# A Study on a New Class of Backward Stochastic Differential Equation

Authors: Ping He; Yong Ren; Defei Zhang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The existence and uniqueness for a new type of backward stochastic differential equation when the generator includes the values of solutions of the past, the present, and the future are obtained in this paper. An important comparison theorem for this sort of BSDEs is also proved.


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## Body

## 1. Introduction

Pardoux and Peng in [1] first provided the famous backward stochastic differential equations (BSDEs). The existence and uniqueness for the BSDEs are proved by them. Since then, BSDEs have been discussed and applied to many fields, e.g., Chen and Epstein [2] and Karoui et al. [3–7]. A lot of research has focused on the assumptions on the generator, such as [8–12]. Recently, Delong and Imkeller in [13, 14] obtained many interesting results about the time-delayed equation in which the generator at time t only depends on the past solution. Peng and Yang in [15] discussed anticipated BSDEs, in which the generator includes present and future solutions.Therefore, the natural questions are as follows: can we discuss the backward stochastic differential equations when the generator includes not only the past and the present but also the future solutions? The comparison theorem for it is still true? Indeed, these questions are answered in the affirmative in this paper. The equation is called delay and anticipated BSDEs (DABSDEs for short). We obtain existence and uniqueness for delay and anticipated BSDEs, which can be seen as a general version of Delong and Imkeller in [13] or Peng and Yang in [15].


## 2. Main Notations

LetΩ,ℱ,P be a probability space equipped with a natural filtration ℱvv≥0 and Bvv≥0 be a standard Brownian motion. We denote the norm in Rm by ⋅. Given T>0, denote the following:(i)
L2ℱT;Rn=ξ:ℱT-measurable and Eξ2<∞.(ii)
Lℱ20,T;Rn=φt: ℱt-adapted and E∫0Tφt2dt<∞.(iii)
Sℱ20,T;Rn=χs:χs∈Lℱ20,T;Rn and continuous, as well as Esup0≤t≤Tχt2<∞.In the casen=1, they are abbreviated to L2ℱT,Lℱ20,T, and Sℱ20,T, respectively.


