Stochastic PDEs and Infinite Horizon Backward Doubly Stochastic Differential Equations

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.

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 𝑑𝑊_𝑡 and a backward stochastic integral 𝑑𝐵_𝑡. 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 (𝑦_𝑡,𝑧_𝑡) in the interval [0,𝑇]: 𝑌_𝑡=𝜉+^𝑇_𝑡𝑓𝑠,𝑌_𝑠,𝑍_𝑠𝑑𝑠+^𝑇_𝑡𝑔𝑠,𝑌_𝑠,𝑍_𝑠𝑑𝐵_𝑠−^𝑇_𝑡𝑍_𝑠𝑑𝑊_𝑠,𝑡∈[0,𝑇]. (1.1) 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 lim_𝑇→∞𝑒^−𝐾𝑇𝑌_𝑇=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 Section 2 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 vector 𝑥∈𝑅^𝑘 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 {𝑊_𝑡}_𝑡≥0 and {𝐵_𝑡}_𝑡≥0 be two mutually independent standard Brownian motions, with values, respectively, in 𝑅^𝑑 and 𝑅^𝑙, defined on (Ω,ℱ,𝑃). Let 𝒩 denote the class of 𝑃-null sets of ℱ. For each 𝑡∈[0,∞), we define ℱ^𝑊_0,𝑡≐𝜎𝑊_𝑟;0≤𝑟≤𝑡𝒩,ℱ^𝐵_𝑡,∞≐𝜎𝐵_𝑟−𝐵_𝑡ℱ;𝑡≤𝑟<∞𝒩,_𝑊0,∞≐_0≤𝑡<∞ℱ^𝑊_0,𝑡,ℱ^𝐵_∞,∞≐_0≤𝑡<∞ℱ^𝐵_𝑡,∞,ℱ_𝑡≐ℱ^𝑊_0,𝑡ℱ^𝐵_𝑡,∞,𝑡∈[0,∞). (2.1) Note that {ℱ^𝑊_0,𝑡;𝑡∈[0,∞)} is an increasing filtration and {ℱ^𝐵_𝑡,∞;𝑡∈[0,∞)} is a decreasing filtration, and the collection {ℱ_𝑡,𝑡∈[0,∞)} is neither increasing nor decreasing.
Suppose ℱ=ℱ_∞≐ℱ^𝑊_0,∞ℱ^𝐵_∞,∞. (2.2)
For any 𝑛∈𝑁, let 𝑆²(𝑅⁺;𝑅^𝑛) denote the space of all {ℱ_𝑡}-measurable 𝑛-dimensional processes 𝑣 with norm of ||𝑣||_𝑆≐[𝐸(sup_𝑠≥0|𝑣(𝑠)|)²]^1/2<∞.
We denote similarly by 𝑀²(𝑅⁺;𝑅^𝑛) the space of all (classes of 𝑑𝑃⊗𝑑𝑡 a.e. equal) {ℱ_𝑡}-measurable 𝑛-dimensional processes 𝑣 with norm of ||𝑣||_𝑀≐[𝐸∫^∞₀|𝑣(𝑠)|²𝑑𝑠]^1/2<∞.
For any 𝑡≥0, let 𝐿²(Ω,ℱ_𝑡,𝑃;𝑅^𝑛) denote the space of all {ℱ_𝑡}-measurable 𝑛-valued random variables 𝜉 satisfying 𝐸|𝜉|²<∞.
We also denote 𝐵²≐(𝑋,𝑌);𝑋∈𝑆²𝑅⁺;𝑅^𝑛,𝑌∈𝑀²𝑅⁺;𝑅^𝑛. (2.3) For each (𝑋,𝑌)∈𝐵², we define the norm of (𝑋,𝑌) by ‖(𝑋,𝑌)‖_𝐵≐‖𝑋‖²_𝑆+‖𝑌‖²_𝑀^1/2. (2.4) Obviously 𝐵² is a Banach space.
Consider the following infinite horizon backward doubly stochastic differential equation: 𝑦_𝑡=𝜉+^∞_𝑡𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^∞_𝑡𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠−^∞_𝑡𝑧_𝑠𝑑𝑊_𝑠,𝑡≥0, (2.5) where 𝜉∈𝐿²(Ω,ℱ,𝑃;𝑅^𝑘) is given. We note that the integral with respect to {𝐵_𝑡} is a backward Itô integral and the integral with respect to {𝑊_𝑡} 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⁺→𝑅^𝑘×𝑅^𝑘×𝑑 is called a solution of BDSDE (2.5), if (𝑦,𝑧)∈𝐵² and satisfies BDSDE (2.5).
Let 𝑓∶Ω×𝑅⁺×𝑅^𝑘×𝑅^𝑘×𝑑→𝑅^𝑘,𝑔∶Ω×𝑅⁺×𝑅^𝑘×𝑅^𝑘×𝑑→𝑅^𝑘×𝑙 (2.6) satisfy the following assumptions: (H1) for any (𝑦,𝑧)∈𝑅^𝑘×𝑅^𝑘×𝑑, 𝑓(⋅,𝑦,𝑧) and 𝑔(⋅,𝑦,𝑧) are {ℱ_𝑡}-progressively measurable processes, such that 𝐸^∞₀𝑓(𝑡,0,0)𝑑𝑡²<∞;𝑔(⋅,0,0)∈𝑀²𝑅⁺;𝑅^𝑘×𝑙, (2.7)(H2)𝑓 and 𝑔 satisfy Lipschitz condition with Lipschitz coefficient 𝑣∶={𝑣(𝑡)}; that is, there exists a positive nonrandom function {𝑣(𝑡)} such that𝑓𝑡,𝑦₁,𝑧₁−𝑓𝑡,𝑦₂,𝑧₂≤𝑣(𝑡)𝑦₁−𝑦₂+𝑧₁−𝑧₂,𝑔𝑡,𝑦₁,𝑧₁−𝑔𝑡,𝑦₂,𝑧₂≤𝑣(𝑡)𝑦₁−𝑦₂+𝑧₁−𝑧₂, (2.8) for all (𝑡,𝑦_𝑖,𝑧_𝑖)∈𝑅⁺×𝑅^𝑘×𝑅^𝑘×𝑑,  𝑖=1,2, (H3)∫^∞₀𝑣²(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 for 𝑓 and 𝑔. For any 𝑇∈[0,∞], let 𝑌^𝑖_𝑇∈𝐿²(Ω,ℱ_𝑇,𝑃;𝑅^𝑘),  (𝑌^𝑖,𝑍^𝑖) and (𝑦^𝑖,𝑧^𝑖)∈𝐵²(𝑖=1,2) satisfy the following equation: 𝑌^𝑖_𝑡=𝑌^𝑖_𝑇+^𝑇_𝑡𝑓𝑠,𝑦^𝑖_𝑠,𝑧^𝑖_𝑠𝑑𝑠+^𝑇_𝑡𝑔𝑠,𝑦^𝑖_𝑠,𝑧^𝑖_𝑠𝑑𝐵_𝑠−^𝑇_𝑡𝑍^𝑖_𝑠𝑑𝑊_𝑠,0≤𝑡≤𝑇≤∞. (3.1) Then there is a constant 𝐶>0, such that, for any 𝜏∈[0,𝑇], 𝑌¹−𝑌²𝐼_[𝜏,𝑇],𝑍¹−𝑍²𝐼_[𝜏,𝑇]²_𝐵≤𝐶𝐸𝑌¹_𝑇−𝑌²_𝑇²+𝑙_[𝜏,𝑇]𝑦¹−𝑦²𝐼_[𝜏,𝑇],𝑧¹−𝑧²𝐼_[𝜏,𝑇]²_𝐵, (3.2) where 𝑙_[𝜏,𝑇]=∫^𝑇_𝜏𝑣²(𝑠)𝑑𝑠 and 𝐼_[𝜏,𝑇](⋅) is an indicator function.

