# Comparison Theorems for the Multidimensional BDSDEs and Applications

Authors: Bo Zhu; Baoyan Han
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

A class of backward doubly stochastic differential equations (BDSDEs) are studied. We
obtain a comparison theorem of these multidimensional BDSDEs. As its applications, we
derive the existence of solutions for this multidimensional BDSDEs with continuous coefficients. We can also prove that this solution is the minimal solution of the BDSDE.


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## Body

## 1. Introduction

Backward stochastic differential equations (BSDEs in short) were first introduced in [1] in order to give a probabilistic interpretation (Feynman-Kac formula) for the solutions of semilinear parabolic PDEs, one can see [2, 3]. Moreover, BSDEs have been considered with great interests because for their connections with mathematical finance [4] as well as stochastic optimal control and stochastic games [5, 6]. As we know, the comparison theorem is a very useful result in the theory of BSDEs, for example, it can be used to study viscosity solutions of partial differential equations. In 1992, Peng [2] gave the comparison of the one-dimensional BSDE. In 1994, Christel Geiß and Ralf Manthey [7] proved the comparison theorems for stochastic differential equations in finite and infinite dimensions.After they introduced the theory of BSDEs, Pardoux and Peng [8] in 1994 brought forward a new kind of BSDEs, that is a class of backward doubly stochastic differential equations (BDSDEs in short) with two different directions of stochastic integrals, that is, the equations involve both a standard (forward) stochastic integral dWt and a backward stochastic integral dBt. They have proved the existence and uniqueness of solutions to BDSDEs under uniformly Lipschitz conditions on coefficients. That is, for a given terminal time T>0, under the uniformly Lipschitz assumptions on coefficients f, g, for any square integrable terminal value ξ, the following BDSDE has a unique solution pair (yt,zt) in the interval [0,T]:(1.1)yt=ξ+∫tTf(s,ys,zs)ds+∫tTg(s,ys,zs)dBs-∫tTzsdWs.
Pardoux and Peng [8] showed that BDSDEs can produce a probabilistic representation for certain quasilinear stochastic partial differential equations (SPDEs). Recently, [9] Zhang and Zhao, [10] Zhu and Han studied the infinite horizon BDSDEs. Many researchers do their work in this area (cf. [11–18] and the references therein). As we know, the comparison theorem is a very useful result in the theory of BDSDEs, for example, it can be used to study viscosity solutions of SPDEs [19]. Shi et al. proved the comparison theorem of these BDSDEs, but they studied the one-dimensional BDSDEs.In this paper, we will prove the comparison theorem of the multidimensional BDSDEs. Then we study the multidimensional BDSDEs with continuous coefficients as an application of the comparison theorem.The paper is organized as follows: in Section2 we introduce some preliminaries and notations; in Section 3 we prove the comparison theorem of multidimensional BDSDEs; at the end, we give the multi-dimension BDSDEs with continuous coefficients in Section 4.


## 2. Preliminaries: The Existence and Uniqueness of BDSDEs and an Extension of Itô Formula

Notation 1.