## 3. Delay and Anticipated Backward Stochastic Differential Equations

We propose a new type of BSDEs as follows:(1)−dYt=gt,Yt−d1t,Zt−d2t,Yt,Zt,Yt+d3t,Zt+d4tdt−ZtdBt,0≤t≤T,Yt=ξt,T≤t≤T+D,Zt=ηt,T≤t≤T+D,where di⋅,i=1,…,4, are four continuous functions s.t.(D1) ∃D≥0 s.t.(2)0≤t−d1t≤t;0≤t−d2t≤t;t+d3t≤T+D;t+d4t≤T+D,0≤t≤T.(D2) ∃L≥0 s.t., for all nonnegative and integrable f⋅,(3)∫tTfv−d1vdv≤L∫tT+Dfvdv;∫tTfv−d2vdv≤L∫tT+Dfvdv,∫tTfv+d3vdv≤L∫tT+Dfvdv;∫tTfv+d4vdv≤L∫tT+Dfvdv.Note that the simple examples fordis satisfying (D1) and (D2) are constant delay and dis=s.We call system (1) delay and anticipated backward stochastic differential equations (delay and anticipated BSDEs).Our aim is to search out a pair of processesY.,Z.∈Sℱ20,D+T;Rm×Lℱ20,D+T;Rm×d which satisfies the delay and anticipated BSDEs (1).For alls∈0,T, we suppose gω,s,ϕ,μ,y,z,ψ,ν:Ω×0,T×Lℱ20,s;Rm×Lℱ20,s;Rm×d×Rm×Rm×d×L2ℱr;Rm×L2ℱr′;Rm×d⟶L2ℱs;Rm, s≤r,r′≤T+D, and satisfies(A1) ∃C>0, ϕ.,ϕ.′∈Lℱ20,s;Rm;μ.,μ.′∈Lℱ20,s;Rm×d;y,y′∈Rm;z,z′∈Rm×d;ψ.,ψ.′∈Lℱ2s,T+D;Rm;ν.,ν.′∈Lℱ2s,T+D;Rm×d;r,t∈0,s;r¯,t¯∈s,T+D, 0≤s≤T, such that(4)gs,ϕr,μt,y,z,ψr¯,νt¯−gs,ϕr′,μt′,y′,z′,ψr¯′,νt¯′≤Cϕr−ϕr′+μt−μt′+y−y′+z−z′+Eψr¯−ψr¯′+νt¯−νt¯′ℱs.(A2) E∫0Tgs,0,0,0,0,0,02<∞.Theorem 1.
Supposeg satisfies (A1) and (A2), as well as dit,i=1,…,4, satisfy (D1) and (D2). Then, the DABSDE (1) has a unique solution for ξ.∈Sℱ2T,D+T;Rm and η.∈Lℱ2T,D+T;Rm×d, namely, there exists unique Y˜.,Z˜.∈Sℱ20,D+T;Rm×Lℱ20,D+T;Rm×d satisfying the DABSDEs.Proof.
We choose suitableβ which satisfies 0<eβT/β≤1/32C22L+1 and define a norm in Lℱ20,D+T;Rm:(5)μ⋅β=E∫0D+Tμt2eβtdt.
We set(6)Yu=ξT+∫uTgv,yv−d1v,zv−d2v,yv,zv,yv+d3v,zv+d4vdv−∫uTZvdBv,0≤u≤T;Yu=ξu,T≤u≤T+D;Zu=ηu,T≤u≤T+D.
We set a mappingMy.,z.:=Y.,Z.:Lℱ20,D+T;Rm×Rm×d⟶Lℱ20,D+T;Rm×Rm×d. Let us show M is a contraction mapping under ⋅β. ∀y.,z.,y′′,z′′∈Lℱ20,T+D;Rm×Rm×d; define Y.,Z.=My.,z., Y.′,Z.′=My.′,z.′, and(7)y^.,z^.=y.−y.′,z.−z.′,Y^.,Z^.=Y.−Y.′,Z.−Z.′.
Using Itô’s lemma forY^u2eβu,0≤u≤T, and taking expectation,(8)Y^02+βE∫0TeβuY^u2ds+E∫0TeβuZ^u2du=E2∫0Teβugu,yu−d1u,zu−d2u,yu,zu,yu+d3u,zu+d4u−gu,yu−d1u′,zs−d2u′,yu′,zu′,yu+d3u′,zu+d4u′Y^udu≤β2E∫0TeβuY^u2du+2βE∫0Teβugu,yu−d1u,zu−d2u,yu,zu,yu+d3u,zu+d4u−gu,yu−d1u′,zu−d2u′,yu′,zu′,yu+d3u′,zu+d4u′2du.
Thus,(9)E∫0Teβuβ2Y^u2+Z^u2du≤2βE∫0Teβugu,yu−d1u,zu−d2u,yu,zu,yu+d3u,zu+d4u−gu,yu−d1u′,zu−d2u′,yu′,zu′,yu+d3u′,zu+d4u′2du.
First, we note that(10)∫0Teβsy^s−dis2ds=∫0Teβs−disy^s−dis2eβdisds≤eβT∫0Teβsy^s2ds.
Becausedis,i=1,...,4, satisfy (D2) and g satisfies (A1), we have(11)E∫0Teβsβ2Y^s2+Z^2ds≤2C2βE∫0Teβsy^s−d1s+z^s−d2s+y^s+z^s+Ey^s+d3s+z^s+d4sℱs2ds≤8C2βE∫0Teβu2y^u−d1u2+2z^u−d2u2+z^u2+y^u2+2y^u+d3u2+2z^u+d4u2du≤8C22L+1eβTβE∫0T+Deβuy^u2+z^u2du.
Sinceβ satisfies eβT/β≤1/32C22L+1, then(12)E∫0T+DeβsY^s2+Z^2ds≤E∫0T+Deβsβ2Y^s2+Z^2ds≤14E∫0T+Deβsy^s2+z^s2ds.
Therefore,(13)Y^.,Z^.β≤12y^.,z^.β.
Consequently,M is a strict contraction mapping. From the fixed point theorem, the DABSDE (1) has a unique solution.
We use Itô’s lemma forYv2eαv,t≤v≤T, where α is an arbitrary constant, and take conditional expectation, similar to Proposition 4.4 in [15]; then,(14)Esup0≤t≤TYt2+∫tTZs2dsℱt≤C′EξT2+∫0Tgs,0,0,0,0,0,02ds+∫TT+Dξv2+ηv2dv,where C′ only depends on C in (A1), L in (D2), and T. Therefore, Y.∈Sℱ20,D+T;Rm.Example 1.
Consider a typical delay and anticipated backward stochastic differential equation(15)−dYu=Zu−π/2+Zu+πdu−ZudBu,0≤u≤T,Yu=∫0TsintdBt,T≤u≤T+D,Zu=sinT,T≤u≤T+D,with YT=∫0TsintdBt. We can get the unique solution of DABSDE (15) which is(16)Yu=∫0usinsdBs−cosu+sinu+cosT+sinT,0≤u≤T,Zu=sinu,0≤u≤T.


