# Averaging Principle for Backward Stochastic Differential Equations

Authors: Yuanyuan Jing; Zhi Li
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## 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.


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## Body

## 1. Introduction

The backward stochastic differential equations (BSDEs) were first studied by Parduox and Peng in [1] and have the following type:(1)Yt=ξ+∫tTgs,Ys,Zsds−∫tTZsdBs,where Bt:0≤t≤T is a d-dimensional Brownian motion defined on the probability space Ω,ℱ,ℙ with the natural filtration ℱt:0≤t≤T, the terminal value ξ is square integrable, and g is a mapping from Ω×0,T×ℝ×ℝd to ℝ. They 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). That is, a solution for such an equation associated with a coefficient g, a terminal value ξ, and a continuous barrier St, is a triple Xt,Yt,Zt0≤t≤T of adapted processes valued on ℝ1+d+1, which satisfies a square integrability condition:(2)Yt=ξ+∫tTgs,Ys,Zsds+KT−Kt−∫tTZsdBs,Yt≥St,K0=0,∫0TYt−StdKt=0.They established that this equation has a unique smooth square integrable solution wheng 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., [11–14]). 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 Z and Y and the function Kt, which is also one of the most challenging tasks in this paper.The remaining part of this paper is organized as follows. In Section2, 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 paperBt:0≤t≤T is a d-dimensional standard Brownian motion defined on a probability space Ω,ℱ,ℙ. Let ℱt:0≤t≤T be the natural filtration of Bt, where ℱ0 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:C1L2Ω,ℱT,ℙ: = ξ:ℱT−measurable random variable,Eξ2<∞.C2S20,T;ℝ: =φ:φis continuous processes with values inℝsuch thatφS22=Esup0≤t≤Tφt2<∞,andφtisℱt−measurable,for anyt∈0,T.C3M20,T;ℝ: =φ:φis jointly measurable processes such thatφM22=E∫0T|φt|2dt<∞,andφtisℱt−measurable,for anyt∈0,T.C3S∞0,T;ℝ: =φ:φis continuous processes with values inℝsuch thatφS∞=supω∈Ωsup0≤t≤Tφtω<∞,andφtisℱt−measurable,for anyt∈0,T.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)
Letg:Ω×0,T×ℝ×ℝd⟶ℝ such that for any y,z∈ℝ×ℝd, g is ℱt-measurable and Ly>0 and Lz>0 are two constants. Then, for all t,yi,zi∈0,T×ℝ×ℝd, i=1,2,(3)gt,y1,z1−gt,y2,z2≤Lyy1−y2+Lzz1−z2.(A2)
ξ∈L2Ω,ℱ,ℙ.(A3)
g·,0,0∈M20,T,ℝ.(A4)
The obstacleS∈S20,T;ℝ.(A5)
Kt∈L2Ω,ℱ,ℙ is continuous and ST<ξ.(A6)ξ is Malliavin differentiable and Dξ∞:=sup0<t<Tsupω∈ΩDtξ<∞, where Dtξ 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 Yt,Ztt∈0,T in S2×M2. And arguing as in [15], one can show that the solution Yt,Ztt∈0,T is bounded. Moreover, we can get the following bounded for Z:(4)ZS∞≤DξS∞2eLyT.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)
Yt,Zt,Kt∈S20,T;ℝ×M20,T;ℝd×L2Ω,P,ℝ+.(ii)
Yt≥St,0≤t≤T.(iii)
Kt is continuous and increasing, K0=0, and ∫0TYt−StdKt=0.(iv)
Yt=ξ+∫tTgs,Ys,Zsds+KT−Kt−∫tTZsdBs,0≤t≤T.In particular, since [6], one can easily see that Yt,Zt,Kt 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):(5)Ytε=ξ+ε∫tTgs,Ysε,Zsεds−ε∫tTZsεdBs.According to the second part, equation (5) also has an adapted unique and square integrable solution. We will examine whether the solution Ytε can be approximated to the solution process Y¯t of the simplified equation:(6)Y¯t=ξ+ε∫tTg¯Y¯s,Z¯sds−ε∫tTZ¯sdBs,whereY¯t,Z¯s has the same properties as Ysε,Zsε and g¯:ℝ×ℝd⟶ℝ is a measurable function satisfying A1 and the additional inequalities.(A7)
For anyt∈0,T1⊂0,T, we have(7)1T1−t∫tT1gs,Y,Z−g¯Y,Z2ds≤φT1−t1+Y2+Z2,whereφT1 is a bounded function.Theorem 1.
