Linear-Quadratic Optimal Control Problem for Mean-Field Stochastic Differential Equations with a Type of Random Coefficients

TL;DR

This paper addresses a linear-quadratic optimal control problem for mean-field stochastic differential equations with random coefficients, providing new insights into solvability under regime switching and independent filtrations.

Contribution

It introduces a novel LQ control framework for mean-field SDEs with regime switching and independent filtrations, extending classical methods to this complex setting.

Findings

Characterization of open-loop and closed-loop solvability

Application of the completing the square method with identified limitations

Extension of LQ control theory to mean-field SDEs with random coefficients

Abstract

Motivated by linear-quadratic optimal control problems (LQ problems, for short) for mean-field stochastic differential equations (SDEs, for short) with the coefficients containing regime switching governed by a Markov chain, we consider an LQ problem for an SDE with the coefficients being adapted to a filtration independent of the Brownian motion driving the control system. Classical approach of completing the square is applied to the current problem and obvious shortcomings are indicated. Open-loop and closed-loop solvability are introduced and characterized.