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

Jiali Wu; Maoning Tang; Qingxin Meng; Jiali Wu; Maoning Tang; Qingxin Meng

doi:10.3934/math.2023202

AIMS Mathematics

2023, Volume 8, Issue 2: 4042-4078. doi: 10.3934/math.2023202

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Research article

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

1.
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
2.
Department of Mathematical Sciences, Huzhou University, Zhejiang 313000, China

Received: 19 June 2022 Revised: 19 October 2022 Accepted: 23 October 2022 Published: 01 December 2022
MSC : 60H10, 93E24

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.
- stochastic linear quadratic optimal control,
- stabilizability,
- open-loop solvability,
- closed-loop solvability,
- algrbraic Riccati equation,
- stabilizing solution,
- closed-loop representation
Citation: Jiali Wu, Maoning Tang, Qingxin Meng. A stochastic linear-quadratic optimal control problem with jumps in an infinite horizon[J]. AIMS Mathematics, 2023, 8(2): 4042-4078. doi: 10.3934/math.2023202

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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.

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