The Euclidean norm of a vectorx∈Rk will be denoted by |x|, and for a d×k matrix A, we define ∥A∥=TrAA*, where A* is the transpose of A. We also define (α,β) as the inner product of α and β.Fora∈Rk, let ai be the ith component of a; c∈Rk×d, let ci be the ith row of c, let ci,j be an element for the ith row and the jth column of c. For a1,a2∈Rk, we define(2.1)a1≥a2⟺aj1≥aj2,j=1,2,…,k.Let(Ω,ℱ,P) be a probability space and let T be an arbitrarily fixed positive constant throughout this paper. Let {Wt;0≤t≤T} and {Bt;0≤t≤T} be two mutually independent standard Brownian motions with values in Rd and Rl, respectively, defined on (Ω,ℱ,P). Let 𝒩 denote the class of P-null sets of ℱ. For each t∈[0,T], we define
(2.2)Ft≐FtW∨Ft,TB,
where for any process {ηt},ℱs,tη=σ{ηr-ηs;s≤r≤t}∨𝒩,ℱtη=ℱ0,tη.Note that the collection{ℱt;t∈[0,T]} is neither increasing nor decreasing, and it does not constitute a filtration.For anyn∈N, let M2(0,T;Rn) denote the set of (classes of dP⊗dt a.e. equal) n-dimensional jointly measurable stochastic processes {φt;t∈[0,T]} which satisfy(i)∥φ∥M22:=E∫0T|φt|2dt<∞;(ii)φt is ℱt-measurable, for a.e. t∈[0,T].Similarly, we denote byS2([0,T];Rn) the set of n-dimensional continuous stochastic processes {φt;t∈[0,T]} which satisfy(iii)∥φ∥S22:=E(sup0≤t≤T|φt|2)<∞;(iv)φt is ℱt-measurable, for any t∈[0,T].Obviously,M2(0,T;Rn) and S2([0,T];Rn) are Hilbert spaces.Let(2.3)f:Ω×[0,T]×Rk×Rk×d⟶Rk,g:Ω×[0,T]×Rk×Rk×d⟶Rk×l
be jointly measurable and such that for any (y,z)∈Rk×Rk×d.(H1)
f(·,y,z)∈M2(0,T;Rk), g(·,y,z)∈M2(0,T;Rk×l).(H2)
There exist constantsC>0 and 0<α<1 such that for any (ω,t)∈Ω×[0,T], (y1,z1), (y2,z2)∈Rk×Rk×d,(2.4)|f(t,y1,z1)-f(t,y2,z2)|2≤C(|y1-y2|2+‖z1-z2‖2)‖g(t,y1,z1)-g(t,y2,z2)‖2≤C|y1-y2|2+α‖z1-z2‖2.
Given ξ∈L2(Ω,ℱT,P;Rk), we consider the following backward doubly stochastic differential equation:(2.5)Yt=ξ+∫tTf(s,Ys,Zs)ds+∫tTg(s,Ys,Zs)dBs-∫tTZsdWs,0≤t≤T.
Note that the integral with respect to {Bt} is a “backward Itô integral” and the integral with respect to {Wt} is a standard forward Itô integral. The forward integrals were defined in [20] Da Prato and Zabczyk and [21] D. Nualart, and E. Pardoux. To see the backward one, for any s≥0 let {h(s)}s≥0 be a stochastic process such that {h(s)} is ℱs measurable, and locally square integrable, that is, for any 0≤a≤b<∞, ∫ab∥h(s)∥2ds<∞ almost surely. Since ℱs is a backward filtration with respect to B, so from the one-dimensional backward Itô’s integral and relation with forward integral, for 0≤T≤T′, we have ∫tTh(s)dBs=-∫T′-TT′-th(T′-s)dB̂s a.s, where B̂s=BT′-s-BT′.Definition 2.1.
A pair of processes(y,z):Ω×[0,T]→Rk×Rk×d is called a solution of BDSDE (2.5), if (y,z)∈S2([0,T];Rk)×M2(0,T;Rk×d) and satisfies BDSDE (2.5).Proposition 2.2.
Under conditions (H1) and (H2), BDSDE (2.5) has a unique solution (Y,Z)∈S2([0,T];Rk)×M2(0,T;Rk×d).Proposition 2.3.
Letα∈S2([0,T];Rk), β∈M2(0,T;Rk), γ∈M2(0,T;Rk×l), δ∈M2(0,T;Rk×d) satisfy
(2.6)αt=α0+∫0tβsds+∫0tγsdBs+∫0tδsdWs,0≤t≤T.
Then
(2.7)|αt|2=|α0|2+2∫0t(αs,βs)ds+2∫0t(αs,γsdBs)+2∫0t(αs,δsdWs)-∫0t‖γs‖2ds+∫0t‖δs‖2ds,E|αt|2=E|α0|2+2E∫0t(αs,βs)ds-E∫0t‖γs‖2ds+E∫0t‖δs‖2ds.This two propositions were derived in [8].


## 3. Comparison Theorem of Multidimensional BDSDEs

In this section, we will prove the comparison theorem of the multidimensional BDSDEs. Firstly, we give the definition of the indicator function for the positive and negative parts.Definition 3.1.
