The Properties For Solutions Of The Infinite Horizon Backward Doubly Stochastic Differential Equations

Since the backward stochastic differential equations was given by Pardoux and Peng, the theory of backward stochastic differential equations have been studied widely.Backward stochastic differential equations are the important basic tool of financial mathematics studies,and they play an significance role for the study of modern financial products ,such as options and futures.They are applied to PDE,SPDE,SDG and non-linear expectation.In 1990,Pardox and Peng[1] introduced the classical backward stochastic differential equation:Under the uniformly Lipschitz assumption on coeffcient g, the equation has a unique solution. Prom then on, a lot of researchers have studied backward stochastic differential equations.In 1994,Pardox and Peng[11] introduced backward doubly stochastic differential equations:where dB_t is backward It(?) integral and dW_t is standerd forward It(?) integral. They have proved the existence and uniqueness of the solutions to BDSDEs under uniformly Lipschitz conditions on coefficients f and g, and applied the results to PDE.From then on ,there are many papers about BDSDEs.At present, the study of the properties for solutions of backward stochastic differential equations is with a fixed terminal time T, when T=∞,i.e.infinite horizon backward doubly stochastic differential equations is also significative.Following paper [6], [17] studied the infinite horizon backward doubly stochastic differential equations:where g is independent of z,the existence and uniqueness of the solutions to BDSDEs was proved under Lipschitz conditions on coefficients f and g.On account of the difficulty of using the It(?) formula under infinite horizon, compared to the finite horizon BDSDE, it is difficult to study the infinite horizon BDSDE with non-Lipschitz coefficients. And the results under the Lipschitz condition are not enough, so in this paper, we mainly study the properties for solutions of the infinite horizon backward doubly stochastic differential equations with Lipschitz conditions.Firstly, the comparison theorem of the infinite horizon BDSDE with Lipschitz conditions is proved, where g is independent of z. Secondly, for the case of g being dependent on z, a class of Lipschitz conditions is given, by approximation of finite interval,using It(?) formula,the existence ,uniqueness and comparison theorem are proved.(Applied mathematics)Zhen Xin Supervised by Sun Xiaojun...