Researches On Anticipated BSDEs,Delayed BSDEs And Related Applications

Backward stochastic differential equation(BSDE)was firstly introduced by Pardoux and Peng in 1990.They proved the existence and uniqueness of adapted solutions.Since then,the theory of BSDEs has played an important role in many fields.Specially,BSDEs have become the important approaches to deal with the stochastic games and optimal control.El-Karoui and Hamadene[8]solved the risk-sensitive control problems which considered the attitude towards the risk of the controller by using the theory of BSDEs.With the development of the theory of BSDEs,many new terms were introduced and studied.In 2007,the anticipated BSDE(ABSDE)was introduced by Peng and Yang[30].Peng and Yang proved the existence and uniqueness of adapted solutions of under proper assumptions and gave some comparison theorems for 1-dimensional ABSDEs.In 2010 Delong and Imkeller[5]introduced a backward stochastic differential equation with time-delayed generator.They introduced the stopping time and proved the comparison theorem on the corresponding intervals.Furthermore,Chen and Huang in[2]gave some preliminary results on delayed BSDEs and the necessary and sufficient conditions of the maximum principle.In this thesis,firstly,we aim to deal with the risk-sensitive zero-sum stochastic differential game,in which the payoff depends on not only the value of xt on[0,T]but also the value after time T.We will obtain the saddle-point by using the theory of the anticipated backward stochastic differential equations with quadratic growth(see[19]).Secondly,similar to the method in[30],we prove the comparison theorem of the delayed BSDEs whose generator does not contain the delayed term of Z.and give the related applications.Finally,we give the necessary condition of the maximum principle of the optimal control for a stochastic anticipated system.The generator of the equation we consider is linear and its coefficients are bounded.