## 4. Comparison Theorem for Delay and Anticipated BSDEs

Next, we deduce the comparison theorem for one-dimensional DABSDEs. Denoteiti≔Yti,i=1,2,3. Let Y.j,Z.jj=1,2 be the solution of the following one-dimensional delay and anticipated BSDEs:(17)−dYsj=gjs,Ys−d1sj,Ysj,Zsj,Ys+d3sjds−ZsjdBs,s∈0,T,Ysj=ξsj,s∈T,D+T.Theorem 2.
Assume thatg1,g2 satisfy (A1) and (A2), ξ.1,ξ.2∈Sℱ2T,D+T,dkt,k=1,3, satisfy (D1), (D2), and ∀y∈R,z∈Rd,g2t,⋅,y,z,⋅ is strictly increasing. If ξ.2≤ξ.1,s∈T,D+T and g1⋅,ϕr,y,z,φr¯≥g2⋅,ϕr,y,z,φr¯,ϕr∈Lℱ20,t,φr¯∈Lℱ2t,T+D,r∈0,t,r¯∈t,D+T, then(18)Yt2≤Yt1,a.e.,a.s.
In particular, we also have the strict comparison theorem:(19)Y01=Y02⟺g1t,Yt−d1t1,Y⋅1,Z⋅1,Yt+d3t1=g2t,Yt−d1t1,Y⋅1,Z⋅1,Yt+d3t1,t∈0,T,ξt1=ξt2,t∈T,D+T.Proof.
Set(20)Yu3=ξT2+∫uTg2s,Ys−d1s1,Ys3,Zs3,Ys+d3s1ds−∫uTZs3dBs,u∈0,T,Yu3=ξu2,u∈T,D+T.
Denoteg^u:=g1u,Yu−d1u1,Yu1,Zu1,Yu+d3u1−g2u,Yu−d1u1,Yu1,Zu1,Yu+d3u1 and ξ^.=ξ.1−ξ.2,y^.:=Y.1−Y.3,z^.:=Z.1−Z.3. Then, y^.,z^. is the solution of the following DABSDEs:(21)y^t=ξ^T+∫tTg^v+avy^v+bvz^vdv−∫tTz^vdBv,0≤t≤T,yt=ξ^t,T≤t≤D+T,where(22)av=g1v,Yv−d1v1,Yv1,Zv1,Yv+d3v1−g2v,Yv−d1v1,Yv3,Zv1,Yv+d3v1Yv1−Yv3,Yv1≠Yv3,0,Yv1=Yv3,bv=g1v,Yv−d1v1,Yv3,Zv1,Yv+d3v1−g2v,Yv−d1v1,Yv3,Zv3,Yv+d3v1Zv1−Zv3,Zv1≠Zv3,0,Zv1=Zv3,
Sinceg2 follows assumption (A1), then av≤C and bv≤C. Denote(23)qs≔exp−12∫0sbv2dv+∫0savdv+∫0sbvdBv.
Using Itô’s lemma forqsy^s and taking expectation, then(24)y^t=Eξ^TqT+∫tTg^sqsdsℱt.
From this, we haveY⋅3≤Y⋅1 a.e., a.s.
Similarly, we set(25)Yu4=∫uTg2s,Ys−d1s3,Ys4,Zs4,Ys+d3s3ds−∫uTZs4dBs+ξT2,0≤u≤T,Yu4=ξu2,T≤u≤D+T.
BecauseYt1≥Yt3 a.e., a.s., and g2⋅ is strictly increasing, from the classical comparison theorem, then Y⋅4≤Y⋅3. When n=5,6,…, we investigate the DABSDEs:(26)Yun=ξT2+∫uTg2v,Yv−d1vn−1,Yvn,Zvn,Yv+d3vn−1dv−∫uTZvndBv,0≤u≤T,Yun=ξu2,T≤u≤T+D.
