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

TL;DR

This paper addresses finite-horizon stochastic LQ control problems with random coefficients and jumps, establishing existence of solutions and a closed-loop control representation without relying on traditional nonsingularity assumptions.

Contribution

It proves the existence of a unique strongly regular solution to the stochastic Riccati equation under less restrictive conditions and constructs the optimal control from the value flow.

Findings

Existence of a unique strongly regular solution to the SRE with jumps.

Closed-loop representation of the open-loop optimal control.

Examples illustrating indefinite weights and jump components.

Abstract

This paper studies finite-horizon stochastic linear-quadratic optimal control problems with random coefficients and Poisson jumps, where the weighting matrices may be random and indefinite. Under a uniform convexity condition on the cost functional, we prove that the associated stochastic Riccati equation (SRE) with jumps admits a unique strongly regular solution. As a consequence, the open-loop optimal control admits a closed-loop representation. The proof does not rely on a global representation of the form $P=\mathbf Y\mathbf X^{-1}$ or on any nonsingularity condition on the jump multiplier $I_n+E$ in the state equation. Instead, we construct $P$ from the stochastic value flow, and derive the strong regularity of the Riccati solution by a small-interval localization method. In addition, sufficient conditions are obtained for uniform convexity, and examples are presented to illustrate indefinite terminal and control weighting matrices and a nonzero jump martingale component in the SRE.