Proof. Firstly, we assume that 𝜏=0,  𝑇=∞.
Set 𝑌_𝑡=𝑌¹_𝑡−𝑌²_𝑡,𝑍_𝑡=𝑍¹_𝑡−𝑍²_𝑡,𝑦_𝑡=𝑦¹_𝑡−𝑦²_𝑡,𝑧_𝑡=𝑧¹_𝑡−𝑧²_𝑡,𝑓_𝑡=𝑓𝑡,𝑦¹_𝑡,𝑧¹_𝑡−𝑓𝑡,𝑦²_𝑡,𝑧²_𝑡,𝑔_𝑡=𝑔𝑡,𝑦¹_𝑡,𝑧¹_𝑡−𝑔𝑡,𝑦²_𝑡,𝑧²_𝑡. (3.3) Then 𝑌_𝑡=𝑌_∞+^∞_𝑡𝑓_𝑠𝑑𝑠+^∞_𝑡𝑔_𝑠𝑑𝐵_𝑠−^∞_𝑡𝑍_𝑠𝑑𝑊_𝑠. (3.4) We define the filtration {𝒢_𝑡}_{0≤𝑡≤𝑇} by 𝒢_𝑡≐ℱ^𝑊_0,𝑡ℱ^𝐵_0,∞. (3.5) Obviously 𝒢_𝑡 is an increasing filtration. Since (𝑌,𝑍)∈𝐵², {∫^𝑡₀𝑍_𝑠𝑑𝑊_𝑠} is a 𝒢_𝑡-martingale, thus from (3.4) it follows that 𝑌_𝑡=𝐸^𝒢𝑡𝑌_∞+^∞_𝑡𝑓_𝑠𝑑𝑠+^∞_𝑡𝑔_𝑠𝑑𝐵_𝑠. (3.6) Note that 𝐸^∞₀|𝑓_𝑠|𝑑𝑠²≤𝐸^∞₀𝑣(𝑠)𝑦_𝑠+𝑣(𝑠)𝑧_𝑠𝑑𝑠²≤2𝐸sup_𝑡≥0𝑦_𝑡²⋅^∞₀𝑣²(𝑠)𝑑𝑠+2𝐸^∞₀𝑣²(𝑠)𝑑𝑠⋅^∞₀𝑧_𝑠²𝑑𝑠=2^∞₀𝑣²(𝑠)𝑑𝑠⋅𝑦,𝑧²_𝐵𝐸<∞,_∞0𝑔_𝑠²𝑑𝑠≤𝐸^∞₀𝑣(𝑠)𝑦_𝑠+𝑣(𝑠)𝑧_𝑠²𝑑𝑠≤2^∞₀𝑣²(𝑠)𝑑𝑠⋅𝑦,𝑧²_𝐵<∞. (3.7) Applying Doob inequality and B-D-G inequality, we can deduce 𝑌²_𝑆=𝐸sup_𝑡≥0𝑌_𝑡²=𝐸sup_𝑡≥0𝐸^𝒢𝑡𝑌_∞+^∞_𝑡𝑓_𝑠𝑑𝑠+^∞_𝑡𝑔_𝑠𝑑𝐵_𝑠²≤2𝐸sup_𝑡≥0𝐸^𝒢𝑡𝑌_∞+^∞_𝑡𝑓_𝑠𝑑𝑠²+2𝐸sup_𝑡≥0𝐸^𝒢𝑡∞_𝑡𝑔_𝑠𝑑𝐵_𝑠²≤8𝐸𝑌_∞+^∞₀𝑓_𝑠𝑑𝑠²+2𝑐₀𝐸^∞₀𝑔_𝑠²𝐸𝑌𝑑𝑠≤16_∞²+𝐸^∞₀𝑓_𝑠𝑑𝑠²+2𝑐₀𝐸^∞₀𝑔_𝑠²≤𝑑𝑠16+2𝑐₀𝐸𝑌_∞²+𝐸^∞₀𝑓_𝑠𝑑𝑠²+𝐸^∞₀𝑔_𝑠²𝑑𝑠, (3.8) where 𝑐₀>0 is a constant.
On the other hand, from (3.4) it follows that 𝑍²_𝑀=𝐸^⋅₀𝑍_𝑠𝑑𝑊_𝑠∞=𝐸𝑌_∞+^∞₀𝑓_𝑠𝑑𝑠+^∞₀𝑔_𝑠𝑑𝐵_𝑠²−𝐸𝑌_∞+^∞₀𝑓_𝑠𝑑𝑠+^∞₀𝑔_𝑠𝑑𝐵_𝑠²≤𝐸𝑌_∞+^∞₀𝑓_𝑠𝑑𝑠+^∞₀𝑔_𝑠𝑑𝐵_𝑠²≤3𝐸𝑌_∞²+^∞₀𝑓_𝑠𝑑𝑠²+^∞₀𝑔_𝑠²𝑑𝑠, (3.9) where ⟨𝑀⟩ is the variation process generated by the martingale 𝑀.
Consequently, (3.8) and (3.9) imply that 𝑌,𝑍²_𝐵=𝑌²_𝑆+𝑍²_𝑀≤19+2𝑐₀𝐸𝑌_∞²+𝐸^∞₀𝑓_𝑠𝑑𝑠²+𝐸^∞₀𝑔_𝑠²≤𝑑𝑠57+6𝑐₀𝐸𝑌_∞²+𝑙_[0,∞]𝑦,𝑧²_𝐵=𝐶𝐸𝑌_∞²+𝑙_[0,∞]𝑦,𝑧²_𝐵, (3.10) where 𝐶=(57+6𝑐₀) is a constant, and 𝑙_[0,∞]=∫^∞₀𝑣²(𝑠)𝑑𝑠.
For any 𝜏,𝑇∈[0,∞], we set 𝑓₁(𝑡,𝑦_𝑡,𝑧_𝑡)=𝑓(𝑡,𝑦_𝑡,𝑧_𝑡)𝐼_[𝜏,𝑇], and 𝑔₁(𝑡,𝑦_𝑡,𝑧_𝑡)=𝑔(𝑡,𝑦_𝑡,𝑧_𝑡)𝐼_[𝜏,𝑇]. Then 𝑓₁, and 𝑔₁ satisfy the assumptions (H1), (H2), and (H3), and their Lipschitz constants are 𝑣𝐼_[𝜏,𝑇].
