# Stochastic PDEs and Infinite Horizon Backward Doubly Stochastic Differential Equations

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

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## Abstract

We give a sufficient condition on the coefficients of a class of infinite horizon BDSDEs, under which the infinite horizon BDSDEs have a unique solution for any given square integrable terminal values. We also show continuous dependence theorem and convergence theorem for this kind of equations. A probabilistic interpretation for solutions to a class of stochastic partial differential equations is given.


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## Body

## 1. Introduction

Pardoux and Peng [1] brought forward a new kind of backward doubly stochastic differential equations (BDSDEs in short); these equations are 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 on finite time interval [0,T]. 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,t∈[0,T].
Pardoux and Peng also showed that BDSDEs can produce a probabilistic representation for certain quasilinear stochastic partial differential equations (SPDEs). Many researchers do their work in this area (refer to, e.g., [2–13] and the references therein). Infinite horizon BDSDEs are also very interesting to produce a probabilistic representation of certain quasilinear stochastic partial differential equations. Recently, Zhang and Zhao [14] got stationary solutions of SPDEs and infinite horizon BDSDEs, but their researches under the assumption that terminal value limT→∞e-KTYT=0. Zhu and Han [15] also give a sufficient condition on the coefficients of a class of infinite horizon BDSDEs, but there the coefficient g is independent of z.This paper studies the existence and uniqueness of BDSDE (1.1) when T=∞. Our method is different from Zhang and Zhao. Due to sufficient utilization of the properties of martingales, this method is essential to the theory of BSDEs. In this paper we give a sufficient condition on coefficients f,  g under which, for any square integrable random variable ξ, BDSDE (1.1) still has a unique solution pair when T=∞. Our conditions are a special kind of Lipschitz conditions, which even include some cases of unbounded coefficients. This allows us to give a probabilistic interpretation for the solutions to a class of stochastic partial differential equations (SPDEs in short).The paper is organized as follows: in Section2 we introduce some preliminaries and notations; in Section 3 we prove the existence and uniqueness theorem of BDSDEs; in Section 4 we discuss continuous dependence theorem and convergence theorem; at the end, we give the connection of the solutions of SPDEs and BDSDEs in Section 5.


