A Stochastic Linear-quadratic Optimal Control Problem With Lévy Processes

This paper is concerned with stochastic linear-quadratic (LQ in short) optimal controlproblem driven by Lévy processes and an independent Brownian motion. It includes three parts.Firstly, we prove the existence and uniqueness of the adapted solution of multi-dimensionalbackward stochastic differential equations driven by Brownian motion and Lévy process byusing predictable representation theorem and the fixed point theorem about contract mapping,and also prove the comparison theory.Secondly, we consider stochastic LQ optimal control problem driven by Lévy processes andBrownian motion. By applying the Ito-Lévy formula, Gronwall’s inequality, Young’sinequality and Cauchy-Schwarz inequality, we obtain the closeness property of the solution ofmulti-dimensional backward stochastic Riccati differential equations (BSRDE in short). Andthen we prove the existence and uniqueness of the adapted solution of one-dimensional BSRDEby making use of the closeness property, we also consider the non-homogeneous stochastic LQoptimal control problem and its application.Thirdly, we solve the mean-variance asset-liability management problem in a continuoustime setting. The price processes of the risky assets are driven by Brownian motion and Lévyprocess, and the price processes of the liabilities are only driven by Brownian motion. Usingembedding techniques, we employ stochastic optimal control theory to analytically solve theasset-liability management problem. More specifically, we derive both the optimal policy and themean-variance efficient frontier.Finally, we study stochastic LQ optimal problem with Poisson processes, which is a class ofspecial stochastic LQ optimal problem driven by Lévy processes. As its application, we solve aclass of mean-variance portfolio selection problems by employing the embedding technology.