Iff:Ω↦R¯, then the positive part of f is defined by the formula f+≐max{f,0}. Similarly, the negative part of f is defined as f-≐max{-f,0}=-min{f,0}Note that bothf+ and f- are nonnegative functions.We consider the followingk-dimensional BDSDEs: (0≤t≤T)(3.1)yt1=ξ1+∫tTf1(s,ys1,zs1)ds+∫tTg(s,ys1,zs1)dBs-∫tTzs1dWs,yt2=ξ2+∫tTf2(s,ys2,zs2)ds+∫tTg(s,ys2,zs2)dBs-∫tTzs2dWs,
where ξ1,ξ2∈L2(Ω,ℱT,P;Rk), f1(ω,t,y,z), f2(ω,t,y,z):Ω×[0,T]×Rk×Rk×d→Rk, g(ω,t,y,z):Ω×[0,T]×Rk×Rk×d→Rk×l. We assume ξ1,ξ2 and f1,f2,g satisfy the following conditions:(i)ξ1≥ξ2P-a.s.;(ii) For allj=1,2,…,k, for all (ω,t)∈Ω×[0,T], y1, y2∈Rk, z1, z2∈Rk×d, yj1=yj2, zj1=zj2, yl1≥yl2, l≠j(3.2)fj1(ω,t,y1,z1)≥fj2(ω,t,y2,z2);(iii) for all(ω,t)∈Ω×[0,T], y1, y2∈Rk, z1, z2∈Rk×d(3.3)|fji(t,y1,z1)-fji(t,y2,z2)|2≤C(|y1-y2|2+‖z1-z2‖2),i=1,2;(iv) for all(ω,t,y,z)∈Ω×[0,T]×Rk×Rk×d, ∃h(ω,t,μ,ν):Ω×[0,T]×R×Rd→Rl such that(3.4)gj(ω,t,y,z)=hj(ω,t,yj,zj),|hj(t,yj1,zj1)-hj(t,yj2,zj2)|2≤C|yj1-yj2|2+α|zj1-zj2|2,whereC>0, 0<α<1 are two constants.From Proposition2.2, under conditions (iii) and (iv), there exist two pairs of measurable processes (y1,z1)∈S2([0,T];Rk)×M2(0,T;Rk×d) and (y2,z2)∈S2([0,T];Rk)×M2(0,T;Rk×d) satisfying BDSDEs (3.1), respectively.Theorem 3.2.
Assume the conditions (i)–(iv) hold, let(y1,z1) and (y2,z2) be solutions of BDSDEs (3.1), respectively. Then yt1≥yt2, a.s. for all t∈[0,T].Proof.
forallɛ>0, we define a function ϕɛ(y):R→R(3.5)ϕɛ(y)={y2,y≤0,y2-16ɛy3,0≤y≤2ɛ,2ɛy-43ɛ2,y≥2ɛ.
Obviously, ϕɛ(y)∈C2(R) and ϕɛ′′(y) is bounded, for all y∈R,
(3.6)ϕɛ(y)⟶|y-|2,ϕɛ′′(y)⟶-2y-,ϕɛ′′(y)⟶2I{y≤0},whenɛ→0.
We set
(3.7)y¯t=yt1-yt2,z¯t=zt1-zt2,ξ¯=ξ1-ξ2.
Then (y¯t,z¯t) satisfy the following BDSDE:
(3.8)y¯t=ξ¯+∫tT[f1(s,ys1,zs1)-f2(s,ys2,zs2)]ds+∫tT[g(s,ys1,zs1)-g(s,ys2,zs2)]dBs-∫tTz¯sdWs.
Applying Itô’s formula to ϕɛ(y¯j(t)), we get
(3.9)ϕɛ(y¯j(t))=ϕɛ(ξ¯j)+∫tTϕɛ′(y¯j(s))[fj1(s,ys1,zs1)-fj2(s,ys2,zs2)]ds+∫tTϕɛ′(y¯j(s))[gj(s,ys1,zs1)-gj(s,ys2,zs2)]dBs-∫tTϕɛ′(y¯j(s))z¯j(s)dWs+12∫tTϕ′ɛ′(y¯j(s))|gj(s,ys1,zs1)-gj(s,ys2,zs2)|2ds-12∫tTϕɛ′′(y¯j(s))|z¯j(s)|2ds.
