Linear Quadratic Optimal Control Of Mean-field Type With Jumps Under Partial Information And Two-person Zero-sum Differential Games Problem

This thesis discusses a kind of linear quadratic optimal control of mean-field type with jumps under partial information and two-person zero-sum differential games problem with deterministic coefficients,which is divided into the following two parts.In the first part,the stochastic linear-quadratic optimal control problem of meanfield type with jumps under partial information is discussed.The state equation which contains affine terms is a SDE with jumps driven by a multidimensional Brownian motion and a Poisson random martingale measure,and the quadratic cost function contains cross terms.In addition,the state and the control as well as their expectations are contained both in the state equation and the cost functional.This is the so-called optimal control problem of mean-field type.Firstly,the existence and uniqueness of the optimal control is proved.Secondly,the adjoint processes of the state equation is introduced,and by using the duality technique,the optimal control is characterized by the stochastic Hamiltonian system.Thirdly,by applying a decoupling technology,we deduce two integro-differential Riccati equations and get the feedback representation of the optimal control under partial information.Fourthly,the existence and uniqueness of the solutions of two Riccati equations are proved.Finally,we discuss a special case,and establish the corresponding feedback representation of the optimal control by means of filtering technique.In the second part,a class of mean-field type stochastic linear-quadratic two-person zero-sum differential games problem under partial information is discussed.Firstly,the stationarity condition satisfied by the open-loop saddle point is obtained;Secondly,two Riccati equations are introduced through the interaction between two players,and by using classical variational technique and the completion of the squares approach,we obtain the explicit feedback representation of open-loop saddle point and the optimal game value function under partial information.Thirdly,the existence and the uniqueness of open-loop saddle point is proved;Finally,we discuss a special case of the problem,and get the corresponding feedback representation of the open-loop saddle point.