Assume that conditions A1–A3 and A7 are satisfied. For a given arbitrarily small numberδ1>0, there exists ε1∈0,ε0 and β∈0,1 such that for all ε∈0,ε1 having(8)supTε1−β≤t≤TEYtε−Y¯t2≤δ1.In order to prove Theorem1, we need the following Lemma 3.Lemma 3.
Under the assumptions of Theorem1, let 0,u⊆0,t⊆0,T be arbitrary, and it holds that(9)E∫uTZsε−Z¯s2ds≤LE∫uTYsε−Y¯s2ds+C1T−u,where L and C1 are two constants depending on u.Proof.
By equations (5) and (6), we get(10)Ytε−Y¯t=ε∫uTgs,Ysε,Zsε−g¯Y¯t,Z¯sds−ε∫uTZsε−Z¯sdBs.
Applying Ito^’s formula to Ytε−Y¯t2 and taking the mathematical expectation, we obtain(11)εE∫uTZsε−Z¯s2ds+EYtε−Y¯t2=2εE∫uTYsε−Y¯sgs,Ysε,Zsε−g¯Y¯s,Z¯sds≤2εE∫uTYsε−Y¯sgs,Ysε,Zsε−gs,Y¯s,Z¯sds+2εE∫uTYsε−Y¯sgs,Y¯s,Z¯s−g¯Y¯s,Z¯sds≕B1+B2.
ForB1, by using the condition A1 and Young’s inequality, for any θ>0, we deduce that(12)B1≤2εE∫uTYsε−Y¯sLyYsε−Y¯s+LzZsε−Z¯sds≤2LyεE∫uTYsε−Y¯s2ds+2LzεE∫uTYsε−Y¯sZsε−Z¯sds≤2LyεE∫uTYsε−Y¯s2ds+Lz1θεE∫uTYsε−Y¯s2ds+LzθεE∫uTZsε−Z¯s2ds=2Ly+LzθεE∫uTYsε−Y¯s2ds+LzθεE∫uTZsε−Z¯s2ds.
ForB2, owing to the condition A2, Ho..lder’s inequality and Young’s inequality, it follows that(13)B2≤2εE∫uTYsε−Y¯s2ds1/2E∫uTgs,Y¯s,Z¯s−g¯Y¯s,Z¯s2ds1/2≤2T−u1/2εE∫uTYsε−Y¯s2ds1/2E1T−u∫uTgs,Y¯s,Z¯s−g¯Y¯s,Z¯s2ds1/2≤2εET−u∫uTYsε−Y¯s2ds1/2supu≤s≤Tφ2s−u1+supu≤s≤TEY¯s2+supu≤s≤TEZ¯s21/2≤2CεET−u∫uTYsε−Y¯s2ds1/2≤CεE∫uTYsε−Y¯s2ds+T−u=CεE∫uTYsε−Y¯s2ds+CεT−u,where C=supu≤s≤Tφ2s−u1+supu≤s≤TEY¯s2+supu≤s≤TEZ¯s21/2. Now, we can choose θ=1/2Lz>0, then(14)εE∫uTZsε−Z¯s2ds≤2Ly+Lzθ+CεE∫uTYsε−Y¯s2ds+LzθεE∫uTZsε−Z¯s2ds−EYtε−Y¯t2+CεT−u≤2Ly+2Lz2+C0εE∫uTYsε−Y¯s2ds+12εE∫uTZsε−Z¯s2ds+CεT−u.
Thus,(15)E∫uTZsε−Z¯s2ds≤LE∫uTYsε−Y¯s2ds+C1T−u,where L=22Ly+2Lz2+C0, C1=2C. The proof is complete.With the help of Lemma3, now we can prove Theorem 1 by some conditions.Proof.
