Averaging Principle for Backward Stochastic Differential Equations

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.


Introduction
e backward stochastic differential equations (BSDEs) were first studied by Parduox and Peng in [1] and have the following type: where B t : 0 ≤ t ≤ T is a d-dimensional Brownian motion defined on the probability space (Ω, F, P) with the natural filtration F t : 0 ≤ t ≤ T , the terminal value ξ is square integrable, and g is a mapping from Ω × [0, T] × R × R d to R. ey proved that equation (1) has a unique adapted and square integrable solution when g is globally Lipschitz. Mao [2] and Wang and Wang [3] both considered the adapted solution to equation (1) when g 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).
at is, a solution for such an equation associated with a coefficient g, a terminal value ξ, and a continuous barrier S t , is a triple (X t , Y t , Z t ) 0≤t≤T of adapted processes valued on R 1+d+1 , which satisfies a square integrability condition: (2) ey established that this equation has a unique smooth square integrable solution when g is Lipschitz in y, z. 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 (y, z). 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][12][13][14]). However, motivated by the above works, the averaging principle for equation (1), even for equation (2), has not introduced at all. e 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 Z and Y and the function K t , which is also one of the most challenging tasks in this paper. e 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. en, 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.

Preliminaries
where F 0 contains all P-null sets of F and let P be the σ-algebra of predictable subsets of Ω × [0, T]. 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.
(A6) ξ is Malliavin differentiable and ‖Dξ‖ ∞ :� sup 0<t<T sup ω∈Ω |D t ξ| < ∞, where D t ξ denotes the Malliavian derivative of ξ. 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 And arguing as in [15], one can show that the solution (Y t , Z t ) t∈[0,T] 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: In particular, since [6], one can easily see that

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 Y ε t can be approximated to the solution process Y t of the simplified equation: where (Y t , Z s ) has the same properties as (Y ε s , Z ε s ) and g: R × R d ⟶ R is a measurable function satisfying A1 and the additional inequalities.
where φ(T 1 ) is a bounded function.

Lemma 3. Under the assumptions of eorem 1, let [0, u]⊆[0, t]⊆[0, T] be arbitrary, and it holds that
Proof. By equations (5) and (6), we get Applying Ito's formula to |Y ε t − Y t | 2 and taking the mathematical expectation, we obtain For B 1 , by using the condition A1 and Young's inequality, for any θ > 0, we deduce that For B 2 , owing to the condition A2, Ho .. lder's inequality and Young's inequality, it follows that Discrete Dynamics in Nature and Society 3 where us, where L � 2(2L y + 2L 2 z + C 0 ), C 1 � 2C. e proof is complete.

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

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

Theorem 2.
Assume that conditions A1-A5 are satisfied. For a given arbitrarily small number δ 2 > 0, there exists ε 1 ∈ (0, ε 0 ] and β ∈ (0, 1) such that for all ε ∈ (0, ε 1 ], Before giving the proof of eorem 2, we need some Lemmas as follows. Proof. If equations (23) and (24) are satisfied, according to the property of K T , K T in A5, then applying Ito's formula to |Y ε t − Y t | 2 and taking the mathematical expectation, we obtain that Since Y ε t ≥ S t , Y t ≥ S t ,