Communications on Applied Mathematics and Computation >

2022 , Vol. 4 >Issue 4: 1386 - 1415

DOI: https://doi.org/10.1007/s42967-021-00181-y

Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equation with Lévy Process

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

Received date: 2021-08-26

  Revised date: 2021-12-11

  Online published: 2022-09-26

Supported by

The authors would like to thank anonymous referees for helpful comments and suggestions which improved the original version of the paper. Q. Meng was supported by the Key Projects of Natural Science Foundation of Zhejiang Province of China (no. Z22A013952) and the National Natural Science Foundation of China (no. 11871121). Maoning Tang was supported by the Natural Science Foundation of Zhejiang Province of China (no. LY21A010001).

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Abstract

This paper investigates a linear-quadratic mean-field stochastic optimal control problem under both positive definite case and indefinite case where the controlled systems are mean-field stochastic differential equations driven by a Brownian motion and Teugels martingales associated with Lévy processes. In either case, we obtain the optimality system for the optimal controls in open-loop form, and by means of a decoupling technique, we obtain the optimal controls in closed-loop form which can be represented by two Riccati differential equations. Moreover, the solvability of the optimality system and the Riccati equations are also obtained under both positive definite case and indefinite case.

Key words： Mean-field Teugels martingales; Linear-quadratic; Optimal control; Riccati equations; Feedback representation

Cite this article

Hong Xiong, Maoning Tang, Qingxin Meng . Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equation with Lévy Process[J]. Communications on Applied Mathematics and Computation, 2022 , 4(4) : 1386 -1415 . DOI: 10.1007/s42967-021-00181-y

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