Let ɛ→0, we get
(3.10)|y¯j(t)-|2=|ξj-¯|2-∫tT2y¯j(s)-[fj1(s,ys1,zs1)-fj2(s,ys2,zs2)]ds-∫tT2y¯j(s)-[gj(s,ys1,zs1)-gj(s,ys2,zs2)]dBs+∫tT2y¯j(s)-z¯j(s)dWs+∫tTI(y¯j(s)≤0)|gj(s,ys1,zs1)-gj(s,ys2,zs2)|2ds-∫tTI(y¯j(s)≤0)|z¯j(s)|2ds.
From (i), we have ξ1-ξ2≥0, a.s., so
(3.11)E|(ξ1-ξ2)-|2=0.
Since (y1,z1) and (y2,z2) are in S2([0,T];Rk)×M2(0,T;Rk×d), it easily follows that
(3.12)E∫tTy¯j(s)-[gj(s,ys1,zs1)-gj(s,ys2,zs2)]dBs=0,(3.13)E∫tTy¯j(s)-z¯j(s)dWs=0.
Let
(3.14)△1=-2∫tTy¯j(s)-[fj1(s,ys1,zs1)-fj2(s,ys2,zs2)]ds]=-2∫tTy¯j(s)-[fj1(y11,…,yj1,…,yk1,z11,…,zj1,…,zk1)-fj1(y11+y¯1-,…,yj2,…,yk1+y¯k-,z11,…,zj2,…,zk1)]ds-2∫tTy¯j(s)-[fj1(y11+y¯1-,…,yj2,…,yk1+y¯k-,z11,…,zj2,…,zk1)-fj2(y12,…,yj2,…,yk2,z12,…,zj2,…,zk2)]ds=△11+△21,
where △11 is the first integral, and △21 is the second integral. Since
(3.15)yl1+y¯l-≥yl2,l≠j
and (ii) we have
(3.16)fj1(y11+y¯1-,…,yj2,…,yk1+y¯k-,z11,…,zj2,…,zk1)≥fj2(y12,…,yj2,…,yk2,z12,…,zj2,…,zk2).
So △21≤0.
From (iii) and Young’s inequality, it follows that(3.17)△11≤∫tT2y¯j-L(|y¯1-|+⋯+|y¯j-1-|+|y¯j|+|y¯j+1-|+⋯+|y¯k-|+|z¯j|)ds=∫tTL(2y¯j-|y¯1-|+⋯+2y¯j-|y¯j-1-|+2y¯j-|y¯j|+2y¯j-|y¯j+1-|+⋯+2y¯j-|y¯k-|+2y¯j-|z¯j|)ds=∫tTL(2y¯j-|y¯1-|+⋯+2y¯j-|y¯j-1-|+2|y¯j-|2+2y¯j-|y¯j+1-|+⋯+2y¯j-|y¯k-|+2y¯j-|z¯j|)ds≤∫tTL(|y¯j-|2+|y¯1-|2+⋯+|y¯j-|2+|y¯j-1-|2+2|y¯j-|2+|y¯j-|2+|y¯j+1-|2+⋯+|y¯j-|2+|y¯k-|2+L1-α|y¯j-|2+1-αLI(y¯j≤0)|z¯j|2)ds=(Lk+L21-α)∫tT|y¯j-|2ds+L∑l=1k∫tT|y¯l-|2ds+(1-α)∫tTI(y¯j≤0)|z¯j|2)ds,
where L>0 only depends on the Lipschitz constant C in (iii). From (iv) we deduce
(3.18)△2=∫tTI(y¯j(s)≤0)|gj(s,ys1,zs1)-gj(s,ys2,zs2)|2ds=∫tTI(y¯j(s)≤0)|hj(s,yj1(s),zj1(s))-hj(s,yj2(s),zj2(s))|2ds≤∫tTI(y¯j(s)≤0)[C|yj1(s)-yj2(s)|2+α|zj1(s)-zj2(s)|2]ds=C∫tT|y¯j-|2ds+α∫tTI(y¯j(s)≤0)|z¯j|2ds.