Sinceg2t,⋅,y,z,⋅ is strictly increasing, we have Yt4≥Yt5≥⋯≥Yt5≥. From the proving method of Theorem 1, Y.n and Z.nn≥4 are Cauchy sequences in Lℱ20,D+T and in Lℱ20,T,n≥4. Write their limits as Y. and Z.; then, Y.,Z.∈Lℱ20,D+T×Lℱ20,T, and when n⟶∞,(27)E∫tTg2s,Ys−d1sn−1,Ysn,Zsn,Ys+d3sn−1−g2s,Ys−d1s,Ys,Zs,Ys+d3s2eβsds≤4C2E∫tTYvn−Yv2+2LYvn−1−Yv2+Zvn−Zv2eβv⟶0.
Therefore,Y.,Z. satisfies the following delay and anticipated BSDEs:(28)Yt=ξT2+∫tTg2s,Ys−d1s,Ys,Zs,Ys+d3sds−∫tTZsdBs,0≤t≤T,Ytn=ξt2,T≤t≤T+D.
By Theorem1, Y⋅=Y⋅2.
SinceYt1≥Yt3≥⋯≥Ytn≥⋯, it holds immediately Y⋅1≥Y⋅2.
Similar to the deducing technique of Peng and Yang [15], next we prove the strict comparison theorem.
IfY01=Y02, then we have(29)g1u,Yu−d1u1,Yu1,Zu1,Yu+d3u1=g2u,Yu−d1u2,Yu1,Zu1,Yu+d3u2,
SinceY01≥Y03≥Y02, then Y01=Y03 and(30)g1t,Yt−d1t1,Yt1,Zt1,Yt+d3t1=g2t,Yt−d1t1,Yt1,Zt1,Yt+d3t1,as well as g2⋅ is strictly increasing, and we can easily get satisfying results.
Ifg1u,Yu−d1u1,Yu1,Zu,Yu+d3u1=g2u,Y∧1u−d1u1,Yu1,Zu1,Yu+d3u1,0≤u≤T, and ξu1=ξu2,T≤u≤T+D, then(31)y^t=Yt1−Yt3=Eξ^TqT+∫tTg^sqsdsℱt≡0.
Therefore, from Theorem1, Y⋅1=Y⋅2; in particular, Y01=Y02, a.e., a.s.The following result is to show the important relation between the delayed time and the solution of delay and anticipated BSDEs.Theorem 3.
SupposeY.l,Z.l,l=1,2, are solutions of two delay and anticipated BSDEs:(32)−dYsl=gs,Ys−d1lsl,Ysl,Zsl,Ys+d3lslds−ZsldBs,s∈0,T,Ysl=ξsl,s∈T,T+D.
Letξ.∈S2T,D+T, d1l and d3l satisfy (D1) and (D2), and the function g⋅ satisfies (A1) and (A2). If d11v≥d12v,d32v≥d31v,0≤v≤T, then ∃λ:=λC,L,T>0 s.t.(33)Yt1−Yt22≤λ∫tTd11v−d12v+d32v−d31vdv×E∫TT+Dξr2dr+ξT2+∫tTgr,0,0,0,02drℱt.Proof.
Denotey.=Y.1−Y.2,z.=Z.1−Z.2, and then apply Itô’s lemma for ys2eβs:(34)yu2+E∫uTβyv2+zv2eβv−udsℱu=E∫uT2ysgv,Yv−d11v1,Yv1,Zv1,Yv+d31v1−gv,Yv−d12v2,Yv2,Zv2,Yv+d32v22eβv−udvℱu,namely,(35)yt2+E∫tTexpβs−tzs2+0.5βys2dsℱt≤2β−1E∫tTgs,Ys−d11s1,Ys1,Zs1,Ys+d31s1−gs,Ys−d12s2,Ys2,Zs2,Ys+d32s22eβs−tdsℱt≤8C2βE∫tTYs−d11s1−Ys−d12s22+ys2+zs2+EYs+d31s1−Ys+d32s22ℱseβs−tdsℱt≤8C2βE∫tT2ys−d11s2+2Ys−d11s2−Ys−d12s22+ys2+zs2+2Eys+d31s2ℱs+2E^Ys+d31s2−Ys+d32s22ℱseβs−tdsℱt≤8C2+24C2LβE∫tTexpβs−tys2dsℱt+8C2βE∫tTexpβs−tzs2dsℱt+16C2βE∫tTE∫s+d31ss+d32sgr,Yr−d1r2,Yr2,Zr2,Yr+d3r2dr2ℱseβs−tdsℱt+16C2βE∫tT∫s−d11ss−d12sgr,Yr−d1r2,Yr2,Zr2,Yr+d3r2dr2eβs−tdsℱt.