Obviously, 𝑌_𝑡𝐼_[𝜏,𝑇]=𝑌_𝑇+^𝑇_𝑡𝑓_𝑠𝐼_[𝜏,𝑇]𝑑𝑠+^𝑇_𝑡𝑔_𝑠𝐼_[𝜏,𝑇]𝑑𝐵_𝑠−^𝑇_𝑡𝑍_𝑠𝐼_[𝜏,𝑇]𝑑𝑊_𝑠. (3.11) Since (𝑌𝐼_[𝜏,𝑇],𝑍𝐼_[𝜏,𝑇])∈𝐵², {∫^𝑡₀𝑍_𝑠𝐼_[𝜏,𝑇]𝑑𝑊_𝑠} is a 𝒢_𝑡-martingale, thus from (3.11) it follows that 𝑌_𝑡𝐼_[𝜏,𝑇]=𝐸^𝒢𝑡𝑌_𝑇+^𝑇_𝑡𝑓_𝑠𝐼_[𝜏,𝑇]𝑑𝑠+^𝑇_𝑡𝑔_𝑠𝐼_[𝜏,𝑇]𝑑𝐵_𝑠. (3.12) Note that 𝐸^𝑇₀𝑓_𝑠𝐼_[𝜏,𝑇]𝑑𝑠²≤𝐸^𝑇₀𝑣(𝑠)𝐼_[𝜏,𝑇]𝑦_𝑠+𝑣(𝑠)𝐼_[𝜏,𝑇]𝑧_𝑠𝑑𝑠²≤2𝐸sup_𝑡≥0𝑦_𝑡𝐼_[𝜏,𝑇]²⋅^𝑇₀𝑣²(𝑠)𝐼_[𝜏,𝑇]𝑑𝑠+2𝐸^𝑇₀𝑣²(𝑠)𝐼_[𝜏,𝑇]𝑑𝑠⋅^𝑇₀𝑧_𝑠𝐼_[𝜏,𝑇]²𝑑𝑠=2^𝑇_𝜏𝑣²(𝑠)𝑑𝑠⋅𝑦𝐼_[𝜏,𝑇],𝑧𝐼_[𝜏,𝑇]²_𝐵𝐸<∞,_𝑇0𝑔_𝑠𝐼_[𝜏,𝑇]²𝑑𝑠≤𝐸^𝑇₀𝑣(𝑠)𝐼_[𝜏,𝑇]𝑦_𝑠+𝑣(𝑠)𝐼_[𝜏,𝑇]𝑧_𝑠²𝑑𝑠≤2^𝑇_𝜏𝑣²(𝑠)𝑑𝑠⋅𝑦𝐼_[𝜏,𝑇],𝑧𝐼_[𝜏,𝑇]²_𝐵<∞. (3.13) Applying Doob inequality and B-D-G inequality, we can deduce 𝑌𝐼_[𝜏,𝑇]²_𝑆=𝐸sup_𝑡≥0|𝑌_𝑡𝐼_[𝜏,𝑇]|²≤2𝐸sup_𝑡≥0𝐸^𝒢𝑡𝑌_𝑇+^𝑇_𝑡𝑓_𝑠𝐼_[𝜏,𝑇]𝑑𝑠²+2𝐸sup_𝑡≥0𝐸^𝒢𝑡𝑇_𝑡𝑔_𝑠𝐼_[𝜏,𝑇]𝑑𝐵_𝑠²≤8𝐸𝑌_𝑇+^𝑇₀𝑓_𝑠𝐼_[𝜏,𝑇]𝑑𝑠²+2𝑐₀𝐸^𝑇₀𝑔_𝑠𝐼_[𝜏,𝑇]²𝐸𝑌𝑑𝑠≤16_𝑇²+𝐸^𝑇₀𝑓_𝑠𝐼_[𝜏,𝑇]𝑑𝑠²+2𝑐₀𝐸^𝑇₀𝑔_𝑠𝐼_[𝜏,𝑇]²≤𝑑𝑠16+2𝑐₀𝐸𝑌_𝑇²+𝐸^𝑇₀𝑓_𝑠𝐼_[𝜏,𝑇]𝑑𝑠²+𝐸^𝑇₀𝑔_𝑠𝐼_[𝜏,𝑇]²𝑑𝑠, (3.14) where 𝑐₀>0 is a constant.
On the other hand, from (3.11) it follows that 𝑍𝐼_[𝜏,𝑇]²_𝑀=𝐸^⋅₀𝑍_𝑠𝐼_[𝜏,𝑇]𝑑𝑊_𝑠∞=𝐸𝑌_𝑇+^𝑇₀𝑓_𝑠𝐼_[𝜏,𝑇]𝑑𝑠+^𝑇₀𝑔_𝑠𝐼_[𝜏,𝑇]𝑑𝐵_𝑠²−𝐸𝑌_𝑇+^𝑇₀𝑓_𝑠𝐼_[𝜏,𝑇]𝑑𝑠+^𝑇₀𝑔_𝑠𝐼_[𝜏,𝑇]𝑑𝐵_𝑠²≤𝐸𝑌_𝑇+^𝑇₀𝑓_𝑠𝐼_[𝜏,𝑇]𝑑𝑠+^𝑇₀𝑔_𝑠𝑑𝐼_[𝜏,𝑇]𝐵_𝑠²≤3𝐸𝑌_𝑇²+^𝑇₀𝑓_𝑠𝐼_[𝜏,𝑇]𝑑𝑠²+^𝑇₀𝑔_𝑠𝐼_[𝜏,𝑇]²𝑑𝑠. (3.15) Consequently, (3.14) and (3.15) imply that (𝑌𝐼_[𝜏,𝑇],𝑍𝐼_[𝜏,𝑇])²_𝐵=𝑌𝐼_[𝜏,𝑇]²_𝑆+𝑍𝐼_[𝜏,𝑇]²_𝑀≤19+2𝑐₀𝐸𝑌_𝑇²+𝐸^𝑇₀𝑓_𝑠𝐼_[𝜏,𝑇]𝑑𝑠²+𝐸^𝑇₀𝑔_𝑠𝐼_[𝜏,𝑇]²≤𝑑𝑠57+6𝑐₀𝐸𝑌_𝑇²+𝑙_[𝜏,𝑇]𝑦𝐼_[𝜏,𝑇],𝑧𝐼_[𝜏,𝑇]²_𝐵=𝐶𝐸𝑌_𝑇²+𝑙_[𝜏,𝑇]𝑦𝐼_[𝜏,𝑇],𝑧𝐼_[𝜏,𝑇]²_𝐵, (3.16) where 𝐶=(57+6𝑐₀) is a constant, and 𝑙_[𝜏,𝑇]=∫^𝑇_𝜏𝑣²(𝑠)𝑑𝑠.

