Linear-Quadratic Optimal Control for Mean-Field Stochastic Differential Equations in Infinite-Horizon with Regime Switching

TL;DR

This paper develops a framework for solving infinite-horizon linear quadratic stochastic control problems with mean-field terms and regime switching, deriving algebraic Riccati equations and BSDEs to characterize optimal strategies.

Contribution

It introduces a novel approach combining orthogonal decomposition with algebraic Riccati equations and BSDEs for mean-field control in regime-switching environments.

Findings

Derived algebraic Riccati equations for the problem

Established solvability conditions for the Riccati equations and BSDEs

Characterized optimal control strategies via solvability of the equations

Abstract

This paper is concerned with stochastic linear quadratic (LQ, for short) optimal control problems in an infinite horizon with conditional mean-field term in a switching regime environment. The orthogonal decomposition introduced in [21] has been adopted. Desired algebraic Riccati equations (AREs, for short) and a system of backward stochastic differential equations (BSDEs, for short) in infinite time horizon with the coefficients depending on the Markov chain have been derived. The determination of closed-loop optimal strategy follows from the solvability of ARE and BSDE. Moreover, the solvability of BSDEs leads to a characterization of open-loop solvability of the optimal control problem.