## 2. Setting of Infinite Horizon BDSDEs

Notation
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.
Let(Ω,ℱ,P) be a completed probability space and let {Wt}t≥0 and {Bt}t≥0 be two mutually independent standard Brownian motions, with values, respectively, in Rd and Rl, defined on (Ω,ℱ,P). Let 𝒩 denote the class of P-null sets of ℱ. For each t∈[0,∞), we define
(2.1)ℱ0,tW≐σ{Wr;0≤r≤t}⋁𝒩,ℱt,∞B≐σ{Br-Bt;t≤r<∞}⋁𝒩,ℱ0,∞W≐⋁0≤t<∞ℱ0,tW,ℱ∞,∞B≐⋂0≤t<∞‍ℱt,∞B,ℱt≐ℱ0,tW⋁ℱt,∞B,t∈[0,∞).
Note that {ℱ0,tW;t∈[0,∞)} is an increasing filtration and {ℱt,∞B;t∈[0,∞)} is a decreasing filtration, and the collection {ℱt,t∈[0,∞)} is neither increasing nor decreasing.
Suppose(2.2)ℱ=ℱ∞≐ℱ0,∞W⋁ℱ∞,∞B.
For anyn∈N, let S2(R+;Rn) denote the space of all {ℱt}-measurable n-dimensional processes v with norm of ||v||S≐[E(sups≥0|v(s)|)2]1/2<∞.
We denote similarly byM2(R+;Rn) the space of all (classes of dP⊗dt a.e. equal) {ℱt}-measurable n-dimensional processes v with norm of ||v||M≐[E∫0∞‍|v(s)|2ds]1/2<∞.
For anyt≥0, let L2(Ω,ℱt,P;Rn) denote the space of all {ℱt}-measurable n-valued random variables ξ satisfying E|ξ|2<∞.
We also denote(2.3)B2≐{(X,Y);X∈S2(R+;Rn),Y∈M2(R+;Rn)}.
For each (X,Y)∈B2, we define the norm of (X,Y) by
(2.4)∥(X,Y)∥B≐(∥X∥S2+∥Y∥M2)1/2.
Obviously B2 is a Banach space.
Consider the following infinite horizon backward doubly stochastic differential equation:(2.5)yt=ξ+∫t∞‍f(s,ys,zs)ds+∫t∞‍g(s,ys,zs)dBs-∫t∞‍zsdWs,t≥0,
where ξ∈L2(Ω,ℱ,P;Rk) is given. We 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. These two types of integrals are particular cases of the Itô-Skorohod integral; see Boufoussi et al. [12]. Our aim is to find some conditions under which BDSDE (2.5) has a unique solution. Now we give the definition of solution of BDSDE (2.5).Definition 2.1.
A pair of processes(y,z):Ω×R+→Rk×Rk×d is called a solution of BDSDE (2.5), if (y,z)∈B2 and satisfies BDSDE (2.5).
Let(2.6)f:Ω×R+×Rk×Rk×d→Rk,g:Ω×R+×Rk×Rk×d→Rk×l
satisfy the following assumptions: (H1)
for any(y,z)∈Rk×Rk×d, f(·,y,z) and g(·,y,z) are {ℱt}-progressively measurable processes, such that
(2.7)E(∫0∞‍f(t,0,0)dt)2<∞;g(·,0,0)∈M2(R+;Rk×l),(H2)
f and g satisfy Lipschitz condition with Lipschitz coefficient v:={v(t)}; that is, there exists a positive nonrandom function {v(t)} such that(2.8)|f(t,y1,z1)-f(t,y2,z2)|≤v(t)(|y1-y2|+∥z1-z2∥),∥g(t,y1,z1)-g(t,y2,z2)∥≤v(t)(|y1-y2|+∥z1-z2∥),
for all (t,yi,zi)∈R+×Rk×Rk×d,  i=1,2,(H3)
∫0∞‍v2(t)dt<∞.


