Stochastic Linear Quadratic Control Problem With Levy Processes And Backward Stochastic Differential Equations With Markov Chains

This dissertation is developed on the theory of linear quadratic(LQ in short) optimal control problem and backward stochastic differential equations(BSDEs in short).And we mainly focus on stochastic LQ control problem driven by Lévy processes and BSDEs with Markov chains.This thesis includes four chapters.In Chapter 1,a brief review of the historical literature in these topics we concerned is given.And also,we present the main results obtained in this thesis.In Chapter 2 and Chapter 3,we will consider stochastic LQ control problem driven by Lévy processes both in finite time horizon and in infinite time horizon.On one hand, as we all know,optimal control theory has great applications in practical problems. Especially,the LQ optimal control problem is one of the most popular problem in optimal control theory.After presented firstly by Kalman[22],the theory of such problem has been well established.Then,stochastic LQ problem with Brownian motion as the noise source in finite time horizon,which was firstly given by Wonham[53,54], has been studied extensively by many researchers,such as[13,5,46,59,45]and their references.On another hand,Lévy process,which has independent,stationary increments and càdlàg trajectories,has received much attention in recent years because of its many applications in finance and other fields.So we consider stochastic LQ control problem driven by Lévy processes in finite time horizon in Chapter 2.And we will show that,if one kind of Riccati equation admits a solution,this LQ control problem is well-posed and optimal feedback controls can be obtained via the solution. We also give some sufficient conditions for the solvability of this Riccati equation.Chapter 3 is a continuation of Chapter 2,in which we will discuss stochastic LQ control problem driven by Lévy processes in infinite time horizon.As we know,for infinite time horizon,stochastic LQ control problem with the system using Brownian motion to describe the noise was widely studied,such as in[59],[46],[47],[55]and their references.In this chapter,we will give the relation between attainability of this LQ problem and the stabilizing solution of one kind of algebraic Riccati equation.And we also study the attainability of this LQ problem via a semi-definite programming (SDP in short) and its dual problem.The last chapter is devoted to the study of BSDEs with Markov chains and homogenization of systems of partial differential equations(PDEs in short).The study of BSDEs stems from the research about stochastic control problem([6]).After the pioneering work of Pardoux and Peng[37]about BSDEs in a general case,BSDEs have been extensively studied recently because of its connections with mathematical finance, stochastic control and PDEs,see among[23,14,15].It is noted that BSDEs provide a probabilistic tool to study the homogenization of PDEs,which is the process of replacing rapidly varying coefficients by new ones thus the solutions are close.In Chapter 4, motivated by a stochastic LQ control problem with Markov jumps,we consider BSDEs with coefficients disturbed by Markov chains and give the uniqueness and existence results about the solution.When the Markov chain has a large state space,to distinguish the fast transitions from the slow transitions among different states,it is known that a small parameter(ε＞0) can be introduced which leads to a singularly perturbed Markov chain involving two time scales,namely,the actual time t and the stretched time t/εThen,using singular perturbation techniques and probabilistic approaches, the asymptotic probability distributions of singularly perturbed Markov chains can be derived asε→0.More details can be found in[61].In this chapter,based on the asymptotic probability distributions,we will consider the weak convergence of BSDEs with singularly perturbed Markov chains.We also give the homogenization result of one kind of PDE based on the connection between BSDEs and PDEs.