic integral w.r.t. Brownian motion on time scales and is a square-integrable martingale with 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 denotes the terminal condition, is predictable, and is a square-integrable martingale with , orthogonal to in a weak suitable sense.

This paper aims to establish the existence and uniqueness of solutions to general BS Es as follows:

A triplet of processes will satisfy the equations (5), where is predictable and is orthogonal to the driving processes . The BS Es driven by the Brownian motion on time scales are similar to traditional BSDEs driven by general c dl g martingales beyond the Brownian setting [14, 23, 24].

The 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.

2. Preliminaries

A time scale is a nonempty closed subset of the real numbers . The distance between the points is defined as the normal distance on : . In this paper, we always suppose is bounded with . The forward jump operator and backward jump operator are, respectively, defined by the following:

We say that is right-scattered (left-scattered, right-dense, left-dense), if (, , ) holds. is called graininess, is called backward graininess. Consider the following:

The set of left-scattered points of a time scale is at most countable.

For with , define the closed interval in by . Other types of intervals are defined similarly. We introduce the set if has a right-scattered minimum , then , otherwise . If , the ∇-derivative of at the point is defined to be the number (provided it exists) with the property that for each , there is a neighborhood (in ) of such as follows:

Now, suppose that . Continuity of is defined in the usual manner. A function is called right-dense continuous (rd-continuous) on if and only if it is continuous at every right-dense point and the left-sided limit exists at every left-dense point. Denote by and by , respectively, if limits exist. It is to be noted that on the right-scattered points, for continuous functions on time scales. If is left-scattered, then , right-scattered, then .

Let be an increasing right-continuous function of finite variation defined on . We denote as the Lebesgue -measure associated with . For any -measurable function , we write for the integral of w.r.t. the measure on . It is seen that the function is c dl g. For details, please refer to [9].

For any continuous function on time scales , with for all , the -exponential function , defined by ([2], Definition 1.38), is the solution of the initial value problem:and , . Denote .

2.1. Stochastic Calculus on Time Scales

Let be the k-dimensional Euclidean space, equipped with the standard inner product , and the Euclidean norm , be the collection of all real matrices, and be the matrix with and , where represents the transpose of . or is denoted as the conditional expectation w.r.t. the filtration . Assume that we are working on a probability space with the filtration satisfying the usual conditions.

Denoted by , the space of square integrable martingales with and consider . Since is a submartingale, following the Doob–Meryer decomposition theorem on time scales [9], there exists uniquely a natural increasing process , such that is an -martingale. The natural increasing process is called characteristic of the martingale .

Define the quadratic co-variation of two processes similar to ( [9], Definition 3.13). If we write for and call it the quadratic variation of . For partitions of with ,

It means is an -martingale, which implies that is also an -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. Let be a d-dimensional Brownian motion. The processes are as follows:are continuous, square-integrable martingales, with the following:where is the classical Lebesgue measure and is the expression for the open subset of as the countable of disjoint open intervals [27]. Furthermore, the vector of martingales is independent of .

Remark 1. The Lévy martingale characterization of Brownian motion fails on time scales, that’s a continuous martingale with , 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 on time scales: let be the space of all real-valued, predictable processes satisfying the following:

Based on [[9], Definition 3.6], define the integral as follows:

The space is actually the space under the measure given by . Now, let us define the multidimensional Stochastic Integral:

Definition 1. Let be a -dimensional Brownian motion on time scales, is matrix process, , . Define the following:to be the multidimensional stochastic integral for , also for short. The i-th component of is as follows:

Clearly it can be seen that belongs to , the -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 and be a d-dimensional semimartingales defined by the following:where , a.s. and . For all , then is a semimartingale and the following formula holds:where for all .

2.2. Summary of Notations

For readers’ convenience, we collect some spaces and notations used in the paper. For any integer , , short for : the space of -valued random vectors that are -measurable and satisfy , , short for : the set of -predictable, -valued integrands processes satisfying , : the continuous -valued square-integral martingales on time scales with , : the subset of , such that for each , there exists , and , , short for : the set of valued, adapted and continuous processes such that , : subset of , the set of all the -dimensional martingales, such that each martingale is orthogonal to that in .