## 3. Existence and Uniqueness Theorem

The following existence and uniqueness theorem is our main result.Theorem 3.1.
Under the above conditions, in particular (H1), (H2), and (H3), (2.5) has unique solution (y,z)∈B2.In order to prove the existence and uniqueness theorem, one first gives an a priori estimate.Lemma 3.2.
Suppose (H1), (H2), and (H3) hold forf and g. For any T∈[0,∞], let YTi∈L2(Ω,ℱT,P;Rk),  (Yi,Zi) and (yi,zi)∈B2(i=1,2) satisfy the following equation:
(3.1)Yti=YTi+∫tT‍f(s,ysi,zsi)ds+∫tT‍g(s,ysi,zsi)dBs-∫tT‍ZsidWs,0≤t≤T≤∞.
Then there is a constant C>0, such that, for any τ∈[0,T],
(3.2)∥((Y1-Y2)I[τ,T],(Z1-Z2)I[τ,T])∥B2≤C[E|YT1-YT2|2+l[τ,T]∥((y1-y2)I[τ,T],(z1-z2)I[τ,T])∥B2],
where l[τ,T]=∫τT‍v2(s)ds and I[τ,T](·) is an indicator function.Proof.
Firstly, we assume thatτ=0,  T=∞.
Set(3.3)Y^t=Yt1-Yt2,Z^t=Zt1-Zt2,y^t=yt1-yt2,z^t=zt1-zt2,f^t=f(t,yt1,zt1)-f(t,yt2,zt2),g^t=g(t,yt1,zt1)-g(t,yt2,zt2).
Then
(3.4)Y^t=Y^∞+∫t∞‍f^sds+∫t∞‍g^sdBs-∫t∞‍Z^sdWs.
We define the filtration {𝒢t}0≤t≤T by
(3.5)𝒢t≐ℱ0,tW⋁ℱ0,∞B.
Obviously 𝒢t is an increasing filtration. Since (Y^,Z^)∈B2, {∫0t‍Z^sdWs} is a 𝒢t-martingale, thus from (3.4) it follows that
(3.6)Y^t=E𝒢t[Y^∞+∫t∞‍f^sds+∫t∞‍g^sdBs].
Note that
(3.7)E(∫0∞‍|f^s|ds)2≤E(∫0∞‍(v(s)|y^s|+v(s)∥z^s∥)ds)2≤2E(supt≥0|y^t|2)·∫0∞‍v2(s)ds+2E(∫0∞‍v2(s)ds·∫0∞‍∥z^s∥2ds)=2∫0∞‍v2(s)ds·∥(y^,z^)∥B2<∞,E∫0∞‍∥g^s∥2ds≤E∫0∞‍(v(s)|y^s|+v(s)∥z^s∥)2ds≤2∫0∞‍v2(s)ds·∥(y^,z^)∥B2<∞.
Applying Doob inequality and B-D-G inequality, we can deduce
(3.8)∥Y^∥S2=E(supt≥0|Y^t|)2=E(supt≥0|E𝒢t[Y^∞+∫t∞‍f^sds+∫t∞‍g^sdBs]|)2≤2E(supt≥0E𝒢t[|Y^∞|+∫t∞‍|f^s|ds])2+2E(supt≥0E𝒢t[|∫t∞‍g^sdBs|])2≤8E(|Y^∞|+∫0∞‍|f^s|ds)2+2c0E∫0∞‍∥g^s∥2ds≤16(E|Y^∞|2+E(∫0∞‍|f^s|ds)2)+2c0E∫0∞‍∥g^s∥2ds≤(16+2c0)(E|Y^∞|2+E(∫0∞‍|f^s|ds)2+E∫0∞‍∥g^s∥2ds),
where c0>0 is a constant.
On the other hand, from (3.4) it follows that
(3.9)∥Z^∥M2=E〈∫0·‍Z^sdWs〉∞=E(Y^∞+∫0∞‍f^sds+∫0∞‍g^sdBs)2-[E(Y^∞+∫0∞‍f^sds+∫0∞‍g^sdBs)]2≤E(Y^∞+∫0∞‍f^sds+∫0∞‍g^sdBs)2≤3E(|Y^∞|2+(∫0∞‍|f^s|ds)2+∫0∞‍∥g^s∥2ds),
where 〈M〉 is the variation process generated by the martingale M.
Consequently, (3.8) and (3.9) imply that
(3.10)∥(Y^,Z^)∥B2=∥Y^∥S2+∥Z^∥M2≤(19+2c0)(E|Y^∞|2+E(∫0∞‍|f^s|ds)2+E∫0∞‍∥g^s∥2ds)≤(57+6c0)(E|Y^∞|2+l[0,∞]∥(y^,z^)∥B2)=C(E|Y^∞|2+l[0,∞]∥(y^,z^)∥B2),