Martingale Representation Theorem [4]
Suppose 𝑌 is a random variable, such that 𝐸|𝑌²|<∞. Note that 𝑀_𝑡=𝐸[𝑌∣𝒢_𝑡] is a square integrable martingale with respect to 𝒢_𝑡 and can be represented using martingale representation theorem as 𝑀_𝑡=𝑀₀+∫^𝑡₀𝑍_𝑠𝑑𝑊_𝑠, where 𝐸∫^∞₀‖𝑍_𝑡‖²𝑑𝑡<∞.
Now we give the proof of the Theorem 3.1.

Proof. The proof of Theorem 3.1 is divided into two steps.
Step 1. We assume 𝑙_[0,∞]=∫^∞₀𝑣²(𝑠)𝑑𝑠≤1/2𝐶. For any (𝑦,𝑧)∈𝐵², let 𝑀_𝑡≐𝐸^𝒢𝑡𝜉+^∞₀𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^∞₀𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠,𝑡≥0. (3.17) We will prove {𝑀_𝑡} is a square integrable 𝒢_𝑡-martingale. From (H1)–(H3), it follows that 𝐸𝜉+^∞₀𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^∞₀𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠²≤𝐸𝜉+^∞₀𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^∞₀𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠²≤3𝐸𝜉²+9𝐸^∞₀𝑓(𝑠,0,0)𝑑𝑠²+9𝐸^∞₀𝑣(𝑠)𝑦_𝑠𝑑𝑠²+9𝐸^∞₀𝑣(𝑠)𝑧_𝑠𝑑𝑠²+9𝐸^∞₀𝑔(𝑠,0,0)²+𝑣²(𝑠)𝑦_𝑠²+𝑣²(𝑠)𝑧_𝑠²𝜉𝑑𝑠≤3𝐸²+9𝐸^∞₀𝑓(𝑠,0,0)𝑑𝑠²+9^∞₀𝑣²(𝑠)𝑑𝑠⋅𝑦²_𝑆+9^∞₀𝑣²(𝑠)𝑑𝑠⋅‖𝑧‖²_𝑀+9𝐸^∞₀𝑔(𝑠,0,0)²𝑑𝑠+9^∞₀𝑣²(𝑠)𝑑𝑠⋅𝑦²_𝑆+9^∞₀𝑣²(𝑠)𝑑𝑠⋅‖𝑧‖²_𝑀<∞, (3.18) which means {𝑀_𝑡} is a square integrable 𝒢_𝑡-martingale. According to the martingale representation theorem, there exists a unique 𝒢_𝑡-progressively measurable process 𝑍_𝑡 with value in 𝑅^𝑘×𝑑 such that 𝐸^∞₀𝑍_𝑡²𝑀𝑑𝑡<∞,_𝑡=𝐸^𝒢0𝜉+^∞₀𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^∞₀𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠+^𝑡₀𝑍_𝑠𝑑𝑊_𝑠,0≤𝑡≤∞. (3.19) Let 𝑌_𝑡≐𝐸^𝒢𝑡𝜉+^∞_𝑡𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^∞_𝑡𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠,0≤𝑡≤∞. (3.20) So 𝑀_𝑡=𝐸^𝒢𝑡𝜉+^∞₀𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^∞₀𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠=𝐸^𝒢𝑡𝜉+^∞_𝑡𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^∞_𝑡𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠+^𝑡₀𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^𝑡₀𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠=𝑌_𝑡+^𝑡₀𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^𝑡₀𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠=𝐸^𝒢0𝜉+^∞₀𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^∞₀𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠+^𝑡₀𝑍_𝑠𝑑𝑊_𝑠. (3.21) Then 𝑌_𝑡=𝜉+^∞_𝑡𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^∞_𝑡𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠−^∞_𝑡𝑍_𝑠𝑑𝑊_𝑠,𝑡≥0. (3.22) We show that {𝑌_𝑡} and {𝑍_𝑡} are in fact ℱ_𝑡-measurable. For 𝑌_𝑡, this is obvious since, for each 𝑡, 𝑌_𝑡=𝐸^𝒢𝑡𝜉+^∞_𝑡𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^∞_𝑡𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠Θ=𝐸ℱ_𝑡⋁ℱ^𝐵_0,𝑡, (3.23) where Θ=𝜉+∫^∞_𝑡𝑓(𝑠,𝑦_𝑠,𝑧_𝑠)𝑑𝑠+∫^∞_𝑡𝑔(𝑠,𝑦_𝑠,𝑧_𝑠)𝑑𝐵_𝑠 is indeed ℱ^𝑊_0,∞⋁ℱ^𝐵_𝑡,∞-measurable. Hence ℱ^𝐵_0,𝑡 is independent of ℱ_𝑡⋁𝜎(Θ), and 𝑌_𝑡Θ=𝐸ℱ_𝑡. (3.24) Now ^∞_𝑡𝑍_𝑠𝑑𝑊_𝑠=𝜉+^∞_𝑡𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^∞_𝑡𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠−𝑌_𝑡 (3.25) and the right side is ℱ^𝑊_0,∞⋁ℱ^𝐵_𝑡,∞-measurable. Hence, from Itô’s martingale representation theorem, {𝑍_𝑠,𝑠>𝑡} is ℱ^𝑊_0,𝑠⋁ℱ^𝐵_𝑡,∞-adapted. Consequently 𝑍_𝑠 is ℱ^𝑊_0,𝑠⋁ℱ^𝐵_𝑡,∞-measurable, for any 𝑡<𝑠, and, thus, 𝑍_𝑡 is ℱ_𝑡-measurable. So (𝑌,𝑍)∈𝐵². Therefore (3.22) has constructed a mapping from 𝐵² to 𝐵², and we denote it by 𝜙, that is, 𝜙∶𝑦,𝑧→(𝑌,𝑍). (3.26) 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 that 𝜙 is a contractive mapping. Supposing that (𝑦^𝑖,𝑧^𝑖)∈𝐵², let (𝑌^𝑖,𝑍^𝑖) be the map 𝜙 of (𝑦^𝑖,𝑧^𝑖),(𝑖=1,2), that is 𝜙𝑦^𝑖,𝑧^𝑖=𝑌^𝑖,𝑍^𝑖,𝑖=1,2. (3.27) We denote 𝑌=𝑌¹−𝑌²,𝑍=𝑍¹−𝑍²,𝑦=𝑦¹−𝑦²,𝑧=𝑧¹−𝑧²,𝑓_𝑡=𝑓𝑡,𝑦¹,𝑧¹−𝑓𝑡,𝑦²,𝑧²,𝑔_𝑡=𝑔𝑡,𝑦¹,𝑧¹−𝑔𝑡,𝑦²,𝑧². (3.28) By Lemma 3.2, we have 𝜙(𝑦¹,𝑧¹)−𝜙(𝑦²,𝑧²)²_𝐵=(𝑌,𝑍)²_𝐵≤𝐶𝑙_[0,∞](𝑦,𝑧)²_𝐵. (3.29)
Due to 𝑙_[0,∞]≤1/2𝐶, it follows that 𝜙 is a contractive mapping from 𝐵² to 𝐵².
Step 2. Since ∫^∞₀𝑣²(𝑡)𝑑𝑡<∞, then there exists a sufficiently large constant 𝑇 such that ^∞_𝑇𝑣²(𝑠)1𝑑𝑠≤.2𝐶 (3.30) Let 𝑓₁𝑡,𝑦,𝑧≐𝐼_[𝑇,∞](𝑡)𝑓𝑡,𝑦,𝑧,𝑔₁𝑡,𝑦,𝑧≐𝐼_[𝑇,∞](𝑡)𝑔𝑡,𝑦,𝑧, (3.31) then (H1)–(H3) hold for 𝑓₁ and 𝑔₁, whose Lipschitz coefficients are 𝑣(𝑡)=𝐼_[𝑇,∞]𝑣(𝑡). Obviously, ^∞₀𝑣²(𝑠)1𝑑𝑠≤.2𝐶 (3.32) By Step 1, there exists a unique (𝑦,𝑧)∈𝐵² satisfying 𝑦_𝑡=𝜉+^∞_𝑡𝑓₁𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^∞_𝑡𝑔₁𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠−^∞_𝑡𝑧_𝑠𝑑𝑊_𝑠,0≤𝑡≤∞. (3.33) For (𝑦_𝑡,𝑧_𝑡) given as above, let us consider the following infinite BDSDE: 𝑦_𝑡=^𝑇_𝑡𝑓𝑠,𝑦_𝑠+𝑦_𝑠,𝑧_𝑠+𝑧_𝑠𝑑𝑠+^𝑇_𝑡𝑔𝑠,𝑦_𝑠+𝑦_𝑠,𝑧_𝑠+𝑧_𝑠𝑑𝐵_𝑠−^𝑇_𝑡𝑧_𝑠𝑑𝑊_𝑠,0≤𝑡≤𝑇,𝑦_𝑡≡0,𝑧_𝑡≡0,𝑡>𝑇. (3.34) 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 𝑦≐𝑦+𝑦,𝑧≐𝑧+𝑧. (3.35) It is easy to check that (𝑦_𝑡,𝑧_𝑡) is the unique solution of (2.5).

