Further Research On Anticipated Backward Stochastic Differential Equations And The Related Zero-sum Stochastic Differential Games

Pardoux-Peng[21]firstly introduced the notion of nonlinear backward stochastic differential equations(BSDEs)in 1990 and proved the existence and uniqueness theo-rem of adapted solutions.Since then,the theory of BSDEs develops rapidly and many new forms have been introduced.Certainly,these forms of BSDEs have got wide ap-plications in many areas.In particular,they have become the important mathematical tools for dealing with zero-sum stochastic differential games to solve various prob-lems such as determining saddle points(see e.g.[14,15,29]),risk-sensitive control problems(see e.g.[10]),and mixed control and stopping problems(see e.g.[16-18]).Peng and Yang introduced a new type of BSDEs called anticipated backward stochastic differential equations(ABSDEs)in[30](see also Yang[37]).They proved the existence and uniqueness of adapted solutions under suitable assumptions and gave the comparison theorems.Besides,they proved a duality between anticipated BSDEs and SDDEs,when the anticipated time and the delay time are both constants.Furthermore,they applied the duality to deal with a related stochastic control problem.The main goal of this thesis is to further study the anticipated backward stochastic differential equations and the related zero-sum stochastic differential games.Firstly,we give a duality between anticipated BSDEs and SDDEs,where the anticipated time and the delay time are both discontinuous time of a specific form.And we apply the duality to solve a related stochastic control problem.Secondly,we discuss the anticipated BSDEs with the generators containing multiple anticipated terms,where the generators contain two anticipated terms of Y.and two anticipated terms of Z..We mainly solve the existence and uniqueness theorem,and give some comparison theorems.Lastly,as an application,we discuss the zero-sum stochastic differential games with three different types of payoffs,which depend on not only the value of xt on[0,T]but also the value after time T.For the three game problems,we prove the existence of a saddle point under appropriate assumptions,respectively.