where C=(57+6c0) is a constant, and l[0,∞]=∫0∞‍v2(s)ds.
For anyτ,T∈[0,∞], we set f1(t,yt,zt)=f(t,yt,zt)I[τ,T], and g1(t,yt,zt)=g(t,yt,zt)I[τ,T]. Then f1, and g1 satisfy the assumptions (H1), (H2), and (H3), and their Lipschitz constants are vI[τ,T].
Obviously,(3.11)Y^tI[τ,T]=Y^T+∫tT‍f^sI[τ,T]ds+∫tT‍g^sI[τ,T]dBs-∫tT‍Z^sI[τ,T]dWs.
Since (Y^I[τ,T],Z^I[τ,T])∈B2, {∫0t‍Z^sI[τ,T]dWs} is a 𝒢t-martingale, thus from (3.11) it follows that
(3.12)Y^tI[τ,T]=E𝒢t[Y^T+∫tT‍f^sI[τ,T]ds+∫tT‍g^sI[τ,T]dBs].
Note that
(3.13)E(∫0T‍|f^sI[τ,T]|ds)2≤E(∫0T‍(v(s)I[τ,T]|y^s|+v(s)I[τ,T]∥z^s∥)ds)2≤2E(supt≥0|y^tI[τ,T]|2)·∫0T‍v2(s)I[τ,T]ds+2E(∫0T‍v2(s)I[τ,T]ds·∫0T∥z^sI[τ,T]∥2‍ds)=2∫τT‍v2(s)ds·∥(y^I[τ,T],z^I[τ,T])∥B2<∞,E∫0T‍∥g^sI[τ,T]∥2ds≤E∫0T‍(v(s)I[τ,T]|y^s|+v(s)I[τ,T]∥z^s∥)2ds≤2∫τT‍v2(s)ds·∥(y^I[τ,T],z^I[τ,T])∥B2<∞.
Applying Doob inequality and B-D-G inequality, we can deduce
(3.14)∥Y^I[τ,T]∥S2=E(supt≥0|Y^tI[τ,T]|)2≤2E(supt≥0E𝒢t[|Y^T|+∫tT‍|f^sI[τ,T]|ds])2+2E(supt≥0E𝒢t[|∫tT‍g^sI[τ,T]dBs|])2≤8E(|Y^T|+∫0T‍|f^sI[τ,T]|ds)2+2c0E∫0T‍∥g^sI[τ,T]∥2ds≤16(E|Y^T|2+E(∫0T‍|f^sI[τ,T]|ds)2)+2c0E∫0T‍∥g^sI[τ,T]∥2ds≤(16+2c0)(E|Y^T|2+E(∫0T‍|f^sI[τ,T]|ds)2+E∫0T‍∥g^sI[τ,T]∥2ds),
where c0>0 is a constant.
On the other hand, from (3.11) it follows that
(3.15)∥Z^I[τ,T]∥M2=E〈∫0·‍Z^sI[τ,T]dWs〉∞=E(Y^T+∫0T‍f^sI[τ,T]ds+∫0T‍g^sI[τ,T]dBs)2-[E(Y^T+∫0T‍f^sI[τ,T]ds+∫0T‍g^sI[τ,T]dBs)]2≤E(Y^T+∫0T‍f^sI[τ,T]ds+∫0T‍g^sdI[τ,T]Bs)2≤3E(|Y^T|2+(∫0T‍|f^sI[τ,T]|ds)2+∫0T‍∥g^sI[τ,T]∥2ds).
Consequently, (3.14) and (3.15) imply that
(3.16)∥(Y^I[τ,T],Z^I[τ,T])∥B2=∥Y^I[τ,T]∥S2+∥Z^I[τ,T]∥M2≤(19+2c0)(E|Y^T|2+E(∫0T‍|f^sI[τ,T]|ds)2+E∫0T‍∥g^sI[τ,T]∥2ds)≤(57+6c0)(E|Y^T|2+l[τ,T]∥(y^I[τ,T],z^I[τ,T])∥B2)=C(E|Y^T|2+l[τ,T]∥(y^I[τ,T],z^I[τ,T])∥B2),
where C=(57+6c0) is a constant, and l[τ,T]=∫τT‍v2(s)ds.Martingale Representation Theorem [4]
SupposeY is a random variable, such that E|Y2|<∞. Note that Mt=E[Y∣𝒢t] is a square integrable martingale with respect to 𝒢t and can be represented using martingale representation theorem as Mt=M0+∫0t‍ZsdWs, where E∫0∞‍∥Zt∥2dt<∞.
Now we give the proof of the Theorem3.1.Proof.
The proof of Theorem3.1 is divided into two steps.
Step 1. We assume l[0,∞]=∫0∞‍v2(s)ds≤1/2C. For any (y,z)∈B2, let
(3.17)Mt≐E𝒢t[ξ+∫0∞‍f(s,ys,zs)ds+∫0∞‍g(s,ys,zs)dBs],t≥0.