Remark 3.3. Suppose 𝑣 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 any 𝜉∈𝐿²(Ω,ℱ,𝑃;𝑅^𝑘) 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 choose 𝑓(𝑠,𝑦_𝑠,𝑧_𝑠)=(1/(1+𝑠))𝑧_𝑠,𝑔(𝑠,𝑦_𝑠,𝑧_𝑠)=0 and any 𝜉∈𝐿²(Ω,ℱ,𝑃;𝑅^𝑘), then the solution of BDSDE 𝑦_𝑡=𝜉+^∞_𝑡1𝑧1+𝑠^𝑠𝑑𝑠−^∞_𝑡𝑧_𝑠𝑑𝑊_𝑠 (3.36) should be 𝑦_𝑡−()∫=𝐸𝜉exp1/2^∞_𝑡(1/(1+𝑠))²𝑑𝑠+∫^∞_𝑡(1/(1+𝑠))𝑑𝑊_𝑡ℱ_𝑡,𝑧_𝑡=𝑑𝑦_𝑡,𝑊_𝑡,𝑑𝑡 (3.37) where ⟨𝑦_𝑡,𝑊_𝑡⟩ is the variation process generated by the semimartingale 𝑦_𝑡 and Brownian motion 𝑊_𝑡.
Thus the assumption (H3) is necessary.

