| The study of Backward Stochastic Differential Equation(BSDE) comes from the research of stochastic control and finance, and the results of the equation theory are applied to many fields such as control, finance and PDE etc. Since the research of BSDE is later than Stochastic Differential Equation (SDE), the study of BSDE are becoming more and more thorough, and the results of BSDE are becoming plentiful. So far, BSDEs have two types: one is Ito integral type BSDE driven by Brown motion[5] which comes directly from stochastic control and it is applied to financial problem later; the other is BSDE with expectation[38] which comes directly from financial problem. Since the conditions of these two types, such as the integrability of parameter and coefficient, the filtration etc. are different, the result of these two types of BSDEs are not covered by each other.With the application and thorough research of the BSDE, the BSDE with jumps has attracted more and more scholars’ interests[14][15] and is widely applied to the stochastic control research. At the same time, Peng has proposed and studied a kind of Backward doubly stochastic differential equation (BDSDE)[17], and applied it to quasi-linear parabolic SPDEs. BDSDE with Possion jumps is the extension of BDSDE[24][39]. Compared to the BSDE, the study of the BDSDE is not enough, and the study of the BDSDE with jumps is just beginning. So it is valuable in mathematical theory and application.In this paper, the properties for the solutions of BDSDE with Possion jumps under non - Lipschitz condition are studied. The main results include: In view of BDSDE with jumps, the existence and uniqueness of the solutions, the comparison theorem, the stability and continuous dependence are studied by using the extension of Ito formula. Besides these, the continuous dependence for the solutions of BDSDE under non-Lipschitz condition is studied. They extend the research on the properties of BSDE. |
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