We will prove {Mt} is a square integrable 𝒢t-martingale. From (H1)–(H3), it follows that
(3.18)E(|ξ+∫0∞‍f(s,ys,zs)ds+∫0∞‍g(s,ys,zs)dBs|2)≤E(|ξ|+∫0∞‍|f(s,ys,zs)|ds+|∫0∞‍g(s,ys,zs)dBs|)2≤3E|ξ|2+9E(∫0∞‍|f(s,0,0)|ds)2+9E(∫0∞‍v(s)|ys|ds)2+9E(∫0∞‍v(s)∥zs∥ds)2+9E∫0∞‍(∥g(s,0,0)∥2+v2(s)|ys|2+v2(s)∥zs∥2)ds≤3E|ξ|2+9E(∫0∞‍|f(s,0,0)|ds)2+9∫0∞‍v2(s)ds·∥y∥S2+9∫0∞‍v2(s)ds·∥z∥M2+9E∫0∞‍∥g(s,0,0)∥2ds+9∫0∞‍v2(s)ds·∥y∥S2+9∫0∞‍v2(s)ds·∥z∥M2<∞,
which means {Mt} is a square integrable 𝒢t-martingale. According to the martingale representation theorem, there exists a unique 𝒢t-progressively measurable process Zt with value in Rk×d such that
(3.19)E∫0∞‍∥Zt∥2dt<∞,Mt=E𝒢0[ξ+∫0∞‍f(s,ys,zs)ds+∫0∞‍g(s,ys,zs)dBs]+∫0t‍ZsdWs,0≤t≤∞.
Let
(3.20)Yt≐E𝒢t[ξ+∫t∞‍f(s,ys,zs)ds+∫t∞‍g(s,ys,zs)dBs],0≤t≤∞.
So
(3.21)Mt=E𝒢t[ξ+∫0∞‍f(s,ys,zs)ds+∫0∞‍g(s,ys,zs)dBs]=E𝒢t[ξ+∫t∞‍f(s,ys,zs)ds+∫t∞‍g(s,ys,zs)dBs]+∫0t‍f(s,ys,zs)ds+∫0t‍g(s,ys,zs)dBs=Yt+∫0t‍f(s,ys,zs)ds+∫0t‍g(s,ys,zs)dBs=E𝒢0[ξ+∫0∞‍f(s,ys,zs)ds+∫0∞‍g(s,ys,zs)dBs]+∫0t‍ZsdWs.
Then
(3.22)Yt=ξ+∫t∞‍f(s,ys,zs)ds+∫t∞‍g(s,ys,zs)dBs-∫t∞‍ZsdWs,t≥0.
We show that {Yt} and {Zt} are in fact ℱt-measurable. For Yt, this is obvious since, for each t,
(3.23)Yt=E𝒢t[ξ+∫t∞‍f(s,ys,zs)ds+∫t∞‍g(s,ys,zs)dBs]=E(Θℱt⋁ℱ0,tB),
where Θ=ξ+∫t∞‍f(s,ys,zs)ds+∫t∞‍g(s,ys,zs)dBs is indeed ℱ0,∞W⋁ℱt,∞B-measurable. Hence ℱ0,tB is independent of ℱt⋁σ(Θ), and
(3.24)Yt=E(Θℱt).
Now
(3.25)∫t∞‍ZsdWs=ξ+∫t∞‍f(s,ys,zs)ds+∫t∞‍g(s,ys,zs)dBs-Yt
and the right side is ℱ0,∞W⋁ℱt,∞B-measurable. Hence, from Itô’s martingale representation theorem, {Zs,s>t} is ℱ0,sW⋁ℱt,∞B-adapted. Consequently Zs is ℱ0,sW⋁ℱt,∞B-measurable, for any t<s, and, thus, Zt is ℱt-measurable. So (Y,Z)∈B2. Therefore (3.22) has constructed a mapping from B2 to B2, and we denote it by ϕ, that is,
(3.26)ϕ:(y,z)→(Y,Z).
If ϕ is a contractive mapping with respect to the norm ∥·∥B, by the fixed point theorem, there exists a unique (y,z)∈B2, satisfying (3.22), which is just the unique solution to BDSDE (2.5).
Now we are in the position to prove thatϕ is a contractive mapping. Supposing that (yi,zi)∈B2, let (Yi,Zi) be the map ϕ of (yi,zi),(i=1,2), that is
(3.27)ϕ(yi,zi)=(Yi,Zi),i=1,2.
We denote
(3.28)Y^=Y1-Y2,Z^=Z1-Z2,y^=y1-y2,z^=z1-z2,f^t=f(t,y1,z1)-f(t,y2,z2),g^t=g(t,y1,z1)-g(t,y2,z2).
By Lemma 3.2, we have