Remark 3.5. The following example shows that if the coefficients 𝑓 and 𝑔 of BDSDE (2.5) satisfy the uniformly Lipschitz, the BDSDE (2.5) has no solution.
For all 𝑇>0, let 𝜉_𝑇=∫^𝑇₀(1/(1+𝑠))𝑑𝑊_𝑠, then the BDSDE 𝑦_𝑡=𝜉+∫^𝑇_𝑡𝑣|𝑧_𝑠|𝑑𝑠−∫^𝑇_𝑡𝑧_𝑠𝑑𝑊_𝑠 has a unique solution pair (𝑦^𝑇_𝑡,𝑧^𝑇_𝑡), 𝑦^𝑇_𝑡=∫𝜉,𝑡>𝑇,_𝑡011+𝑠𝑑𝑊_𝑠+∫^𝑇_𝑡𝑣𝑧1+𝑠𝑑𝑠,𝑡≤𝑇,_𝑇𝑡=10,𝑡>𝑇,1+𝑡,𝑡≤𝑇. (3.38)
When 𝑇→∞, 𝜉_𝑇→∫^∞₀(1/(1+𝑠))𝑑𝑊_𝑠 and 𝑧^𝑇_𝑡→1/(1+𝑡) in 𝐿²(Ω,ℱ,𝑃), but 𝑧_𝑡=1/(1+𝑡) is not the solution of the following infinite horizon BDSDE: 𝑦_𝑡=^∞₀11+𝑠𝑑𝑊_𝑠+^∞_𝑡𝑣𝑧_𝑠𝑑𝑠−^∞_𝑡𝑧_𝑠𝑑𝑊_𝑠 (3.39) 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. Suppose 𝜉_𝑖∈𝐿²(Ω,ℱ,𝑃;𝑅^𝑘),(𝑖=1,2), and consider (H1)–(H3). Let (𝑦^𝑖,𝑧^𝑖) 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 𝑦¹−𝑦²,𝑧¹−𝑧²²_𝐵≤𝐶𝐸𝜉₁−𝜉₂². (4.1)

Proof. Set 𝑦∶=𝑦¹−𝑦²,  𝑧∶=𝑧¹−𝑧². Since ∫^∞₀𝑣²(𝑠)𝑑𝑠<∞, we can choose a strictly increasing sequence 0=𝑡₀<𝑡₁<⋯<𝑡_𝑛<𝑡_𝑛+1=∞ such that 𝑙_{[𝑡𝑖,𝑡𝑖+1]}=^𝑡𝑖+1_𝑡𝑖𝑣²(𝑠)1𝑑𝑠≤2𝐶,𝑖=0,1,…,𝑛. (4.2) Applying Lemma 3.2, we have 𝑦,𝑧𝐼_{[𝑡𝑖,𝑡𝑖+1]}²_𝐵≤𝐶𝐸𝑦_𝑡𝑖+1²+𝐶𝑙_{[𝑡𝑖,𝑡𝑖+1]}𝑦,𝑧𝐼_{[𝑡𝑖,𝑡𝑖+1]}²_𝐵≤𝐶𝐸𝑦_𝑡𝑖+1²+12𝑦,𝑧𝐼_{[𝑡𝑖,𝑡𝑖+1]}²_𝐵. (4.3) Thus (𝑦,𝑧)𝐼_{[𝑡𝑖,𝑡𝑖+1]}²_𝐵≤2𝐶𝐸𝑦_𝑡𝑖+1²≤2𝐶𝐸sup_{𝑡𝑖+1≤𝑠≤𝑡𝑖+2}𝑦_𝑠²+^𝑡𝑖+2_𝑡𝑖+1𝑧_𝑠²(𝑑𝑠=2𝐶𝑦,𝑧)𝐼_{[𝑡𝑖+1,𝑡𝑖+2]}²_𝐵,𝑖=0,1,…,𝑛−1. (4.4) In particular, we have (𝑦,𝑧)𝐼_{[𝑡𝑛,∞]}²_𝐵≤2𝐶𝐸𝜉₁−𝜉₂². (4.5) From (4.4) and (4.5), it follows that 𝑦,𝑧²_𝐵≤^𝑛_𝑖=0(𝑦,𝑧)𝐼_{[𝑡𝑖,𝑡𝑖+1]}²_𝐵≤2𝐶+(2𝐶)²+⋯+(2𝐶)^𝑛+1𝐸𝜉₁−𝜉₂²=2𝐶(2𝐶)^𝑛+1−1𝐸𝜉2𝐶−1¹−𝜉₂²=𝐶𝐸𝜉₁−𝜉₂². (4.6) Thus the desired result is obtained.

Now we can assert the following convergence theorem for infinite horizon BDSDEs.

