Research Article | Open Access
Stochastic PDEs and Infinite Horizon Backward Doubly Stochastic Differential Equations
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  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 and a backward stochastic integral . They have proved the existence and uniqueness of solutions to BDSDEs under uniformly Lipschitz conditions on coefficients on finite time interval . That is, for a given terminal time , under the uniformly Lipschitz assumptions on coefficients , for any square integrable terminal value , the following BDSDE has a unique solution pair in the interval : 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  got stationary solutions of SPDEs and infinite horizon BDSDEs, but their researches under the assumption that terminal value . Zhu and Han  also give a sufficient condition on the coefficients of a class of infinite horizon BDSDEs, but there the coefficient is independent of .
This paper studies the existence and uniqueness of BDSDE (1.1) when . 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 , under which, for any square integrable random variable , BDSDE (1.1) still has a unique solution pair when . 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
The Euclidean norm of a vector will be denoted by , and, for a matrix , we define , where is the transpose of .
Let be a completed probability space and let and be two mutually independent standard Brownian motions, with values, respectively, in and , defined on . Let denote the class of -null sets of . For each , we define Note that is an increasing filtration and is a decreasing filtration, and the collection is neither increasing nor decreasing.
For any , let denote the space of all -measurable -dimensional processes with norm of .
We denote similarly by the space of all (classes of a.e. equal) -measurable -dimensional processes with norm of .
For any , let denote the space of all -measurable -valued random variables satisfying .
We also denote For each , we define the norm of by Obviously is a Banach space.
Consider the following infinite horizon backward doubly stochastic differential equation: where is given. We note that the integral with respect to is a backward Itô integral and the integral with respect to is a standard forward Itô integral. These two types of integrals are particular cases of the Itô-Skorohod integral; see Boufoussi et al. . 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 is called a solution of BDSDE (2.5), if and satisfies BDSDE (2.5).
Let satisfy the following assumptions: (H1) for any , and are -progressively measurable processes, such that (H2) and satisfy Lipschitz condition with Lipschitz coefficient ; that is, there exists a positive nonrandom function such that for all , , (H3).
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 .
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 and . For any , let , and satisfy the following equation: Then there is a constant , such that, for any , where and is an indicator function.
Proof. Firstly, we assume that , .
Set Then We define the filtration by Obviously is an increasing filtration. Since , is a -martingale, thus from (3.4) it follows that Note that Applying Doob inequality and B-D-G inequality, we can deduce where is a constant.
On the other hand, from (3.4) it follows that where is the variation process generated by the martingale .
Consequently, (3.8) and (3.9) imply that where is a constant, and .
For any , we set , and . Then , and satisfy the assumptions (H1), (H2), and (H3), and their Lipschitz constants are .
Obviously, Since , is a -martingale, thus from (3.11) it follows that Note that Applying Doob inequality and B-D-G inequality, we can deduce where is a constant.
On the other hand, from (3.11) it follows that Consequently, (3.14) and (3.15) imply that where is a constant, and .
Martingale Representation Theorem 
Suppose is a random variable, such that . Note that is a square integrable martingale with respect to and can be represented using martingale representation theorem as , where .
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 . For any , let We will prove is a square integrable -martingale. From (H1)–(H3), it follows that which means is a square integrable -martingale. According to the martingale representation theorem, there exists a unique -progressively measurable process with value in such that Let So Then We show that and are in fact -measurable. For , this is obvious since, for each , where is indeed -measurable. Hence is independent of , and Now and the right side is -measurable. Hence, from Itô’s martingale representation theorem, is -adapted. Consequently is -measurable, for any , and, thus, is -measurable. So . Therefore (3.22) has constructed a mapping from to , and we denote it by , that is, If is a contractive mapping with respect to the norm , by the fixed point theorem, there exists a unique , satisfying (3.22), which is just the unique solution to BDSDE (2.5).
Now we are in the position to prove that is a contractive mapping. Supposing that , let be the map of , that is We denote By Lemma 3.2, we have
Due to , it follows that is a contractive mapping from to .
Step 2. Since , then there exists a sufficiently large constant such that Let then (H1)–(H3) hold for and , whose Lipschitz coefficients are . Obviously, By Step 1, there exists a unique satisfying For given as above, let us consider the following infinite BDSDE: According to the results of Pardoux and Peng , the above BDSDE has a unique solution in , thus the above BDSDE has a unique solution such that for every . Let It is easy to check that is the unique solution of (2.5).
Remark 3.4. The condition (H3) is usually necessary. That is, if for any and hold in (H1) and (H2), BDSDE (2.5) has a unique solution in , then the (H3) is necessary.
In fact, let us choose and any , then the solution of BDSDE should be where is the variation process generated by the semimartingale and Brownian motion .
Thus the assumption (H3) is necessary.
Remark 3.5. The following example shows that if the coefficients and of BDSDE (2.5) satisfy the uniformly Lipschitz, the BDSDE (2.5) has no solution.