(3.29)∥ϕ(y1,z1)-ϕ(y2,z2)∥B2=∥(Y^,Z^)∥B2≤Cl[0,∞]∥(y^,z^)∥B2.
Due tol[0,∞]≤1/2C, it follows that ϕ is a contractive mapping from B2 to B2.
Step 2. Since ∫0∞‍v2(t)dt<∞, then there exists a sufficiently large constant T such that
(3.30)∫T∞‍v2(s)ds≤12C.
Let
(3.31)f1(t,y,z)≐I[T,∞](t)f(t,y,z),g1(t,y,z)≐I[T,∞](t)g(t,y,z),
then (H1)–(H3) hold for f1 and g1, whose Lipschitz coefficients are v-(t)=I[T,∞]v(t). Obviously,
(3.32)∫0∞‍v-2(s)ds≤12C.
By Step 1, there exists a unique (y~,z~)∈B2 satisfying
(3.33)y~t=ξ+∫t∞‍f1(s,y~s,z~s)ds+∫t∞‍g1(s,y~s,z~s)dBs-∫t∞‍z~sdWs,0≤t≤∞.
For (y~t,z~t) given as above, let us consider the following infinite BDSDE:
(3.34)y-t=∫tT‍f(s,y-s+y~s,z-s+z~s)ds+∫tT‍g(s,y-s+y~s,z-s+z~s)dBs-∫tT‍z-sdWs,0≤t≤T,y-t≡0,z-t≡0,t>T.
According to the results of Pardoux and Peng [1], the above BDSDE has a unique solution (y-,z-) in [0,T], thus the above BDSDE has a unique solution such that (y-,z-)≡(0,0) for every t>T. Let
(3.35)y≐y-+y~,z≐z-+z~.
It is easy to check that (yt,zt) is the unique solution of (2.5).Remark 3.3.
Supposev is a constant, if we choose v(t)=vI[0,T](t), then Theorem 3.1 is the main theorem in the paper by Pardoux and Peng [1].Remark 3.4.
The condition (H3) is usually necessary. That is, if for anyξ∈L2(Ω,ℱ,P;Rk) and f,g hold in (H1) and (H2), BDSDE (2.5) has a unique solution in B2, then the (H3) is necessary.
In fact, let us choosef(s,ys,zs)=(1/(1+s))zs,g(s,ys,zs)=0 and any ξ∈L2(Ω,ℱ,P;Rk), then the solution of BDSDE
(3.36)yt=ξ+∫t∞‍11+szsds-∫t∞‍zsdWs
should be
(3.37)yt=E[ξexp(-(1/2)∫t∞‍(1/(1+s))2ds+∫t∞‍(1/(1+s))dWt)ℱt],zt=d〈yt,Wt〉dt,
where 〈yt,Wt〉 is the variation process generated by the semimartingale yt and Brownian motion Wt.
Thus the assumption (H3) is necessary.Remark 3.5.
The following example shows that if the coefficientsf and g of BDSDE (2.5) satisfy the uniformly Lipschitz, the BDSDE (2.5) has no solution.
For allT>0, let ξT=∫0T‍(1/(1+s))dWs, then the BDSDE yt=ξ+∫tT‍v|zs|ds-∫tT‍zsdWs has a unique solution pair (ytT,ztT),
(3.38)ytT={ξ,t>T,∫0t‍11+sdWs+∫tT‍v1+sds,t≤T,ztT={0,t>T,11+t,t≤T.
WhenT→∞, ξT→∫0∞‍(1/(1+s))dWs and ztT→1/(1+t) in L2(Ω,ℱ,P), but zt=1/(1+t) is not the solution of the following infinite horizon BDSDE:
(3.39)yt=∫0∞‍11+sdWs+∫t∞‍v|zs|ds-∫t∞‍zsdWs
because ∫0∞‍(v/(1+s))ds=∞.


