Linear Quadratic Stochastic Optimal Control Of Mean-Field Backward Stochastic Differential Equations And Nonzero Sum Differential Games

Based on the theory of forward-backward stochastic differential equations (FBSDEs), mean-field forward-backward stochastic differential equations (mean-field FBSDEs) and the optimal control, we study a new type of initial coupled mean-field FBSDEs. Then, we study mean-field linear quadratic optimal control problems and the nonzero-sum differential games of such equations. In the end we study the near-optimal control problems of mean-field linear forward-backw-ard stochastic systems.In this paper, firstly, we prove that there exists a unique solution of initial coupled mean-field FBSDEs under the monotonic conditions: Then we give the adjoint equation corresponding to the forward stochastic equa-tion. With the help of the solutions of mean-field FBSDEs, we get the explicit form of the optimal control for the linear quadratic optimal control problems and the open-loop Nash equilibrium point of nonzero-sum differential games.Secondly, we study the near-optimal control problems of mean-field linear forward-backward stochastic systems. Inspired by Zhou [25] we give the definit-ion of the near-optimality, and establish the sufficient condition and the necess--ary condition of the near-optimality in the form of Pontryagin stochastic maxi--mum principle.