We setβ=8C2; then,(36)yt2≤1+3LeβT−tE∫tTys2dsℱt+2E∫tTE∫s+d31ss+d32sgr,Yr−d1r2,Yr2,Zr2,Yr+d3r2dr2ℱseβs−tdsℱt+2E∫tT∫s−d11ss−d12sgr,Yr−d1r2,Yr2,Zr2,Yr+d3r2dr2eβs−tdsℱt≤1+3LeβT−tE∫tTys2dsℱt+2E∫tTd32s−d31sE∫s+d31ss+d32sgr,Yr−d1r2,Yr2,Zr2,Yr+d3r22drℱseβs−tdsℱt+2E∫tTd11s−d12s∫s−d11ss−d12sgr,Yr−d1r2,Yr2,Zr2,Yr+d3r22dreβs−tdsℱt≤eβT−t1+3LE∫tTys2dsℱt+8∫tTd11s−d12s+d32s−d31seβs−tds⋅E∫tTC2Yr−d1r22+C2Yr22+C2Zr22+C2Yr+d3r22+gr,0,0,0,02drℱt≤1+3LeβT−tE∫tTys2dsℱt+8∫tTeβs−td11s−d12s+d32s−d31sds⋅E∫tTC22L+1Yr22+C2Zr22+gr,0,0,0,02dr+∫TT+Dξr2drℱt.
Therefore, there existsλ′:=λ′C,L,K>0 s.t.(37)yt2≤λ′E∫tTys2dsℱt+λ′∫tTd11s−d12s+d32s−d31sds×EξT2+∫TD+Tξr2dr+∫tTgr,02drℱt.
Thus, from Gronwall inequality,(38)yt2≤λ∫tTd11s−d12s+d32s−d31sds×EξT2+∫tTgr,0,0,0,02dr+∫TT+Dξr2drℱt.Remark 1.
Ifgv,Yv−d1v,Zv−d2v,Yv,Zv,Yv+d3v,Zv+d4v=fv,Yv,Zv,Yv+d3v,Zv+d4v, then the delay and anticipated BSDE (1) is anticipated BSDEs in [15], and it is the result of Peng and Yang in [15].Remark 2.
In fact, the delay in (1) can go below t=0, namely, the DABSDE could have the following form:(39)Yv=θv′,v∈−τ,0,Zv=χv′,v∈−τ,0,−dYv=gv,Yv−d1v,Zt−d2v,Yv,Zv,Yv+d3v,Zv+d4vdv−ZvdBv,v∈0,T,Yv=θv,v∈T,D+T,Zv=χv,v∈T,D+T,where τ>0,D>0, and di⋅,i=1,…,4, are four positive continuous functions s.t.(D3) ∃D≥0 s.t.(40)−τ≤v−d1v≤v;−τ≤v−d2v≤v;v+d3v≤T+D;v+d4v≤T+D,v∈0,T.(D4) ∃K≥0 s.t., for all nonnegative and integrable functions, f⋅ satisfy(41)∫tTfu−d1udu≤K∫−τD+Tfudu;∫tTfu−d2udu≤K∫−τD+Tfudu;∫tTfu+d3udu≤K∫tD+Tfudu;∫tTfu+d4udu≤K∫tD+Tfudu,
The existence and uniqueness of equation (39) can be obtained by the similar approach of proving Theorem 1.



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