Indefinite Stochastic Linear-Quadratic Optimal Control Problems with Random Coefficients: Closed-Loop Representation of Open-Loop Optimal Controls

TL;DR

This paper studies stochastic linear-quadratic control problems with random, indefinite coefficients, establishing conditions for optimal control existence, uniqueness, and a closed-loop representation using a Hilbert space approach.

Contribution

It introduces new sufficient conditions for the uniform convexity of the cost functional, extending classical positive definiteness requirements.

Findings

Necessary convexity condition for optimal control existence

Sufficient uniform convexity condition ensuring unique solution

Closed-loop representation of open-loop optimal controls

Abstract

This paper is concerned with a stochastic linear-quadratic optimal control problem in a finite time horizon, where the coefficients of the control system are allowed to be random, and the weighting matrices in the cost functional are allowed to be random and indefinite. It is shown, with a Hilbert space approach, that for the existence of an open-loop optimal control, the convexity of the cost functional (with respect to the control) is necessary; and the uniform convexity, which is slightly stronger, turns out to be sufficient, which also leads to the unique solvability of the associated stochastic Riccati equation. Further, it is shown that the open-loop optimal control admits a closed-loop representation. In addition, some sufficient conditions are obtained for the uniform convexity of the cost functional, which are strictly general than the classical conditions that the weighting matrix-valued processes are positive (semi-)definite.