@article{Wu2023, 
author = {Jiali Wu and Maoning Tang and Qingxin Meng},
title = {A stochastic linear-quadratic optimal control problem with jumps in an infinite horizon},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {2},
pages = {4042-4078},
keywords = {stochastic linear quadratic optimal control, stabilizability, open-loop solvability, closed-loop solvability, algrbraic Riccati equation, stabilizing solution, closed-loop representation},
url = {https://www.sciopen.com/article/10.3934/math.2023202},
doi = {10.3934/math.2023202},
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    s  t  a  b  i  l  i  z  i  n  g    s  o  l  u  t  i  o  n 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.}
}