Research On Anticipated Backward Stochastic Differential Equations And Related Applications

Backward stochastic differential equation(BSDE)of the general form was firstly studied by Pardoux-Peng[30]in 1990.Since then,the theory of BSDEs has been studied with great interest due to its wide applications in many areas,such as in the pricing/hedging problem,in the theory of(stochastic)partial differential equations,in the stochastic control problem and stochastic differential game theory,and so on.The comparison theorem is one of the achievements of this theory due to Peng[36].And then Pardoux-Peng[31]and El Karoui-Peng-Quenez[17]generalized the comparison theorem.We can use it to compare the solutions of two BSDEs whenever we can compare the terminal conditions and the generators.Recently,in 2007 Yang[46](see also Peng-Yang[40]),studied a new type of BSDEs called anticipated backward stochastic differential equations(ABSDEs).For these equations,they studied the existence and uniqueness of the adapted solutions,and gave a comparison theorem.Moreover as an application,they dealt with a related stochastic control problem.The objective of this thesis is to study a class of anticipated BSDEs and related applications.Firstly,we give another proof of the comparison theorem in Peng-Yang[40]by the linearization method.And also we give a more general result.Secondly,we study a duality between the stochastic differential delay equations and the antic-ipated BSDEs with the functional anticipated time ?(t),And as an application,we solve a related stochastic control problem.Lastly,we study the anticipated BSDEs under weaker conditions.Then we study their applications to the zero-sum stochastic differential game and get the saddle-point strategy when the Isaacs' condition holds.