Mean-Field Forward-Backward Stochastic Differential Equations And The Related Questions

In1973, Bisnmt [4] studied the linear backward stochastic differential equations (BSDEs). Pardoux and Peng [56](1990), gave the wellposedness of nonlinear BSDEs. From then on. the theory of BSDEs and its applications-have been improved significantly, e.g.. in mathematical finance and stochastic-control problems. In2009Buckdalni, Li and Peng [10], Buckdalnu Djehiche Li and Peng [8] first studied the mean-field BSDE, which begins to attract moi and more attention. The objective of this thesis is to study the Mean-field Forward-Backward Stochastic Differential Equations and the related stochas-tic control problems, and also the optimal control problems of anticipated backward linear quadratic stochastic.This thesis consists of six chapters as follows.Chapter1Introduction. We give briefly introduction of the theory of the classical backward stochastic differential equations and the stochastic optimal control problems. The main ideas will be used in this thesis repeatedly.Chapter2PreliminaryWe present some basic results of the classical backward stochastic differ-ential equationsChapter3Mean-field Forward-Backward Stochastic. Differential systemsWe discuss new types of stochastic differential equations which we call fully-coupled Mean-field Forward-Backward Stochastic Differential Equations (MFFBSDEs for short) which the coefficients depend on the state of the solu-tion processes as well as the expected values of the states. We show that these MFFBSDEs have unique solutions under some certain "monotonicity" condi-tions. Then, we give the continuity property of the solution on parameters. Finally, we discuss the stochastic optimal control problems of MFFBSDEs. The stochastic maximum principles are derived and the related mean-field lin-ear quadratic optimal control problems are also extensively discussed here.Chapter4A type of general FBSDE and ABLQ.We prove the existence and the uniqueness of the solution of a type of general forward-backward stochastic differential equations(FBSDE) with Ito stochastic delayed equations as the forward equation and anticipated back stochastic differential equations as the backward part, under some certain "monotonicity" conditions. Using the solutions of the FBSDEs we discuss the anticipated backward linear quadratic optimal stochastic control problem.Chapter5Mean-field time-delay SDEs and the related stochastic control problemsWe extend the theories of classical time-delay SDEs and anticipated BS-DEs To the mean-field type. Then, we discuss the stochastic optimal control problems of mean-field stochastic differential delayed equations. The stochas-tic maximum principle of necessary and sufficient conditions are derived and the related mean-field linear quadratic delayed optimal control problems are also extensively investigated here.Chapter6Further research.We show several research directions, we will study in the future.