## 4. Continuous Dependence Theorem

In this section we will discuss the convergence of solutions of infinite horizon BDSDEs. First we give the following continuous dependence theorem.Theorem 4.1.
Supposeξi∈L2(Ω,ℱ,P;Rk),(i=1,2), and consider (H1)–(H3). Let (yi,zi) be the solutions of BDSDE (2.5) corresponding to the terminal data ξ=ξ1,ξ=ξ2, respectively. Then there exists a constant C->0 such that
(4.1)∥(y1-y2,z1-z2)∥B2≤C-E|ξ1-ξ2|2.Proof.
Sety^:=y1-y2,  z^:=z1-z2. Since ∫0∞‍v2(s)ds<∞, we can choose a strictly increasing sequence 0=t0<t1<⋯<tn<tn+1=∞ such that
(4.2)l[ti,ti+1]=∫titi+1‍v2(s)ds≤12C,i=0,1,…,n.
Applying Lemma 3.2, we have
(4.3)∥(y^,z^)I[ti,ti+1]∥B2≤CE|y^ti+1|2+Cl[ti,ti+1]∥(y^,z^)I[ti,ti+1]∥B2≤CE|y^ti+1|2+12∥(y^,z^)I[ti,ti+1]∥B2.
Thus
(4.4)∥(y^,z^)I[ti,ti+1]∥B2≤2CE|y^ti+1|2≤2CE((supti+1≤s≤ti+2|y^s|)2+∫ti+1ti+2‍∥z^s∥2ds)=2C∥(y^,z^)I[ti+1,ti+2]∥B2,i=0,1,…,n-1.
In particular, we have
(4.5)∥(y^,z^)I[tn,∞]∥B2≤2CE|ξ1-ξ2|2.
From (4.4) and (4.5), it follows that
(4.6)∥(y^,z^)∥B2≤∑i=0n‍∥(y^,z^)I[ti,ti+1]∥B2≤(2C+(2C)2+⋯+(2C)n+1)E|ξ1-ξ2|2=2C((2C)n+1-1)2C-1E|ξ1-ξ2|2=C-E|ξ1-ξ2|2.
Thus the desired result is obtained.Now we can assert the following convergence theorem for infinite horizon BDSDEs.Theorem 4.2.
Supposeξ,ξi∈L2(Ω,ℱ,P;Rk),(i=1,2,…), (H1)–(H3) hold for f and g. Let (yi,zi) be the solutions of the following BDSDE:
(4.7)yti=ξi+∫t∞‍f(s,ysi,zsi)ds+∫t∞‍g(s,ysi,zsi)dBs-∫t∞‍zsidWs,t≥0.
If E|ξi-ξ|2→0 as i→∞, then there exists a pair (y,z)∈B2 such that ||(yi-y,zi-z)||B→0 as i→∞. Furthermore, (y,z) is the solution of the following BDSDE:
(4.8)yt=ξ+∫t∞‍f(s,ys,zs)ds+∫t∞‍g(s,ys,zs)dBs-∫t∞‍zsdWs,0≤t<∞.Proof.
For anyn,m≥1, let (yn,zn) and (ym,zm) be the solutions of (4.7) corresponding to ξn and ξm, respectively. Due to Theorem 4.1, there exists a constant C->0 such that
(4.9)∥(yn-ym,zn-zm)∥B2≤C-E|ξn-ξm|2≤2C-(E|ξn-ξ|2+E(ξm-ξ)2)→0,asn,m→∞,
which means that {(yi,zi),i=1,2,…} is a Cauchy sequence in B2. Thus there exists a pair (y,z)∈B2 such that ||(yi-y,zi-z)||B→0 as i→∞. Since
(4.10)E|∫t∞‍(f(s,ysi,zsi)-f(s,ys,zs))ds|2≤E(∫0∞‍(v(s)|ysi-ys|+v(s)∥zsi-zs∥)ds)2≤2∫0∞‍v2(s)ds·∥(yi-y,zi-z)∥B2→0,asi→∞,E|∫t∞‍(g(s,ysi,zsi)-g(s,ys,zs))ds|2≤2∫0∞‍v2(s)ds·∥(yi-y,zi-z)∥B2→0,asi→∞.
Thus for any t∈R+,∫t∞‍f(s,ysi,zsi)ds→∫t∞‍f(s,ys,zs)ds and ∫t∞‍g(s,ysi,zsi)dBs→∫t∞‍g(s,ys,zs)dBs in L2(Ω,ℱ,P). Taking the limit on both sides of (4.7), we deduce that (y,z) is the solution to BDSDE (4.8). The desired result is obtained.The following corollary shows the relation between the solution of infinite horizon BDSDE (2.5) and  the following finite time BDSDE:
(4.11)yt=EℱT[ξ]+∫tT‍f(s,ys,zs)ds+∫tT‍g(s,ys,zs)dBs-∫tT‍zsdWs,0≤t≤T<∞.Corollary 4.3.
Assumeξ∈L2(Ω,ℱ,P;Rk), (H1)–(H3) hold for f and g. Let (y,z) be the solution of BDSDE (2.5). For any T>0, let (yT,zT) be the solutions of the finite time interval BDSDE (4.11), then(yT,zT)→(y,z) in B2 as T→∞.Proof.
Note thatEℱT[ξ]→ξ in L2(Ω,ℱ,P;Rk) as T→∞. The proof is straightforward from Theorem 4.2.


