A stochastic linear-quadratic optimal control problem with jumps in an infinite horizon

Abstract

In this paper, a stochastic linear-quadratic (LQ, for short) optimal control problem with jumps in an infinite horizon is studied, where the state system is a controlled linear stochastic differential equation containing affine term driven by a one-dimensional Brownian motion and a Poisson stochastic martingale measure, and the cost functional with respect to the state process and control process is quadratic and contains cross terms. Firstly, in order to ensure the well-posedness of our stochastic optimal control of infinite horizon with jumps, the $ L^2 $-stabilizability of our control system with jump is introduced. Secondly, it is proved that the $ L^2 $-stabilizability of our control system with jump is equivalent to the non-emptiness of the admissible control set for all initial state and is also equivalent to the existence of a positive solution to some integral algebraic Riccati equation (ARE, for short). Thirdly, the equivalence of the open-loop and closed-loop solvability of our infinite horizon optimal control problem with jumps is systematically studied. The corresponding equivalence is established by the existence of a $ stabilizing\ solution $ of the associated generalized algebraic Riccati equation, which is different from the finite horizon case. Moreover, any open-loop optimal control for the initial state $ x $ admiting a closed-loop representation is obatined.

Keywords

• stochastic linear quadratic optimal control
• stabilizability
• open-loop solvability
• closed-loop solvability
• algrbraic riccati equation
• stabilizing solution
• closed-loop representation