Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equations with Jumps

TL;DR

This paper addresses linear-quadratic optimal control problems for mean-field stochastic differential equations with jumps, deriving the optimal control via duality, Riccati equations, and feedback representation.

Contribution

It introduces a novel approach to solve mean-field LQ control problems with jumps, including existence, uniqueness, and explicit feedback control formulation.

Findings

Existence and uniqueness of optimal control established.

Derived the stochastic Hamilton system with jumps.

Proved the optimal control has a state feedback form.

Abstract

In this paper, we study a linear-quadratic optimal control problem for mean-field stochastic differential equations driven by a Poisson random martingale measure and a multidimensional Brownian motion. Firstly, the existence and uniqueness of the optimal control is obtained by the classic convex variation principle. Secondly, by the duality method, the optimality system, also called the stochastic Hamilton system which turns out to be a linear fully coupled mean-field forward-backward stochastic differential equation with jumps, is derived to characterize the optimal control. Thirdly, applying a decoupling technique, we establish the connection between two Riccati equation and the stochastic Hamilton system and then prove the optimal control has a state feedback representation.