## 5. BDSDEs and Systems of Quasilinear SPDEs

In this section, we study the link between BDSDEs and the solution of a class of SPDEs.Let us first give some notations.Ck(Rp;Rq),  Cl,bk(Rp;Rq),  Cpk(Rp;Rq) will denote, respectively, the set of functions of classes from Rp into Rq, the set of those functions of class Ck whose partial derivatives of order less than or equal to k are bounded (and hence the function itself grows at most linearly at infinity), and the set of those functions of class Ck which, together with all their partial derivatives of order less than or equal to k, grow at most like a polynomial function of the variable x at infinity.Fors≥t, let Xst,x be a diffusion process given by the solution of
(5.1)Xst,x=x+∫ts‍b(Xμt,x)dμ+∫ts‍σ(Xμt,x)dWμ,
where b∈Cl,b3(Rd;Rd),σ∈Cl,b3(Rd;Rd×Rd), and, for 0≤s<t, we regulate Xst,x=x.It is well known that the solution defines a stochastic flow of diffeomorphismXst,·:Rd→Rd and denotes by X^st,· the inverse flow (see e.g., [15]). The random field Xst,x;  s≥t,  x∈Rd has a version which is a.s. of class C2 in x, the function and its derivatives being a.s. continuous with respect to (t,s,x).Now the coefficients of the BDSDE will be of the form (with an obvious abuse of notations):(5.2)f(s,y,z)=f(s,Xst,x,y,z);g(s,y,z)=g(s,Xst,x,y,z),
where f:[0,T]×Rd×Rk×Rk×d→Rk;  g:[0,T]×Rd×Rk×Rk×d→Rk×l.We assume that for anys≥t,  (x,y,z)→(f(s,x,y,z),g(s,x,y,z)) is of class C3, and all derivatives are bounded on [0,∞)×Rd×Rk×Rk×d.We assume again that (H1), (H2), and (H3) hold, then the following BDSDE has a unique solution:(5.3)Yst,x=ξ+∫s∞‍f(r,Xrt,x,Yrt,x,Zrt,x)dr+∫s∞‍g(r,Xrt,x,Yrt,x,Zrt,x)dBr-∫s∞‍Zrt,xdWr,t≥0.
Let (Yst,x,Zst,x) denote the unique solution of (4.11). We shall define Xst,x,Yst,x, and Zst,x for all (s,t)∈[0,∞)×[0,∞) by letting Xst,x=Xs∨tt,x,  Yst,x=Ys∨tt,x,  and Zst,x=0 for s<t.We now relate our BDSDE to the following system of quasilinear backward stochastic partial differential equations:(5.4)dκ(t,x)=-[ℒκ(t,x)+f(x,κ(t,x),σ*(x)Dκ(t,x))]dt-g(x,κ(t,x),σ*(x)Dκ(t,x))dBt,t≥0.ℒ is the infinitesimal generator of a diffusion process Xst,x (solution of (5.1)) given by
(5.5)ℒ=12∑i,j=1d‍aij(x)∂2∂xi∂xj+∑i=1d‍bi(x)∂∂xi,
where (ai,j(x))=σσ*(x).Theorem 5.1.
Letκ(t,x);t≥0,x∈Rd be a random field such that κ(t,x) is ℱt,∞B-measurable for each (t,x),κ∈C0,2([0,∞)×Rd;Rk) a.s., and κ satisfies (5.4). Then κ(t,x)=Ytt,x, where (Yst,x,Zst,x;s≥t)t≥0 solves the BDSDE (5.3).Proof.
We can apply the extension of the Itô formula [5] to the solution κ of (5.4):
(5.6)dκ(s,Xst,x)=ℒκ(s,Xst,x)ds+σ*Dκ(s,Xst,x)dWs-[ℒ(κ(s,Xst,x))+f(s,Xst,x,κ(s,Xst,x),σ*Dκ(s,Xst,x))]ds-g(s,Xst,x,κ(s,Xst,x),σ*Dκ(s,Xst,x))dBs.
We can see that (κ(s,Xst,x),σ*Dκ(s,Xst,x)) coincides with the unique solution of (5.3). It follows that κ(t,x)=Ytt,x.We have also a converse to Theorem5.1.Theorem 5.2.
Letf and g satisfy (H1), (H2), and (H3). Then {κ(t,x)=Ytt,x;t≥0,x∈Rd} is the unique classical solution of the system of backward SPDEs (5.3).
We can finish the proof exactly as in Theorem 3.2 of Hu and Ren [13].



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*Source: 582645-2012-12-18.xml*