Indefinite linear-quadratic optimal control of mean-field stochastic differential equation with jump diffusion: an equivalent cost functional method

TL;DR

This paper develops a novel method using equivalent cost functionals to solve indefinite linear-quadratic optimal control problems for mean-field stochastic differential equations with jump diffusion, establishing solution existence, uniqueness, and feedback control representations.

Contribution

It introduces an equivalent cost functional approach to handle indefinite MF-LQJ problems, connecting them to positive-definite cases and solving associated Riccati equations.

Findings

Established solvability conditions for stochastic Hamiltonian systems.

Derived explicit state feedback controls via Riccati equations.

Proved existence and uniqueness of solutions to mean field FBSDEs with jumps.

Abstract

In this paper, we consider a linear-quadratic optimal control problem of mean-field stochastic differential equation with jump diffusion, which is also called as an MF-LQJ problem. Here, cost functional is allowed to be indefinite. We use an equivalent cost functional method to deal with the MF-LQJ problem with indefinite weighting matrices. Some equivalent cost functionals enable us to establish a bridge between indefinite and positive-definite MF-LQJ problems. With such a bridge, solvabilities of stochastic Hamiltonian system and Riccati equations are further characterized. Optimal control of the indefinite MF-LQJ problem is represented as a state feedback via solutions of Riccati equations. As a by-product, the method provides a new way to prove the existence and uniqueness of solution to mean field forward-backward stochastic differential equation with jump diffusion (MF-FBSDEJ, for short), where existing methods in literature do not work. Some examples are provided to illustrate our results.