Using the elementary inequality and the isometry property, we derive that(16)EYtε−Y¯t2=Eε∫uTgs,Ysε,Zsε−g¯Y¯s,Z¯sds−ε∫uTZsε−Z¯sdBs2≤2ε2E∫uTgs,Ysε,Zsε−g¯Y¯s,Z¯sds2+2εE∫uTZsε−Z¯sdBs2≤4ε2E∫uTgs,Ysε,Zsε−gs,Y¯s,Z¯sds2+4ε2E∫uTgs,Y¯s,Z¯s−g¯Y¯s,Z¯sds2+2εE∫uTZsε−Z¯s2ds≕I1+I2+I3.
Applying Ho..lder’s inequality and the assumption A1, we obtain(17)I1≤4T−uε2E∫uTgs,Ysε,Zsε−gs,Y¯s,Z¯s2ds≤4T−uε2E∫uTLyYsε−Y¯s+LzZsε−Z¯s2ds≤8Ly2T−uε2E∫uTYsε−Y¯s2ds+8Lz2T−uε2E∫uTZsε−Z¯s2ds.
Then, together with Ho..lder’s inequality and the assumption A7, we get(18)I2≤4T−uε2E∫uTgs,Y¯s,Z¯s−g¯Y¯s,Z¯s2ds≤4T−uε2E∫uTgs,Y¯s,Z¯s−g¯Y¯s,Z¯s2ds≤4T−u2ε2supu≤s≤Tφ2s−u1+supu≤s≤TEY¯s2+supu≤s≤TEZ¯s2=C2T−u2ε2,where C2=4supu≤s≤Tφ2s−u1+supu≤s≤TEY¯s2+supu≤s≤TEZ¯s2. According to Lemma 3, plug equations (16)–(18), and we refer that(19)supu≤t≤TEYtε−Y¯t2≤8Ly2T−uε2supu≤t≤TE∫uTYsε−Y¯s2ds+8Lz2T−uε2+2εsupu≤t≤TE∫uTZsε−Z¯s2ds+C2T−u2ε2≤8Ly2T−uε2∫uTsupu≤t≤TEYsε−Y¯s2ds+L8Lz2T−uε2+2ε∫uTsupu≤t≤TEYsε−Y¯s2ds+C2T−u2ε2+8C1Lz2T−u2ε2+2C1T−uε≤8Ly2T−uε2+8LLz2T−uε2+2Lε∫uTsupu≤s1≤sEYs1ε−Y¯s12ds+C3T−u2ε2+2C1T−uε,where C3=C2+8C1Lz2. Thanks to Gronwall’s inequality, we obtain(20)supu≤t≤TEYtε−Y¯t2≤C3T−u2ε2+2C1T−uεeT−u8Ly2T−uε2+8LLz2T−uε2+2Lε.
Obviously, the above estimate implies that there existsβ∈0,1, for every t∈Tε1−β,T,(21)supTε1−β≤t≤TEYtε−Y¯t2≤C4ε,in which C4=C3T21−ε1−β2ε+2C1T1−ε1−βeT1−ε1−β8Ly2T1−ε1−βε2+8LLz2T1−ε1−βε2+2Lε is a constant.
Consequently, for any numberδ1>0, we can choose ε1∈0,ε0 making(22)supTε1−β≤t≤TEYtε−Y¯t2≤δ1,where ε∈0,ε1. 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):(23)Ytε=ξ+ε∫tTgs,Ysε,Zsεds+KT−Kt−ε∫tTZsεdBs,Ytε≥St,K0=0,∫0TYtε−StdKt=0.In fact, according to Section2 (Lemma 1), it is easy to find that equation (23) also has a unique solution. Then, we consider the simplified system:(24)Y¯t=ξ+ε∫tTg¯Y¯s,Z¯sds+K¯T−K¯t−ε∫tTZ¯sdBs,Y¯t≥St,K¯0=0,∫0TY¯t−StdK¯t=0,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 numberδ2>0, there exists ε1∈0,ε0 and β∈0,1 such that for all ε∈0,ε1,(25)supTε1−β≤t≤TEYtε−Y¯t2≤δ2.Before giving the proof of Theorem2, we need some Lemmas as follows.Lemma 4.
Under the assumptions of Theorem2, let 0,u⊆0,t⊆0,T be arbitrary, for any t∈0,u, there exists a constant L such that(26)E∫uTZsε−Z¯s2ds≤LE∫uTYsε−Y¯s2ds+C1T−u.Proof.
