The Delayed Doubly Stochastic Linear Quadratic Optimal Control Problem

Yan Chen,Jie Xu

doi:10.1155/2020/2759580

Abstract

In this paper, the delayed doubly stochastic linear quadratic optimal control problem is discussed. It deduces the expression of the optimal control for the general delayed doubly stochastic control system which contained time delay both in the state variable and in the control variable at the same time and proves its uniqueness by using the classical parallelogram rule. The paper is concerned with the generalized matrix value Riccati equation for a special delayed doubly stochastic linear quadratic control system and aims to give the expression of optimal control and value function by the solution of the Riccati equation.

Highlights

As is known to all, the stochastic differential equation and stochastic analysis have developed rapidly. e theory of the stochastic differential equation is widely used in economy, biology, physics, financial mathematics, and other fields. e latest research on the insurance model was given in [1,2,3]. e social optimal mean field control problem was discussed in [4]
Zhu and Shi [6, 7] were concerned with a class of partial information control problems for backward doubly stochastic systems and gave the maximum principle and its applications for the system
Shi and Zhu [8] studied a type of forwardbackward doubly stochastic differential equations driven by Brownian motions and the Poisson process and applied the result to backward doubly stochastic linear quadratic nonzero sum differential games with random jumps to get the explicit form of the open-loop Nash equilibrium point by the solution of this kind of equation

Summary

Introduction

As is known to all, the stochastic differential equation and stochastic analysis have developed rapidly. e theory of the stochastic differential equation is widely used in economy, biology, physics, financial mathematics, and other fields. e latest research on the insurance model was given in [1,2,3]. e social optimal mean field control problem was discussed in [4]. Zhu and Shi [6, 7] were concerned with a class of partial information control problems for backward doubly stochastic systems and gave the maximum principle and its applications for the system. Wu and Wang [10] studied the optimal control problem of the backward stochastic differential delay equation under partial information. Wang and Wu [12] were concerned with the optimal control problems of forward-backward delay systems involving impulse controls and established the stochastic maximum principle for this kind of systems. Huang et al [16] were concerned with one kind of delayed forward-backward linear quadratic stochastic control problems and derived the explicit form of the optimal control. We use the solution of the Riccati equation to show the optimal control for the delayed doubly stochastic LQ problem. We indicate the objective function by the solution of the Riccati equation and the initial value of the state variable

Preliminaries

Main Results

Applications