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

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.

Funding

National Natural Science Foundation of China(11871121); Natural Science Foundation of Zhejiang Province(LY21A010001); Natural Science Foundation of Zhejiang Province(Z22A013952)