Averaging Principle for Backward Stochastic Differential Equations

Abstract

The averaging principle for BSDEs and one-barrier RBSDEs, with Lipschitz coefficients, is investigated. An averaged BSDEs for the original BSDEs is proposed, as well as the one-barrier RBSDEs, and their solutions are quantitatively compared. Under some appropriate assumptions, the solutions to original systems can be approximated by the solutions to averaged stochastic systems in the sense of mean square.

1. Introduction

The backward stochastic differential equations (BSDEs) were first studied by Parduox and Peng in [1] and have the following type:where is a -dimensional Brownian motion defined on the probability space with the natural filtration , the terminal value is square integrable, and is a mapping from to . They proved that equation (1) has a unique adapted and square integrable solution when is globally Lipschitz. Mao [2] and Wang and Wang [3] both considered the adapted solution to equation (1) when is non-Lipschitz. Besides, Lepeltier and Martin [4] studied the case of continuous coefficients, while the locally Lipschitz coefficient was investigated by Bahlali in [5].

Since then, in [6], El Karoui et al. began to introduce the notion of a backward stochastic differential equation reflected to one continuous lower barrier (RBSDEs in short). That is, a solution for such an equation associated with a coefficient , a terminal value , and a continuous barrier , is a triple of adapted processes valued on , which satisfies a square integrability condition:

They established that this equation has a unique smooth square integrable solution when is Lipschitz in , . After that, many scholars have studied the solutions of equation (2) under different conditions, such as Matoussi [7] considered the case of continuous and at most linear growth in . Hamadene [8] studied the case of a right-continuous with left limits barrier and Lepeltier, and Xu [9] investigated the case of discontinuous barrier. For the monotonicity, general increasing growth conditions were investigated by Lepeltier et al. [10].

On the contrary, averaging principle, which is usually used to approximate dynamical systems under random fluctuations, has long and rich history in multiscale problems (see, e.g., [11–14]). However, motivated by the above works, the averaging principle for equation (1), even for equation (2), has not introduced at all. The main motivation of our work is to seek an answer to the following interesting question: compared with the general stochastic differential equations, do the backward stochastic differential equations have the averaging principle of solutions? So, in this paper, we will consider this issue under Lipschitz conditions. But, due to the characteristics of the equations of BSDEs and RBSEDs with a barrier, we should first consider the relationship among the random variables and and the function , which is also one of the most challenging tasks in this paper.

The remaining part of this paper is organized as follows. In Section 2, we present some preliminaries and assumptions for the later use. In Section 3, we investigate the averaging principle for the BSDEs under some proper conditions. Then, the averaging principle for the RBSDEs with a barrier will be given in Section 4. Finally, in Section 5, we design two examples to demonstrate the efficiency of the proposed method.

2. Preliminaries

Throughout, this paper is a -dimensional standard Brownian motion defined on a probability space . Let be the natural filtration of , where contains all -null sets of and let be the -algebra of predictable subsets of . In addition, we define the following: : = . : . : . : .

In order to study the qualitative properties of the solution to equations (1) and (2), we impose some assumptions on the coefficient functions, which will enable us to solve it.(A1)Let such that for any , is -measurable and and are two constants. Then, for all , ,(A2).(A3).(A4)The obstacle .(A5) is continuous and . (A6) is Malliavin differentiable and , where denotes the Malliavian derivative of .

Lemma 1. It is known, since Pardoux and Peng in [1], that under the assumptions A1–A3, the BSDEs (1) have an adapted unique and square integrable solution in . And arguing as in [15], one can show that the solution is bounded. Moreover, we can get the following bounded for Z:

Lemma 2. It has been noticed that equation (2) has a unique solution in [6] under the assumptions A1–A5 and the following conditions:(i).(ii).(iii) is continuous and increasing, , and .(iv). In particular, since [6], one can easily see that is bounded.

3. Averaging Principle for BSDEs

 In this section, we are going to investigate the averaging principle for the BSDEs under Lipschitz coefficients. Let us consider the standard form of equation (1): According to the second part, equation (5) also has an adapted unique and square integrable solution. We will examine whether the solution can be approximated to the solution process of the simplified equation: where has the same properties as and is a measurable function satisfying A1 and the additional inequalities.(A7)For any , we have where is a bounded function.

