Nonzero Sum Differential Game Of Backward Doubly Stochastic System And Near Optimal Control Problem

In this thesis,the optimal control problems and the near optimal control problems of doubly stochastic system are mainly studied.We discuss whether the state process of the doubly stochastic system contains time delay variables and time delay variables that are different.The nonzero sum differential problems of the doubly stochastic system under different conditions are discussed,and on this basis,when the optimal control of the system does not exist or is not easy to obtain,the near optimal control problems of the system are discussed.Firstly,backward doubly stochastic systems are studied,nonzero sum differential game problems for such systems are explored.The corresponding value functionals are introduced,and the classical convex variational technique and dual method are used to give the necessary condition for the existence of Nash equilibrium point.Using the definition of the value functional and Nash equilibrium point,combined with the variational equation corresponding to the system,the corresponding adjoint equations of the system are given,and then the necessary condition for the existence of the Nash equilibrium point is obtained.Finally,we apply the results to a special class of linear backward doubly stochastic system differential game problems and financial examples,the specific form of the Nash equilibrium point corresponding to the systems are given.Taking into account the effect of time delay,For the situation that the controlled system contains time delay variables and the time delay variables are different,we construct a new adjoint equation consisting of a doubly stochastic differential equation and three simple differential equations.Under the appropriate premise of the existence of such equation solutions,the dual method and convex variational technique are used to derive the necessary conditions and verification theorems for Nash equilibrium points similar to the random maximum principle.The nonzero sum differential games of doubly stochastic systems with time delay are mainly explored.Finally,the obtained results are apply to nonzero sum game problem of a class of linear delay backward doubly stochastic systems and financial examples to verify the validity of results.To reduce the effect of the sum of randomness on the system,taking the mean field backward doubly stochastic systems with delay as the research object,nonzero sum differential games for such systems are studied.We continue the method in the third part to discuss the necessary and sufficient conditions for the existence of Nash equilibrium points when the system contains different time delay variables,that is,the verification theorem.The obtained results are applied to financial examples to verify the validity of results.The near optimal control problems of linear time delay doubly stochastic systems with convex control domain are investigated at the end of the thesis.In the process of research,During the study,it was discussed that all the time delay variables were different,then the maximum value principle of control approximation optimal control is obtained.