Theorem 4.2. Suppose 𝜉,𝜉_𝑖∈𝐿²(Ω,ℱ,𝑃;𝑅^𝑘),(𝑖=1,2,…), (H1)–(H3) hold for 𝑓 and 𝑔. Let (𝑦^𝑖,𝑧^𝑖) be the solutions of the following BDSDE: 𝑦^𝑖_𝑡=𝜉_𝑖+^∞_𝑡𝑓𝑠,𝑦^𝑖_𝑠,𝑧^𝑖_𝑠𝑑𝑠+^∞_𝑡𝑔𝑠,𝑦^𝑖_𝑠,𝑧^𝑖_𝑠𝑑𝐵_𝑠−^∞_𝑡𝑧^𝑖_𝑠𝑑𝑊_𝑠,𝑡≥0. (4.7) If 𝐸|𝜉_𝑖−𝜉|²→0 as 𝑖→∞, then there exists a pair (𝑦,𝑧)∈𝐵² such that ||(𝑦^𝑖−𝑦,𝑧^𝑖−𝑧)||_𝐵→0 as 𝑖→∞. Furthermore, (𝑦,𝑧) is the solution of the following BDSDE: 𝑦_𝑡=𝜉+^∞_𝑡𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^∞_𝑡𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠−^∞_𝑡𝑧_𝑠𝑑𝑊_𝑠,0≤𝑡<∞. (4.8)

Proof. For any 𝑛,𝑚≥1, let (𝑦^𝑛,𝑧^𝑛) and (𝑦^𝑚,𝑧^𝑚) be the solutions of (4.7) corresponding to 𝜉_𝑛 and 𝜉_𝑚, respectively. Due to Theorem 4.1, there exists a constant 𝐶>0 such that (𝑦^𝑛−𝑦^𝑚,𝑧^𝑛−𝑧^𝑚)²_𝐵≤𝐶𝐸𝜉_𝑛−𝜉_𝑚²≤2𝐶𝐸𝜉_𝑛−𝜉²+𝐸𝜉_𝑚−𝜉²→0,as𝑛,𝑚→∞, (4.9) which means that {(𝑦^𝑖,𝑧^𝑖),𝑖=1,2,…} is a Cauchy sequence in 𝐵². Thus there exists a pair (𝑦,𝑧)∈𝐵² such that ||(𝑦^𝑖−𝑦,𝑧^𝑖−𝑧)||_𝐵→0 as 𝑖→∞. Since 𝐸^∞_𝑡𝑓𝑠,𝑦^𝑖_𝑠,𝑧^𝑖_𝑠−𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠²≤𝐸^∞₀𝑣(𝑠)𝑦^𝑖_𝑠−𝑦_𝑠+𝑣(𝑠)𝑧^𝑖_𝑠−𝑧_𝑠𝑑𝑠²≤2^∞₀𝑣²(𝑠)𝑑𝑠⋅𝑦^𝑖−𝑦,𝑧^𝑖−𝑧²_𝐵𝐸→0,as𝑖→∞,_∞𝑡𝑔𝑠,𝑦^𝑖_𝑠,𝑧^𝑖_𝑠−𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠²≤2^∞₀𝑣²(𝑠)𝑑𝑠⋅𝑦^𝑖−𝑦,𝑧^𝑖−𝑧²_𝐵→0,as𝑖→∞. (4.10) Thus for any 𝑡∈𝑅⁺,∫^∞_𝑡𝑓(𝑠,𝑦^𝑖_𝑠,𝑧^𝑖_𝑠)𝑑𝑠→∫^∞_𝑡𝑓(𝑠,𝑦_𝑠,𝑧_𝑠)𝑑𝑠 and ∫^∞_𝑡𝑔(𝑠,𝑦^𝑖_𝑠,𝑧^𝑖_𝑠)𝑑𝐵_𝑠→∫^∞_𝑡𝑔(𝑠,𝑦_𝑠,𝑧_𝑠)𝑑𝐵_𝑠 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: 𝑦_𝑡=𝐸^ℱ𝑇𝜉+^𝑇_𝑡𝑓𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝑠+^𝑇_𝑡𝑔𝑠,𝑦_𝑠,𝑧_𝑠𝑑𝐵_𝑠−^𝑇_𝑡𝑧_𝑠𝑑𝑊_𝑠,0≤𝑡≤𝑇<∞. (4.11)

Corollary 4.3. Assume 𝜉∈𝐿²(Ω,ℱ,𝑃;𝑅^𝑘), (H1)–(H3) hold for 𝑓 and 𝑔. Let (𝑦,𝑧) be the solution of BDSDE (2.5). For any 𝑇>0, let (𝑦^𝑇,𝑧^𝑇) be the solutions of the finite time interval BDSDE (4.11), then(𝑦^𝑇,𝑧^𝑇)→(𝑦,𝑧) in 𝐵² as 𝑇→∞.

Proof. Note that 𝐸^ℱ𝑇[𝜉]→𝜉 in 𝐿²(Ω,ℱ,𝑃;𝑅^𝑘) 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. 𝐶^𝑘(𝑅^𝑝;𝑅^𝑞),  𝐶^𝑘_𝑙,𝑏(𝑅^𝑝;𝑅^𝑞),  𝐶^𝑘_𝑝(𝑅^𝑝;𝑅^𝑞) will denote, respectively, the set of functions of classes from 𝑅^𝑝 into 𝑅^𝑞, the set of those functions of class 𝐶^𝑘 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 𝐶^𝑘 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.

For 𝑠≥𝑡, let 𝑋_𝑠^𝑡,𝑥 be a diffusion process given by the solution of 𝑋_𝑠^𝑡,𝑥=𝑥+^𝑠_𝑡𝑏𝑋_𝜇^𝑡,𝑥𝑑𝜇+^𝑠_𝑡𝜎𝑋_𝜇^𝑡,𝑥𝑑𝑊_𝜇, (5.1) where 𝑏∈𝐶³_𝑙,𝑏(𝑅^𝑑;𝑅^𝑑),𝜎∈𝐶³_𝑙,𝑏(𝑅^𝑑;𝑅^𝑑×𝑅^𝑑), and, for 0≤𝑠<𝑡, we regulate 𝑋_𝑠^𝑡,𝑥=𝑥.