Theorem 1. Assume that conditions A1–A3 and A7 are satisfied. For a given arbitrarily small number , there exists and such that for all having

In order to prove Theorem 1, we need the following Lemma 3.

Lemma 3. Under the assumptions of Theorem 1, let be arbitrary, and it holds thatwhere and are two constants depending on u.

Proof. By equations (5) and (6), we getApplying It’s formula to and taking the mathematical expectation, we obtainFor , by using the condition A1 and Young’s inequality, for any , we deduce thatFor , owing to the condition A2, Hlder’s inequality and Young’s inequality, it follows thatwhere . Now, we can choose , thenThus,where , . The proof is complete.

With the help of Lemma 3, now we can prove Theorem 1 by some conditions.

Proof. Using the elementary inequality and the isometry property, we derive thatApplying Hlder’s inequality and the assumption A1, we obtainThen, together with Hlder’s inequality and the assumption A7, we getwhere . According to Lemma 3, plug equations (16)–(18), and we refer thatwhere . Thanks to Gronwall’s inequality, we obtainObviously, the above estimate implies that there exists , for every ,in which is a constant.
Consequently, for any number , we can choose makingwhere . Hence, the proof is complete.

4. Averaging Principle for RBSDEs with a Barrier

In this section, we will continue to study the averaging principle for the RBSDEs under Lipschitz condition. Firstly, let us consider the standard form of equation (2):

In fact, according to Section 2 (Lemma 1), it is easy to find that equation (23) also has a unique solution. Then, we consider the simplified system:where is a continuous function satisfying A5 and Lemma 1.

Theorem 2. Assume that conditions A1–A5 are satisfied. For a given arbitrarily small number , there exists and such that for all ,

Before giving the proof of Theorem 2, we need some Lemmas as follows.

Lemma 4. Under the assumptions of Theorem 2, let be arbitrary, for any , there exists a constant such that

Proof. If equations (23) and (24) are satisfied, according to the property of , in A5, then applying It’s formula to and taking the mathematical expectation, we obtain thatSince , , , and , thenIt follows thatRecalling equations (11)–(14), it is easy to obtain the result of Lemma 3.

Lemma 5. Under the assumptions of Theorem 2, let be arbitrary, for any , it holds thatwhere is a constant.

Proof. By equations (23) and (24), we obtain thatApplying Hlder’s inequality, the isometry property to and taking the mathematical expectation, it follows thatwhere we used the fact that . Owing to the assumptions A1 and A7, we havewhere . Meanwhile, from Lemma 4, we getwhere . The proof is completed.

Motivated by the above results, now we can prove Theorem 2.

Proof. Together with the elementary equality and the isometry property, we deduce thatRecalling (17) and (18), we haveFor , combining the isometry property with Lemma 4 producesBy virtue of Lemma 5, we obtainNow, plug equations (36)–(38) into (35) for any , and we getwhere . In terms of Gronwall’s inequality, we obtainSimilarly, we can choose and such that for every , thenwhere . Indeed, for any number , we can select such that for every and , the inequalityholds. Hence, it is proved.

5. Examples

Now, two examples will be presented to illustrate the procedure of the obtained theory.

Example 1. Consider the following BSDE for any :where andand we can define a new simplified BSDE asIt is easy to check that assumptions A1–A3 and A7 are satisfied for and ; therefore, Theorem 1 and Lemma 3 are satisfied; that is,

Example 2. we consider the following RBSDE for any :where , , for any andand the simplified system can be obtained aswhere for any . We can check that all the assumptions A1–A5 and A7 hold for and ; therefore, Theorem 2 and Lemmas 4 and 5 are satisfied; that is,

6. Conclusions

Backward stochastic differential equations are widely used in finance and optimal control problems. The contribution of this paper is that, different from traditional stochastic differential equations, the averaging principle of backward stochastic differential equations under Lipschitz condition is given.

Data Availability

The data used in this article are available from the authors upon request

Conflicts of Interest

The authors declare that they have no conflicts of interest.