So there ∃M>0 only depends on C,α,k,d, such that
(3.19)△1+△2≤M∑l=1k∫tT|y¯l-|2ds+∫tTI(y¯j(s)≤0)|z¯j|2ds.
Taking expectation on both sides of (3.10) and noting (3.11)–(3.19), we get
(3.20)E|y¯j(t)-|2≤ME∑l=1k∫tT|y¯l-|2ds.
We sum from 1 to k and get ∑j=1kE|y¯j(t)-|2≤MkE∑l=1k∫tT|y¯l-|2ds. That is,
(3.21)E∑j=1k|y¯j(t)-|2≤Mk∫tTE∑j=1k|y¯j-|2ds.
By Gronwall’s inequality, it follows that
(3.22)E∑j=1k|y¯j(t)-|2=0,∀t∈[0,T].dt-a.s.
So y¯j(t)≥0dt⊗dP-a.s.j=1,2,…,k.Example 3.3.
If conditions (ii) and (iv) of the theorem are replaced by the following: (ii′) for allj=1,2,…,k, ∀(ω,t)∈Ω×[0,T], y∈Rk, z1, z2∈Rk×d, zj1=zj2,
(3.23)fj1(ω,t,y,z1)≥fj2(ω,t,y,z2),
and (iv′) for all (ω,t,y,z)∈Ω×[0,T]×Rk×Rk×d,
(3.24)‖g(t,y1,z1)-g(t,y2,z2)‖2≤C|y1-y2|2+α‖z1-z2‖2,
where C>0,0<α<1 are two constants. By the following example we can see, the comparison theorem is untenable at the above conditions.
Ifm=2, f1(y,z)=f2(y,z)=f(y,z)=Ay, g(y,z)=0, where A2×2 is a constant coefficient matrix, obviously f,g satisfy (ii′) and (iv′). We consider the following k-dimensional BDSDEs: (0≤t≤T)(3.25)yt1=ξ1+∫tTAys1ds-∫tTzs1dWs,yt2=ξ2+∫tTAys2ds-∫tTzs2dWs,
then(3.26)yt1=E[eA(T-t)ξ1Ftw],yt2=E[eA(T-t)ξ2Ftw].
Already know ξ1≥ξ2, P a.s., to be sure yt1≥yt2, dt⊗dP a.s., this requires
(3.27)eA(T-t)ξ1≥eA(T-t)ξ2,dt⊗dPa.s.,
then
(3.28)eA(T-t)(ξ1-ξ2)≥0,dt⊗dPa.s.
Let
(3.29)A=[0-110],
then
(3.30)eA(T-t)=[cos⁡(T-t)sin⁡(T-t)-sin⁡(T-t)cos⁡(T-t)].
Here f, g satisfy (ii′) and (iv′) does not satisfy (ii) and (iv). Let T=2π,t∈[0,π/2](3.31)eA(T-t)=[cos⁡t-sin⁡tsin⁡tcos⁡t].
Obviously we cannot guarantee, for all ξ1≥ξ2, there is yt1≥yt2, dt⊗dP a.s.


## 4. The Multidimensional BDSDEs with Continuous Coefficients

In this section, we do not assume a Lipschitz property. Our main interest in this section is to study the multidimensional BDSDEs with continuous coefficients as an application of the comparison theorem obtained in Section3. First, we give the definition of quasimonotonously increasing mapping [22].Definition 4.1.
A mappingf:Rk→Rk is said to be quasi-monotonously increasing mapping, if for each j=1,2,…,k,fj(x)≤fj(y) provided xj=yj, and xl≤yl, l≠j.We give our main result.Theorem 4.2.
Assumef:Ω×[0,T]×Rk×Rk×d→Rk and g:Ω×[0,T]×Rk×Rk×d→Rk×l are jointly measurable functions and satisfy.
(1) There existsr(ω,t,y,γ):Ω×[0,T]×Rk×Rd→Rk, such that for all (ω,t,y,z)∈Ω×[0,T]×Rk×Rk×d,
(4.1)fj(ω,t,y,z)=rj(ω,t,y,zj).
