Linear Quadratic Stochastic Optimal Control of Forward Backward Stochastic Control System Associated with Lévy Process

Hong Huang,Xiangrong Wang,Ting Hou,Lu Xu

doi:10.1155/2017/2541687

Abstract

This paper analyzes one kind of linear quadratic (LQ) stochastic control problem of forward backward stochastic control system associated with Lévy process. We obtain the explicit form of the optimal control, then prove it to be unique, and get the linear feedback regulator by introducing one kind of generalized Riccati equation. Finally, we discuss the solvability of the generalized Riccati equation, and its existence and uniqueness of the solutions are proved in a special case.

Highlights

linear quadratic (LQ) stochastic optimal control is a kind of special optimal control problem, which can be used to model many linear optimal problems practically, and can reasonably be used to approach and solve many nonlinear problems
On the basis of proving the existence and uniqueness of solutions of a kind of forward backward stochastic differential equation with Poisson jumps (FBSDEP), Wu and Wang [11] got the explicit form of the optimal control for LQ stochastic control problem where the state variable was described by a stochastic differential equation with a Poisson process (SDEP)
El Otmani [18] considered a kind of generalized BSDE (GBSDE) associated with Teugels martingale and Brownian motion associated with a pure jump-independent Levy process

Summary

Introduction

LQ stochastic optimal control is a kind of special optimal control problem, which can be used to model many linear optimal problems practically, and can reasonably be used to approach and solve many nonlinear problems. El Otmani [18] considered a kind of generalized BSDE (GBSDE) associated with Teugels martingale and Brownian motion associated with a pure jump-independent Levy process They got the existence and uniqueness theory of this GBSDE when the coefficient verifies some conditions of Lipschitz. We consider one kind of LQ stochastic control problem where the controlled system is driven by a fully coupled linear forward backward stochastic differential equation associated with Levy process (FBSDEL). We extend the result of Shi and Wu [12] to the fully coupled linear forward backward stochastic control system driven by Brownian motion and an independent Teugels martingale.

Preliminaries and Notations

Linear Quadratic Stochastic Optimal

Solvability of the Generalized Riccati Equation

Conclusion