Further Research On Anticipated Backward Doubly Stochastic Differential Equations

In 1990,Pardoux and Peng[24]firstly introduced nonlinear backward stochas-tic differential equations(BSDEs)and proved the existence and uniqueness theorem of adapted solutions under Lipschitz condition on the generator f.Since then,the theory of BSDEs has developed rapidly and many new forms have been introduced.In 1994,Pardoux and Peng[25]proposed the notion of backward doubly stochas-tic differential equations(BDSDEs)and got the existence and uniqueness of adapted solutions under Lipschitz condition.Furthermore,a new type of BSDEs called antic-ipated backward stochastic differential equations(ABSDEs)was considered by Peng and Yang[31],and for these equations with Lipschitz generator,they obtained the ex-istence/uniqueness theorem and the comparison theorem.Motivated by those works,Xu[36]studied the existence and uniqueness of solution to anticipated backward dou-bly stochastic differential equations(anticipated BDSDEs)while the generator f and coefficient g both satisfy the Lipschitz condition.The main purpose of this thesis is to study the anticipated BDSDEs under several conditions which are weaker than the Lipschitz condition.Inspired by non-Lipschitz condition(see Mao[22])and monotonicity condition(see Pardoux[23]),we put for-ward the existence and uniqueness result under a new kind of weak condition and give some comparison theorems for one-dimensional case.Besides,we extend the weak monotonicity condition from Fan and Jiang[9]for BSDEs to anticipated BDS-DEs,for which we prove the existence/uniqueness theorem under weak monotonicity condition and give the comparison theorems for one-dimensional case.