(2) Linear growth: for all(ω,t,y,z)∈Ω×[0,T]×Rk×Rk×d, ∃K>0, such that
(4.2)|f(ω,t,y,z)|≤K(1+|y|+‖z‖).
(3) Continuous: for fixedω and t,f(ω,t,·,·) is continuity.
(4) Quasimonotonously increasing: for fixedω,t and z,f(ω,t,·,z) is quasi-monotonously increase, that is for all j=1,2,…,k, such that
(4.3)fj(ω,t,y1,z)≥fj(ω,t,y2,z),∀y1,y2∈Rk,yj1=yj2,yl1≥yl2,l≠j;
(5) Functiong satisfies the (iv) of Theorem 3.2, that is forall(ω,t,y,z)∈Ω×[0,T]×Rk×Rk×d, ∃h(ω,t,μ,ν):Ω×[0,T]×R×Rd→R such that gj(ω,t,y,z)=hj(ω,t,yj,zj) and
(4.4)|hj(t,yj1,zj1)-hj(t,yj2,zj2)|2≤C|yj1-yj2|2+α|zj1-zj2|2,
where C>0, 0<α<1 are two constants. Then, if ξ∈L2(Ω,ℱT,P;Rk), BDSDE (2.5) has a solution (y,z)∈S2([0,T];Rk)×M2(0,T;Rk×d).We give a lemma, this method was first introduced by [23] Barlow and Perkins. For fixed (ω,t)∈Ω×[0,T], we define the sequence fn(ω,t,y,z):Ω×[0,T]×Rk×Rk×d→Rk as the following: j=1,2,…,k.(4.5)fjn(ω,t,y,z)≐inf⁡x∈Rk,γ∈Rk×d{fj(ω,t,x,γ)+∑l=1,l≠jkn(yl-xl)++n|yj-xj|+n‖z-γ‖}.Lemma 4.3.
Letn≥K and fn be define as in (4.5), then fn satisfies the following properties.
(1) For all(ω,t,y,z)∈Ω×[0,T]×Rk×Rk×d,
(4.6)fjn(ω,t,y,z)≤fj(ω,t,y,z).
(2) Uniformly linear growth: for all(ω,t,y,z)∈Ω×[0,T]×Rk×Rk×d,
(4.7)|fjn(ω,t,y,z)|≤K(1+|y|+|zj|).
(3) Quasimonotonous increase with respect toy: for all (ω,t)∈Ω×[0,T],z1,z2∈Rk×d(4.8)fjn(ω,t,y1,z1)≥fjn(ω,t,y2,z2),∀yj1=yj2,zj1=zj2,yl1≥yl2,l≠j.
(4) Monotonous increase with respect ton: for all (ω,t,y,z)∈Ω×[0,T]×Rk×Rk×d,
(4.9)fjn(ω,t,y,z)≤fjn+1(ω,t,y,z).
(5) Lipschitz condition: for all(ω,t)∈Ω×[0,T], (y1,z1), (y2,z2)∈Rk×Rk×d,
(4.10)|fjn(ω,t,y1,z1)-fjn(ω,t,y2,z2)|≤n(|y1-y2|+‖z1-z2‖),
(6) Strong convergence: for all(ω,t)∈Ω×[0,T], (y,z), (yn,zn)∈Rk×Rk×d, n=1,2,…, if yn→y, zn→z, when n→∞. Then
(4.11)fn(ω,t,yn,zn)⟶f(ω,t,y,z).We consider the followingk-dimensional BDSDEs(4.12)ytn=ξ+∫tTfn(s,ysn,zsn)ds+∫tTg(s,ysn,zsn)dBs-∫tTzsndWs,(4.13)Ut=ξ+∫tTF(s,Us,Vs)ds+∫tTg(s,Us,Vs)dBs-∫tTVsdWs,
where F:Ω×[0,T]×Rk×Rk×d→Rk, and(4.14)Fj(ω,t,y,z)=K(1+|y|+|zj|),j=1,2,…,k.From Lemma4.3, for all (ω,t)∈Ω×[0,T], (y1,z1), (y2,z2)∈Rk×Rk×d, yj1=yj2, zj1=zj2, yl1≥yl2, l≠j, we have(4.15)fjn(t,y2,z2)≤fjn+1(t,y2,z2)≤fjn+1(t,y1,z1)≤K(1+|y1|+|zj1|)=Fj(t,y1,z1),n=K,K+1,….From Pardoux and Peng [8], BDSDE (4.12) and (4.13) has a unique solution(4.16)(yn,zn)∈S2([0,T];Rk)×M2(0,T;Rk×d),(U,V)∈S2([0,T];Rk)×M2(0,T;Rk×d).From the comparison Theorem3.2, for all n≥m≥K,(4.17)yjm(t)≤yjn(t)≤Uj(t),dt⊗dP-a.s.Lemma 4.4.