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 any , consider a partition of inductively by letting and for , set as follows:

The partition is given in [3, 9]. On time scales, the size of the interval will not converge to zero if . 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 is a right-continuous martingale (submartingale) on bounded time scales with a last element and are two bounded stopping times with on , then

Proof. Let be a time scale, and let be a partition of , where (20). Consider the following sequence of random times:and the similarly defined sequences . These are stopping times. For every fixed integer , both and take on a countable number of values and we also have . Therefore, by the discrete optional sampling theorem, we have for every . implies , the preceding inequality also holds for every .
The discrete martingale results show that the sequence of random variables is uniformly integrable, and the same is of course true for . and hold for a.e. . It follows from uniform integrability that , are integrable and that holds for every .

Via the identity , each can be identified with its terminal value (in general the terminal variables can be extended to if exists). becomes a Hilbert space isomorphic to , if endowed with the following inner product:

Indeed, if is a Cauchy sequence for , then the sequence is Cauchy in and so goes to a limit in this space; then if is the martingale with terminal variable , it belongs to and . The set of all continuous elements of denoted by , is a closed subspace of the Hilbert space . Now, define a measure on by the following:

Similar to Lemma 2.2 chapter 3 in [26], we have the following:

Lemma 3. The space is a closed space with the norm .

Proof. We define a Hilbert space . Obviously, is a subspace of . Now, we prove that it is closed. Suppose that is a convergent sequence in with limit . Thus, the sequence has a convergent subsequence which converges almost surely under , also denoted by . Therefore , ; thus, is -measurable.
Restricted on for , repeating the above procedure, by the uniqueness of convergence, we can get that is -measurable. Therefore, is predictable and belongs to . The proof is complete.

Now, define an inner product on by . Recall the inner product on , . Consider the mapping from to . From the definition of the stochastic integral on time scales, the mapping is injective. This mapping preserves following inner products:

Denote . Since any convergent sequence in is also Cauchy, its preimage sequence in must have a limit in . It follows that is closed in . Let us denote by the subset of which consists of stochastic integrals , of processes :

The following result is the “fundamental decomposition theorem” for the martingales w.r.t Brownian motion on time scales.

Theorem 2. For every , with ., we have the following decomposition:where , with and is orthogonal to every element of .

Proof. We have to show the existence of a process such that , where has the propertySuch a decomposition is unique (up to indistinguishability); indeed, if we have with and both and satisfy the property, thenis in with and . Then from [Lemma 2]. The decomposition is unique up to indistinguishability.
Since is a closed subspace of , we can denote its orthogonal complement by . The random variable is in , so it admits the following decomposition:where and satisfies . We construct a martingale through by . Obviously . Taking conditional expectation under on , we obtain the following:It remains to show that is orthogonal to every square-integrable martingale of the form , or equivalently, that is a martingale. It is to be noted that each martingale has a right continuous modification. So, now, we suppose that is right continuous.
According to Lemma 2, we only need to prove the following:holds for every stopping time of the filtration , with (since ). The integral has , where is a process in . Therefore, by Lemma 2,The proof is complete.

Remark 2. (1)Note at left-scattered points for continuous function on time scales, is continuous, but at left-scattered points.(2)Let , (Karatzas [26], Proposition 4.14 one-dimension decomposition), and actually coincide, the component in the decomposition is actually . 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 by , and do not coincide, see the following example.(4)Let (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 Theorem 2 is the weak orthogonality condition in continuous-time, but we call it strong orthogonality condition on time scales. That is, for scattered points : .

Example 1. For a martingale on time scales , we have the following:For , , , ,with .

4. BS Es Driven by Brownian Motion on Time Scales

In this section, we denote to be a probability space equipped with a complete filtration generated by a d-dimensional Brownian motion on time scales, and augmented by all the -null sets in .

For simplicity, we consider the following general BS Es on time scales:

We call the driver of the BS Es and the pair the data of the BS Es.

A solution to BS E is a triple of process satisfying (35), such that is a -valued, continuous and adapted process, is a -valued and predictable process and is a martingale orthogonal to . For terminal condition and generator , we make the following assumptions:

Assumption 1. We assume that for any inter , (H1) is defined as , such that , is progressively measurable, (H2) (H3) satisfies Lipschitz condition w.r.t. : there exists a constant , , , s.t. (H4) .

