Indefinite Backward Stochastic Linear-Quadratic Optimal Control Problems

TL;DR

This paper investigates backward stochastic linear-quadratic optimal control problems with indefinite weights, providing solvability conditions and a method to construct optimal controls via Riccati equations.

Contribution

It introduces a general approach for solving backward stochastic LQ problems with indefinite weights, including solvability criteria and control construction methods.

Findings

Established necessary and sufficient conditions for problem solvability.

Connected backward stochastic LQ problems with forward stochastic LQ problems.

Developed a method to solve Riccati-type equations under weak conditions.

Abstract

This paper is concerned with a backward stochastic linear-quadratic (LQ, for short) optimal control problem with deterministic coefficients. The weighting matrices are allowed to be indefinite, and cross-product terms in the control and state processes are present in the cost functional. Based on a Hilbert space method, necessary and sufficient conditions are derived for the solvability of the problem, and a general approach for constructing optimal controls is developed. The crucial step in this construction is to establish the solvability of a Riccati-type equation, which is accomplished under a fairly weak condition by investigating the connection with forward stochastic LQ optimal control problems.