For all , let , then the BDSDE has a unique solution pair ,
When , and in , but is not the solution of the following infinite horizon BDSDE: because .
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 , and consider (H1)–(H3). Let be the solutions of BDSDE (2.5) corresponding to the terminal data , respectively. Then there exists a constant such that
Proof. Set , . Since , we can choose a strictly increasing sequence such that Applying Lemma 3.2, we have Thus In particular, we have From (4.4) and (4.5), it follows that Thus the desired result is obtained.
Now we can assert the following convergence theorem for infinite horizon BDSDEs.
Theorem 4.2. Suppose , (H1)–(H3) hold for and . Let be the solutions of the following BDSDE: If as , then there exists a pair such that as . Furthermore, is the solution of the following BDSDE:
Proof. For any , let and be the solutions of (4.7) corresponding to and , respectively. Due to Theorem 4.1, there exists a constant such that which means that is a Cauchy sequence in . Thus there exists a pair such that as . Since Thus for any and in . Taking the limit on both sides of (4.7), we deduce that 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:
Proof. Note that in as . 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. , , will denote, respectively, the set of functions of classes from into , the set of those functions of class whose partial derivatives of order less than or equal to are bounded (and hence the function itself grows at most linearly at infinity), and the set of those functions of class which, together with all their partial derivatives of order less than or equal to , grow at most like a polynomial function of the variable at infinity.
For , let be a diffusion process given by the solution of where , and, for , we regulate .
It is well known that the solution defines a stochastic flow of diffeomorphism and denotes by the inverse flow (see e.g., ). The random field ; , has a version which is a.s. of class in , the function and its derivatives being a.s. continuous with respect to .
Now the coefficients of the BDSDE will be of the form (with an obvious abuse of notations): where ; .
We assume that for any , is of class , and all derivatives are bounded on .
We assume again that (H1), (H2), and (H3) hold, then the following BDSDE has a unique solution: Let denote the unique solution of (4.11). We shall define , and for all by letting , , and for .
We now relate our BDSDE to the following system of quasilinear backward stochastic partial differential equations: is the infinitesimal generator of a diffusion process (solution of (5.1)) given by where .
We have also a converse to Theorem 5.1.
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).
- E. Pardoux and S. Peng, “Backward doubly stochastic differential equations and systems of quasilinear SPDEs,” Probability Theory and Related Fields, vol. 98, no. 2, pp. 209–227, 1994.
- V. Bally and A. Matoussi, “Weak solutions for SPDEs and backward doubly stochastic differential equations,” Journal of Theoretical Probability, vol. 14, no. 1, pp. 125–164, 2001.
- R. Buckdahn and J. Ma, “Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I,” Stochastic Processes and their Applications, vol. 93, no. 2, pp. 181–204, 2001.
- R. Buckdahn and J. Ma, “Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II,” Stochastic Processes and their Applications, vol. 93, no. 2, pp. 205–228, 2001.
- E. Pardoux, Stochastic Partial Differential Equations, Fudan lecture notes, 2007.
- S. Peng and Y. Shi, “A type of time-symmetric forward-backward stochastic differential equations,” Comptes Rendus de l'Académie des Sciences—Series I, vol. 336, no. 9, pp. 773–778, 2003.
- H. Kunita, Stochastic Flows and Stochastic Differential Equations, vol. 24 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.
- Y. Ren, A. Lin, and L. Hu, “Stochastic PDIEs and backward doubly stochastic differential equations driven by Lévy processes,” Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 901–907, 2009.
- Q. Zhu, Y. Shi, and X. Gong, “Solutions to general forward-backward doubly stochastic differential equations,” Applied Mathematics and Mechanics, vol. 30, no. 4, pp. 517–526, 2009.
- B. Zhu and B. Y. Han, “Backward doubly stochastic differential equations with non-Lipschitz coefficients,” Acta Mathematica Scientia Series A, vol. 28, no. 5, pp. 977–984, 2008.
- B. Zhu and B. Han, “Comparison theorems for the multidimensional BDSDEs and applications,” Journal of Applied Mathematics, vol. 2012, Article ID 304781, 14 pages, 2012.
- B. Boufoussi, J. van Casteren, and N. Mrhardy, “Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions,” Bernoulli, vol. 13, no. 2, pp. 423–446, 2007.
- L. Hu and Y. Ren, “Stochastic PDIEs with nonlinear Neumann boundary conditions and generalized backward doubly stochastic differential equations driven by Lévy processes,” Journal of Computational and Applied Mathematics, vol. 229, no. 1, pp. 230–239, 2009.
- Q. Zhang and H. Zhao, “Stationary solutions of SPDEs and infinite horizon BDSDEs,” Journal of Functional Analysis, vol. 252, no. 1, pp. 171–219, 2007.
- B. Zhu and B. Han, “Backward doubly stochastic differential equations with infinite time horizon,” Applications of Mathematics, vol. 57, no. 6, pp. 641–653, 2012.
Copyright © 2012 Bo Zhu and Baoyan Han. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.