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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.

Pardoux and Peng [

This paper studies the existence and uniqueness of BDSDE (

The paper is organized as follows: in Section

The Euclidean norm of a vector

Let

Suppose

For any

We denote similarly by

For any

We also denote

Consider the following infinite horizon backward doubly stochastic differential equation:

A pair of processes

Let

for any

The following existence and uniqueness theorem is our main result.

Under the above conditions, in particular (H1), (H2), and (H3), (

In order to prove the existence and uniqueness theorem, one first gives an a priori estimate.

Suppose (H1), (H2), and (H3) hold for

Firstly, we assume that

Set

On the other hand, from (

Consequently, (

For any

Obviously,

On the other hand, from (

Suppose

Now we give the proof of the Theorem

The proof of Theorem

Now we are in the position to prove that

Due to

Suppose

The condition (H3) is usually necessary. That is, if for any

In fact, let us choose

Thus the assumption (H3) is necessary.

The following example shows that if the coefficients

For all

When

In this section we will discuss the convergence of solutions of infinite horizon BDSDEs. First we give the following continuous dependence theorem.

Suppose

Set

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

Suppose

For any

The following corollary shows the relation between the solution of infinite horizon BDSDE (

Assume

Note that

In this section, we study the link between BDSDEs and the solution of a class of SPDEs.

Let us first give some notations.

For

It is well known that the solution defines a stochastic flow of diffeomorphism

Now the coefficients of the BDSDE will be of the form (with an obvious abuse of notations):

We assume that for any

We assume again that (H1), (H2), and (H3) hold, then the following BDSDE has a unique solution:

We now relate our BDSDE to the following system of quasilinear backward stochastic partial differential equations:

Let

We can apply the extension of the Itô formula [

We have also a converse to Theorem

Let

We can finish the proof exactly as in Theorem 3.2 of Hu and Ren [

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).