Some Related Problems Of Backward Doubly Stochastic Differential Equations

In 1994, E.Pardoux and S.Peng firstly studied a certain class of backward doubly stochastic differential equations (BDSDEs in short) in [28], which involved two differ-ent directions of stochastic integral:the forward Ito integral dWt and the backward Ito integral dB,. In this paper, the existence and uniqueness of the solutions for BDSDEs under uniformly Lipchitz conditions were proved and the more important results are that Pardoux and Peng obtained a probabilistic representation of certain quasilinear backward stochastic partial differential equations (SPDEs in short), and extended the Feynman-Kac formula for linear SPDEs, which promote the development of the filed of SPDEs. After that, many researchers pay their attention on this field and they also gave lots of helpful conclusions (see [1,9,20,31,40]). So in this paper, we will study some basic problems on the backward doubly stochastic differential equations (BDSDEs in short). In Section 3, we will obtain the Girsanov theorem for doubly stochastic differential equations and we will give the Girsanov theorem on the forward Ito integral and backward Ito integral respectively. And in Section 4, we will give a concept of weak solutions of BDSDEs and some propositions on BDSDEs through the Girsanov theorem that we obtained in Sec-tion 3. Finally, we will apply some conclusions of BDSDEs to the zero-sum stochastic differential games. And depending on some conditions and theorems we got in Section 3 and 4, we will obtain that the game's payoff J(u, v) can be an expectation:J(u, v)= EY0, where Y is the solution to a certain BDSDE.