If equations (23) and (24) are satisfied, according to the property of KT, K¯T in A5, then applying Ito^’s formula to Ytε−Y¯t2 and taking the mathematical expectation, we obtain that(27)εE∫uTZsε−Z¯s2ds+EYtε−Y¯t2=2εE∫uTYsε−Y¯sgs,Ysε,Zsε−g¯Y¯s,Z¯sds+2E∫uTYsε−Y¯sdKs−K¯s.
SinceYtε≥St, Y¯t≥St, ∫0TY¯t−StdK¯t=0, and ∫0TYtε−StdKt=0, then(28)2E∫uTYsε−Y¯sdKs−K¯s=2E∫uTYsε−StdKs−K¯s−2E∫uTY¯s−StdKs−K¯s=2E∫uTYsε−StdKs−2E∫uTYsε−StdK¯s−2E∫uTY¯s−StdKs+2E∫uTY¯s−StdK¯s≤2E∫uTYsε−StdKs+2E∫uTY¯s−StdK¯s≤2E∫0TYsε−StdKs+2E∫0TY¯s−StdK¯s=0.
It follows that(29)εE∫uTZsε−Z¯s2ds+EYtε−Y¯t2=2εE∫uTYsε−Y¯sgs,Ysε,Zsε−g¯Y¯s,Z¯sds.
Recalling equations (11)–(14), it is easy to obtain the result of Lemma 3.Lemma 5.
Under the assumptions of Theorem2, let u∈0,T be arbitrary, for any 0,u⊆0,t⊆0,T, it holds that(30)EKt−K¯t2ds≤3ε24Ly2T−u+4LLz2T−u+LεE∫uTYsε−Y¯s2ds+C6T−u2ε2+3C1T−uε,where C3 is a constant.Proof.
By equations (23) and (24), we obtain that(31)Kt−K¯t=ε∫uTgs,Ysε,Zsε−g¯Y¯s,Z¯sds−ε∫uTZsε−Z¯sdBs+KT−K¯T.
Applying Ho..lder’s inequality, the isometry property to Ktε−K¯t and taking the mathematical expectation, it follows that(32)EKt−K¯t2≤3ε2E∫uTgs,Ysε,Zsε−g¯Y¯s,Z¯sds2+3εE∫uTZsε−Z¯sdBs2≤3ε2E∫uTgs,Ysε,Zsε−g¯Y¯s,Z¯sds2+3εE∫uTZsε−Z¯s2ds≤3T−uε2E∫uTgs,Ysε,Zsε−g¯Y¯s,Z¯s2ds+3εE∫uTZsε−Z¯s2ds≤6T−uε2E∫uTgs,Ysε,Zsε−gs,Y¯s,Z¯s2ds+6T−u2ε2E1T−u∫uTgs,Y¯s,Z¯s−g¯Y¯s,Z¯s2ds+3εE∫uTZsε−Z¯s2ds,where we used the fact that EKT−K¯T2=0. Owing to the assumptions A1 and A7, we have(33)EKt−K¯t2≤6T−uε2E∫uTLyYsε−Y¯s+LzZsε−Z¯s2ds+6T−u2ε2supu≤s≤Tφ2s1+supu≤s≤TEY¯s2+supu≤s≤TEZ¯s2+3εE∫uTZsε−Z¯s2ds≤12Ly2T−uε2E∫uTYsε−Y¯s2ds+12Lz2T−uε2E∫uTZsε−Z¯s2ds+C5T−u2ε2+3εE∫uTZsε−Z¯s2ds,where C5=6supu≤s≤Tφ2s1+supu≤s≤TEY¯s2+supu≤s≤TEZ¯s2. Meanwhile, from Lemma 4, we get(34)EKt−K¯t2≤12Ly2T−uε2+12LLz2T−uε2+3LεE∫uTYsε−Y¯s2ds+C5T−u2ε2+12C1Lz2T−u2ε2+3C1T−uε≤3ε24Ly2T−u+4LLz2T−u+LεE∫uTYsε−Y¯s2ds+C6T−u2ε2+3C1T−uε,where C6=C5+12C1Lz2. The proof is completed.Motivated by the above results, now we can prove Theorem2.Proof.
