Linear-Quadratic Optimal Control for Backward Stochastic Differential Equations with Random Coefficients

TL;DR

This paper develops a framework for solving linear-quadratic optimal control problems for backward stochastic differential equations with random coefficients, introducing a stochastic Riccati equation to explicitly construct optimal controls.

Contribution

It introduces a new stochastic Riccati-type equation for backward stochastic control with random coefficients and proves the explicit construction of optimal controls.

Findings

Existence of a (possibly non-unique) adapted solution to the Riccati equation.

Decoupling of the optimality system via the Riccati solution.

Explicit expression for the optimal control.

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.