Linear-quadratic optimal control for backward stochastic differential equations with random coefficients

¹ Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, PR China.
² Department of Mathematics, National University of Singapore, Singapore 119076, Singapore.

^*** Corresponding author: hxwang14@fudan.edu.cn

Received: 11 March 2020
Accepted: 3 May 2021

Abstract

This paper is concerned with a linear-quadratic (LQ, for short) optimal control problem for backward stochastic differential equations (BSDEs, for short), where the coefficients of the backward control system and the weighting matrices in the cost functional are allowed to be random. By a variational method, the optimality system, which is a coupled linear forward-backward stochastic differential equation (FBSDE, for short), is derived, and by a Hilbert space method, the unique solvability of the optimality system is obtained. In order to construct the optimal control, a new stochastic Riccati-type equation is introduced. It is proved that an adapted solution (possibly non-unique) to the Riccati equation exists and decouples the optimality system. With this solution, the optimal control is obtained in an explicit way.