Linear-Quadratic Optimal Control Problem For Mean-Field Stochastic Differential Equation With Lévy Process

This thesis investigates a linear-quadratic mean-field stochastic optimal control problem under both positive definite case and indefinite case where the controlled systems are mean-field stochastic differential equations driven by a Brownian motion and Teugels martingales associated with Levy processes.In either case,we obtain the optimal system for the optimal controls in open-loop form,and by means of a decoupling technique,we obtain the optimal controls in closed-loop form which can be represented by two Riccati differential equations.Moreover,the solvability of the optimal system and the Riccati equations are also obtained under both positive definite case and indefinite case.As an application paper of financial engineering,the dynamic mean-variance problem can be solved within the framework of the indefinite mean-field linear quadratic problem.This paper is divided into the following chapters:In the first chapter,we introduce some backgrounds,status of the research and some basic definitions.And we list the main results.In the second chapter,we study linear-quadratic mean-field stochastic optimal control problem under positive definite case where the controlled systems are driven by Brownian motion and Teugels martingales associated with Levy processes.We firstly obtain the optimality system and derive two Riccati differential equations which are uniquely solvable;secondly,we obtain a feedback representation of the optimal control.In the third chapter,we study linear-quadratic mean-field stochastic optimal control problem under indefinite case where the controlled systems are driven by Brownian motion and Teugels martingales associated with Levy processes.We obtain the optimality system and derive two Riccati differential equations under indefinite case.In the fourth chapter,we study the dynamic mean-variance problem which can be solved within the framework of the indefinite mean-field linear quadratic prroblem.We obtain an explicit representation of optimal investment strategy.