Indefinite Stochastic LQ Optimal Control for Jump-Diffusion Systems with Random Coefficients

TL;DR

This paper develops a comprehensive theory for indefinite stochastic LQ control of jump-diffusion systems with random coefficients, including existence, uniqueness, and feedback representation of optimal controls, with applications to financial portfolio optimization.

Contribution

It introduces a novel approach that avoids relaxation and invertibility assumptions, extending classical LQ theory to more general jump-diffusion systems with random coefficients.

Findings

Established existence and uniqueness of optimal controls under uniform convexity.

Derived a stochastic Riccati equation with jumps (SREJ) characterizing the value function.

Applied results to a financial portfolio problem with explicit parametric conditions for optimality.

Abstract

This paper studies indefinite stochastic linear-quadratic (LQ) optimal control for jump-diffusion systems with random coefficients. We construct an algebraic inverse flow from the zero-control base system, extract the semimartingale kernel of the value function, and prove that it satisfies a generalized stochastic Riccati equation with jumps (SREJ). Under a uniform convexity condition, we establish the existence and uniqueness of open-loop optimal controls for any initial pair and show that the associated matrix $\mathscr{N}(t)$ is uniformly positive definite, yielding an exact closed-loop feedback representation of the optimal control via the SREJ. A distinguishing feature of our approach is that it requires neither relaxation techniques (as in the compensator method) nor additional invertibility assumptions on the optimal state process, and it accommodates the general case where the control enters the jump part ($F \neq 0$). As an application, we analyze a financial portfolio problem with a jump-diffusion risky asset whose excess return is zero, where the investor minimizes a cost functional with a negative terminal wealth weight. The uniform convexity condition reduces to an explicit inequality among the risk aversion coefficient, volatility, jump magnitude, and risk-free rate, thereby delineating the parametric region in which an optimal strategy exists. These results extend classical indefinite LQ theory to jump-diffusion systems with random coefficients.