The Property For Solutions Of Backward Doubly Stochastic Differential Equations

The study of Backward Stochastic Differential Equation(BSDE) come from the ?research of stochastic control and finance etc.;On the contrary,the results of the equation theory are applied to the control,finance and PDE etc. fields of math. Relative to Stochastic Differential Equation(SDE),the research of BSDE started later and the results are not so rich . At the same time,owing to the character of BSDE, it is difficult that stopping time and localization etc. technique be applied to BSDE directly. So the study of BSDE has its own character. So far, BSDEs have two types:One is Ito integral type BSDE driven by Brown motion which come directly from stochastic control and it is applied to financial problem later;the other is BSDE with expectation which come directly from financial problem. The conditions of these two types of BSDEs are different ,such as the integrability of parameter and coefficient, the filtration etc.. the two types of BSDEs are not covered by each other.Along with the BSDE widespread application and the thorough research, the backward stochastic differential equation with jumps and its application have attracted the more and more many scholars' interests and is widely applied to the stochastic control research. At the same time, Pardoux and Peng proposed and has studied a kind of Backward doubly stochastic differential equation (BDSDE), and applies to quasi-linear parabolic SPDEs.Relative to BSDE, the study for the solution of BDSDE under non-Lipschitz condition is absence,especially when the uniqueness of the solution can not be guar- anteed,the existence of minimal and maximal solutions of BDSDE is not be stud-ied;Simultaneously,there are not any research results of the BDSDE with jumps so far.This article studies the property for the solution of BDSDE. The main result includes: In view of BDSDE,the existence and uniqueness of the solution under non-Lipschitz condition, comparison theorem are established. Under weaker condition, the existence of minimal and maximal solution is constructively proved using comparison theorem and the monotone iteration technique. Meanwhile the existence and uniqueness for the solution of BDSDE with jumps is also researched.