Theorem 3. (Main Result) If Assumption 1 is satisfied, BS E(35) admits a unique triple of solution .

Remark 3. If , Theorem 3 degenerates to the classical BSDEs. The deterministic integral part could also be or . For simplicity, we only consider the case of , which corresponding to the explicit discrete backward difference equations [22].

To prove the main theorem, first consider real-valued and not depend on .

Lemma 4. Assume that and is a progressively measurable and -predictable process which satisfies . Then, for BS E(4) with parameters and , there exists a unique triple solution . Moreover,where is a positive constant with for all , so .

Proof. Existence: We denote . Apparently, is a square integrable martingale. Following the martingale decomposition theorem on time scales, there exists a unique predictable process and , s.t. , . Then . As also , denote . It can be seen that , we obtain the existence.
Uniqueness: Applying Itô’s formula (Theorem 1) to on , where , or the differential form: . We have the following:whereThus, we haveSince is -predictable, the above integrals with respect to and all belong to . By taking the expectation with respect to , we obtain the following:So, we obtain the inequality.
Now, we suppose there are two solutions and . Define , , , then,Then, satisfy BSDE(35) with and . By the inequality,By the continuity of -a.s.. The uniqueness is proved. □

Introduce the new norms on spaces:for any positive integer . Apparently, for each , is equivalent to which is the original norm on the corresponding space. Now, we start to prove Theorem 3:

Proof. For any fixed , it follows from Lemma 4 that it admits a unique triple solution satisfyingHence, we can define an operator as follows:We can prove that forms a contraction mapping on the Banach space . Take any , we denote the following:and , , . By equation (38), we obtain the following:We obtain as follows:Let ,We can obtain that the operator is contractive. Hence, in this case, there exists a unique fixed point for the map . 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. Let be a bounded martingale satisfying for any . SetThen, satisfies the Doleans exponential equation

Lemma 5. Let be predictable bounded processes, let be the exponential martingale of the martingale , define , . Suppose that is a positive uniformly integrable martingale; .If the linear backward equationhas a solution , then is given by

Proof. By Girsanov’s theorem on time scales [30], is a -martingale, the probability measure which has density with respect to on . Under the measure , the equation becomes . Applying Itô’s formula to the process , we obtain the following:where . Since and are martingales,and consequently,This concludes the proof.

4.2. Comparison Theorem

The linear case allows us to obtain a comparison theorem on time scales.

Theorem 4. Let , be the corresponding solutions to the data , and suppose satisfy conditions (H2) (H3). Assume that and satisfy (H4), . Then, for any , we have the following:

Proof. We define , , . Then, solves the following linear BS EwhereAccording to the linear BS Es, if the coefficients satisfy the suitable conditions, we have the following:Assume now that , and for any , . Then, for any , .

Example 3. Consider , the BS E:The solution on equals to the classical solution on continuous-time with terminal condition and . For right-scattered point ,In the conditional expectation form:Recall (Example 1), in a particular case, when ,

5. Applications to the Financial Market

Consider the financial market on general time scales, such as , the market is no longer complete with any time gap. Let us consider a riskless bond solution towith and a risky asset where is a standard Brownian motion on a probability space . The solution to equation (71) is given in [10]. We will denote by the wealth stochastic process representing the total value of the investor’s portfolio at time , given an initial wealth . In particular, the investor, at a given time , holds share of the risky stock. The trading strategy is called self-financing ifor equivalently,where is a predictable process.

In incomplete markets, Föllmer [31] introduced the broader concept of the mean-self-financing strategy. The cost process is defined by the difference . The hedging strategy against contingent claims is called mean-self-financing if the corresponding cost process is a martingale. That is,where is a martingale orthogonal to . The process 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 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 , we assume the risk-free rate , the trend and the volatility 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 of the portfolio at initial time is given by (57) under some measure :

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. where the turn to be the unique equivalent probability measure. Let .(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.

Data Availability

The data used to support the findings of this study are included within the article.

Disclosure

This work was preprinted on arxiv named “Martingale Decomposition and BSDE on Time Scales” [34].

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Key R&D Program of China (Grant no. 2018YFA0703900).

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