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Volume 10
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Article

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

by
Zixuan Li
and
Jingtao Shi
*
School of Mathematics, Shandong University, Jinan 250100, China
*
Author to whom correspondence should be addressed.
Mathematics 2022, 10(21), 4062; https://doi.org/10.3390/math10214062
Submission received: 30 September 2022 / Revised: 27 October 2022 / Accepted: 27 October 2022 / Published: 1 November 2022
(This article belongs to the Special Issue Stochastic Control Systems: Theory and Applications)
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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 stochastic linear–quadratic optimal control; Poisson random measure; backward stochastic differential equation with Poisson jumps; Riccati integral–differential equation; closed-loop solvability

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MDPI and ACS Style

Li, Z.; Shi, J. Closed-Loop Solvability of Stochastic Linear-Quadratic Optimal Control Problems with Poisson Jumps. Mathematics 2022, 10, 4062. https://doi.org/10.3390/math10214062

AMA Style

Li Z, Shi J. Closed-Loop Solvability of Stochastic Linear-Quadratic Optimal Control Problems with Poisson Jumps. Mathematics. 2022; 10(21):4062. https://doi.org/10.3390/math10214062

Chicago/Turabian Style

Li, Zixuan, and Jingtao Shi. 2022. "Closed-Loop Solvability of Stochastic Linear-Quadratic Optimal Control Problems with Poisson Jumps" Mathematics 10, no. 21: 4062. https://doi.org/10.3390/math10214062

APA Style

Li, Z., & Shi, J. (2022). Closed-Loop Solvability of Stochastic Linear-Quadratic Optimal Control Problems with Poisson Jumps. Mathematics, 10(21), 4062. https://doi.org/10.3390/math10214062

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