Together with the elementary equality and the isometry property, we deduce that(35)supu≤t≤TEYtε−Y¯t2=supu≤t≤TEε∫uTgs,Ysε,Zsε−g¯Y¯s,Z¯sds−ε∫uTZsε−Z¯sdBs+KT−K¯T+Kt−K¯t2≤4ε2supu≤t≤TE∫uTgs,Ysε,Zsε−g¯Y¯s,Z¯sds2+4εsupu≤t≤TE∫uTZsε−Z¯sdBs2+4supu≤t≤TEKt−K¯t2≕J1+J2+J3.
Recalling (17) and (18), we have(36)J1≤16T−uε2Ly2+LLz2supu≤t≤TE∫uTYsε−Y¯s2ds+2C2T−u2ε2+16C1Lz2T−u2ε2.
ForJ2, combining the isometry property with Lemma 4 produces(37)J2≤4εsupu≤t≤TE∫uTZsε−Z¯s2ds≤4Lεsupu≤t≤TE∫uTYsε−Y¯s2ds+4C1T−uε.
By virtue of Lemma5, we obtain(38)J3≤12ε24Ly2T−u+4LLz2T−u+Lεsupu≤t≤TE∫uTYsε−Y¯s2ds+4C6T−u2ε2+12C1T−uε.
Now, plug equations (36)–(38) into (35) for any u≤t≤T, and we get(39)supu≤t≤TEYtε−Y¯t2≤64Ly2T−u+64LLz2T−u+16Lεε2supu≤t≤TE∫uTYsε−Y¯s2ds+C7T−u2ε2+16C1T−uε≤64Ly2T+64LLz2T−u+16Lεε2∫uTsupu≤t≤TEYs1ε−Y¯s12ds+C7T−u2ε2+16C1T−uε,where C7=16C1Lz2+2C2+4C6. In terms of Gronwall’s inequality, we obtain(40)supu≤t≤TEYtε−Y¯t2≤C7T−u2ε2+16C1T−uεe64Ly2T−u2ε2+64LLz2T−u2ε2+16T−uLε.
Similarly, we can chooseβ∈0,1 and L>0 such that for every t∈Tε1−β,T, then(41)supTε1−β≤t≤TEYtε−Y¯t2≤C8ε1−β,where C8=C7T21−ε1−β2ε+16C1T1−ε1−βeT21−ε1−β2ε264Ly2+64LLz2+16L/Tε1−ε1−β. Indeed, for any number δ2, we can select ε1∈0,ε0 such that for every ε∈0,ε1 and t∈Tε1−β,T, the inequality(42)supTε1−β≤t≤TEYtε−Y¯t2≤δ2holds. 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 anyt∈0,T:(43)Yε=ξ+ε∫tTYεsin2s−Zεcos2sds−ε∫tTZεdBs,where gt,Yε,Zε=Yεsin2s−Zεcos2s and(44)g¯Yε,Zε=1T−t∫tTgs,Yε,Zεds=1T−t∫tTYεsin2s−Zεcos2sds=Yε21+sin2t2π−t−Zε21−sin2t2π−t,and we can define a new simplified BSDE as(45)Y¯=ξ+ε∫tTg¯Y¯,Z¯ds−ε∫tTZ¯dBs.
It is easy to check that assumptions A1–A3 and A7 are satisfied forg and g¯; therefore, Theorem 1 and Lemma 3 are satisfied; that is,(46)supTε1−β≤t≤TEYtε−Y¯t2≤δ1.Example 2.
we consider the following RBSDE for anyt∈0,T:(47)Yε=ξ+ε∫tTYεcos2s−Zεsin2sds−ε∫tT12ZεdBs+KT−Kt,where gt,Yε,Zε=Yεcos2s−Zεsin2s, St=0, for any t∈0,T and(48)g¯Yε,Zε=1T−t∫tTgs,Yε,Zεds=1π−t∫tπYεcos2s−Zεsin2sds=Yε21−sin2t2π−t−Zε21+sin2t2π−t,and the simplified system can be obtained as(49)Y¯t=ξ+ε∫tTg¯Y¯s,Z¯sds−ε∫tT12Z¯sdBs+K¯T−K¯t,where S¯t=0 for any t∈0,T. We can check that all the assumptions A1–A5 and A7 hold for g and g¯; therefore, Theorem 2 and Lemmas 4 and 5 are satisfied; that is,(50)supTε1−β≤t≤TEYtε−Y¯t2≤δ2.


## 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.



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