It is well known that the solution defines a stochastic flow of diffeomorphism 𝑋_𝑠^𝑡,⋅∶𝑅^𝑑→𝑅^𝑑 and denotes by 𝑋_𝑠^𝑡,⋅ the inverse flow (see e.g., [15]). The random field 𝑋_𝑠^𝑡,𝑥;  𝑠≥𝑡,  𝑥∈𝑅^𝑑 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): 𝑓𝑠,𝑦,𝑧=𝑓𝑠,𝑋_𝑠^𝑡,𝑥,𝑦,𝑧;𝑔𝑠,𝑦,𝑧=𝑔𝑠,𝑋_𝑠^𝑡,𝑥,𝑦,𝑧, (5.2) where 𝑓∶[0,𝑇]×𝑅^𝑑×𝑅^𝑘×𝑅^𝑘×𝑑→𝑅^𝑘;  𝑔∶[0,𝑇]×𝑅^𝑑×𝑅^𝑘×𝑅^𝑘×𝑑→𝑅^𝑘×𝑙.

We assume that for any 𝑠≥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,∞)×𝑅^𝑑×𝑅^𝑘×𝑅^𝑘×𝑑.

We assume again that (H1), (H2), and (H3) hold, then the following BDSDE has a unique solution: 𝑌_𝑠^𝑡,𝑥=𝜉+^∞_𝑠𝑓𝑟,𝑋_𝑟^𝑡,𝑥,𝑌_𝑟^𝑡,𝑥,𝑍_𝑟^𝑡,𝑥𝑑𝑟+^∞_𝑠𝑔𝑟,𝑋_𝑟^𝑡,𝑥,𝑌_𝑟^𝑡,𝑥,𝑍_𝑟^𝑡,𝑥𝑑𝐵_𝑟−^∞_𝑠𝑍_𝑟^𝑡,𝑥𝑑𝑊_𝑟,𝑡≥0. (5.3) Let (𝑌_𝑠^𝑡,𝑥,𝑍_𝑠^𝑡,𝑥) denote the unique solution of (4.11). We shall define 𝑋_𝑠^𝑡,𝑥,𝑌_𝑠^𝑡,𝑥, and 𝑍_𝑠^𝑡,𝑥 for all (𝑠,𝑡)∈[0,∞)×[0,∞) by letting 𝑋_𝑠^𝑡,𝑥=𝑋^𝑡,𝑥_𝑠∨𝑡,  𝑌_𝑠^𝑡,𝑥=𝑌^𝑡,𝑥_𝑠∨𝑡,  and 𝑍_𝑠^𝑡,𝑥=0 for 𝑠<𝑡.

We now relate our BDSDE to the following system of quasilinear backward stochastic partial differential equations: 𝑑𝜅(𝑡,𝑥)=−ℒ𝜅(𝑡,𝑥)+𝑓𝑥,𝜅(𝑡,𝑥),𝜎^∗(𝑥)()()𝐷𝜅𝑡,𝑥𝑑𝑡−𝑔𝑥,𝜅𝑡,𝑥,𝜎^∗(𝑥)𝐷𝜅(𝑡,𝑥)𝑑𝐵_𝑡,𝑡≥0. (5.4)ℒ is the infinitesimal generator of a diffusion process 𝑋_𝑠^𝑡,𝑥 (solution of (5.1)) given by 1ℒ=2^𝑑_𝑖,𝑗=1𝑎_𝑖𝑗(𝑥)𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗+^𝑑_𝑖=1𝑏_𝑖(𝑥)𝜕𝜕𝑥_𝑖, (5.5) where (𝑎_𝑖,𝑗(𝑥))=𝜎𝜎^∗(𝑥).

Theorem 5.1. Let 𝜅(𝑡,𝑥);𝑡≥0,𝑥∈𝑅^𝑑 be a random field such that 𝜅(𝑡,𝑥) is ℱ^𝐵_𝑡,∞-measurable for each (𝑡,𝑥),𝜅∈𝐶^0,2([0,∞)×𝑅^𝑑;𝑅^𝑘) a.s., and 𝜅 satisfies (5.4). Then 𝜅(𝑡,𝑥)=𝑌_𝑡^𝑡,𝑥, where (𝑌_𝑠^𝑡,𝑥,𝑍_𝑠^𝑡,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): 𝑑𝜅𝑠,𝑋_𝑠^𝑡,𝑥=ℒ𝜅𝑠,𝑋_𝑠^𝑡,𝑥𝑑𝑠+𝜎^∗𝐷𝜅𝑠,𝑋_𝑠^𝑡,𝑥𝑑𝑊_𝑠−ℒ𝜅𝑠,𝑋_𝑠^𝑡,𝑥+𝑓𝑠,𝑋_𝑠^𝑡,𝑥,𝜅𝑠,𝑋_𝑠^𝑡,𝑥,𝜎^∗𝐷𝜅𝑠,𝑋_𝑠^𝑡,𝑥𝑑𝑠−𝑔𝑠,𝑋_𝑠^𝑡,𝑥,𝜅𝑠,𝑋_𝑠^𝑡,𝑥,𝜎^∗𝐷𝜅𝑠,𝑋_𝑠^𝑡,𝑥𝑑𝐵_𝑠. (5.6) We can see that (𝜅(𝑠,𝑋_𝑠^𝑡,𝑥),𝜎^∗𝐷𝜅(𝑠,𝑋_𝑠^𝑡,𝑥)) coincides with the unique solution of (5.3). It follows that 𝜅(𝑡,𝑥)=𝑌_𝑡^𝑡,𝑥.

We have also a converse to Theorem 5.1.

Theorem 5.2. Let 𝑓 and 𝑔 satisfy (H1), (H2), and (H3). Then {𝜅(𝑡,𝑥)=𝑌_𝑡^𝑡,𝑥;𝑡≥0,𝑥∈𝑅^𝑑} 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].

Acknowledgments

This work is supported by the Colleges and Universities Outstanding Young Teacher Domestic Visiting Scholar of Shandong Province Project (2012) and the Nature Science Foundation of Shandong Province of China (Grant no. ZR2010AL014).