General indefinite backward stochastic linear-quadratic optimal control problems

¹ Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China
² Department of Mathematics and SUSTech International center for Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China

^* This author is supported by National Natural Science Foundation of China (Grant No. 11901280) and Guangdong Basic and Applied Basic Research Foundation (Grant No. 2021A1515010031).
^*** This author is supported by National Natural Science Foundation of China (Grant No. 12101291), Natural Science Foundation of Guangdong Province of China (Grant No. 2214050003543), and SUSTech start-up fund (Grant No. Y01286233).
^**** This author is supported partially by National Natural Science Foundation of China (Grants No. 61873325 and 11831010), and SUSTech start-up fund (Grant No. Y01286120).
^** Corresponding author: wenjq@sustech.edu.cn

Received: 19 September 2021
Accepted: 17 April 2022

Abstract

A general backward stochastic linear-quadratic optimal control problem is studied, in which both the state equation and the cost functional contain the nonhomogeneous terms. The main feature of the problem is that the weighting matrices in the cost functional are allowed to be indefinite, and the cross-product terms in the control and the state processes are present. Necessary and sufficient conditions for the solvability of the problem are obtained, and a characterization of the optimal control in terms of forward-backward stochastic differential equations is derived. By a Riccati equation approach, a general procedure for constructing optimal controls is developed and the value function is obtained explicitly.