Closed-Loop Solvability of Stochastic Linear-Quadratic Optimal Control Problems with Poisson Jumps

Abstract

The stochastic linear–quadratic optimal control problem with Poisson jumps is addressed in this paper. The coefficients in the state equation and the weighting matrices in the cost functional are all deterministic but are allowed to be indefinite. The notion of closed-loop strategies is introduced, and the sufficient and necessary conditions for the closed-loop solvability are given. The optimal closed-loop strategy is characterized by a Riccati integral–differential equation and a backward stochastic differential equation with Poisson jumps. A simple example is given to demonstrate the effectiveness of the main result.

Keywords

• stochastic linear–quadratic optimal control
• Poisson random measure
• backward stochastic differential equation with Poisson jumps
• Riccati integral–differential equation
• closed-loop solvability