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

TL;DR

This paper develops a comprehensive framework for solving linear quadratic optimal control problems involving mean-field backward stochastic differential equations with jumps, providing existence, characterization, and explicit solutions.

Contribution

It introduces a novel approach to solve LQ control problems for MF-BSDEs with jumps, including new Riccati equations and explicit control representations.

Findings

Existence and uniqueness of optimal control established.

Optimal control characterized by a linear stochastic Hamilton system.

Explicit control formulas derived using Riccati equations.

Abstract

This paper is concerned with a linear quadratic (LQ, for short) optimal control problem for mean-field backward stochastic differential equations (MF-BSDE, for short) driven by a Poisson random martingale measure and a Brownian motion. Firstly, by the classic convex variation principle, the existence and uniqueness of the optimal control is established. Secondly, the optimal control is characterized by the stochastic Hamilton system which turns out to be a linear fully coupled mean-field forward-backward stochastic differential equation with jumps by the duality method. Thirdly, in terms of a decoupling technique, the stochastic Hamilton system is decoupled by introducing two Riccati equations and a MF-BSDE with jumps. Then an explicit representation for the optimal control is obtained.