General Indefinite Backward Stochastic Linear-Quadratic Optimal Control Problems

TL;DR

This paper investigates a broad class of backward stochastic linear-quadratic optimal control problems with indefinite weights and cross-terms, providing solvability conditions, explicit solutions, and a Riccati equation-based control construction.

Contribution

It introduces necessary and sufficient conditions for solvability and develops a Riccati equation approach for constructing optimal controls in complex backward stochastic LQ problems.

Findings

Derived explicit value function expressions.

Established solvability criteria for indefinite weighting matrices.

Provided a Riccati equation-based control construction method.

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 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.