There exists constantA>0 depending only on K,C,α,T and ξ, such that
(4.18)∀n≥K,‖yn‖S2≤A,‖zn‖M2≤A,‖U‖S2≤A,‖V‖M2≤A.Lemma 4.5.
{(yn,zn)}n=1∞ converges in S2([0,T];Rk)×M2(0,T;Rk×d).The proof of the Lemmas4.4 and 4.5 is similar to Lemma  4.2 and Lemma  4.3 in [19], so we omit it.We give the proof of Theorem4.2.Proof.
For alln≥n0≥K, we have yn0≤yn≤U, and yn converges in S2([0,T];Rk),dt⊗dP-a.s. to y∈S2([0,T];Rk).
On the other hand, sincezn converges in M2(0,T;Rk×d) to z, we can assume, choosing a subsequence if needed, that zn→zdt⊗dP-a.s. and G¯t=supn∥ztn∥ is dt⊗dP integrable.
Therefore, from (3.11) and (2.5) of the Lemma 4.3, we get for almost all ω,
(4.19)fn(t,ytn,ztn)⟶f(t,yt,zt),(n→∞)dt-a.e.|fn(t,ytn,ztn)|≤K(1+sup⁡n|ytn|+sup⁡n‖ztn‖)=K(1+sup⁡n|ytn|+G¯t)∈L1([0,T];dt).
Thus, for almost ω and uniformly in t, it holds that
(4.20)∫tTfn(s,ysn,zsn)ds⟶∫tTf(s,ys,zs)ds,(n→∞).
From the continuity properties of the stochastic integral, it follows that
(4.21)sup⁡0≤t≤T|∫tTzsndWs-∫tTzsdWs|⟶0inprobability,sup⁡0≤t≤T|∫tTg(s,ysn,zsn)dBs-g(s,ys,zs)dBs|⟶0inprobability.
Choosing, again, a subsequence, we can assume that the above convergence is P-a.s. Finally,
(4.22)|ytn-ytm|≤∫tT|fn(s,ysn,zsn)-fm(s,ysm,zsm)|ds+|∫tT(g(s,ysn,zsn)-g(s,ysm,zsm))dBs|+|∫tT(zsn-zsm)dWs|
and taking limits in m and supremum over t, we get
(4.23)sup⁡0≤t≤T|ytn-yt|≤∫0T|fn(s,ysn,zsn)-f(s,ys,zs)|ds+sup⁡0≤t≤T|∫tT(g(s,ysn,zsn)-g(s,ys,zs))dBs|+sup⁡0≤t≤T|∫tT(zsn-zs)dWs|P-a.s.
From which it follows that yn converges uniformly in t to y (in particular, y is a continuous process). Note that yn is monotone, therefore, we actually have the uniform convergence for the entire sequence and not just for a subsequence. Taking limits in (4.12), we deduce that (y,z) is the solution of (2.5).From the comparison theorem, we can get a result.Lemma 4.6.
y is the minimal solution of BDSDE (2.5).Proof.
If let(ŷ,ẑ)∈S2([0,T];Rk)×M2(0,T;Rk×d) be any solution of (2.5), from the comparison Theorem 3.2, we get that yn≤ŷ, for all n∈N, and therefore y≤ŷ. That is, y is the minimal solution of BDSDE (2.5).



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*Source: 304781-2012-04-17.xml*