Infinite Horizon Linear Quadratic Mean Field Problems with Common Noise and Regime Switching via Conditional McKean-Vlasov FBSDEs

TL;DR

This paper develops a theoretical framework for infinite horizon linear quadratic mean field problems with common noise and regime switching, using conditional McKean-Vlasov FBSDEs, and establishes conditions for optimal control and Nash equilibria.

Contribution

It introduces a novel analysis of coupled FBSDEs with Markovian switching for mean field problems, providing well-posedness and optimality conditions.

Findings

Established well-posedness of conditional McKean-Vlasov FBSDEs with switching

Derived necessary and sufficient conditions for optimal control and Nash equilibria

Extended the LQ mean field framework to include common noise and regime switching

Abstract

This paper studies infinite horizon linear quadratic (LQ) mean field problems with common noise and regime switching, covering both control and game formulations. To establish a theoretical foundation for the LQ framework, we first analyze fully coupled forward-backward stochastic differential equations (FBSDEs) of conditional McKean-Vlasov type with Markovian switching and establish its well-posedness under a generalized domination-monotonicity condition. Building upon this solvability result, we then derive necessary and sufficient conditions for both the open-loop optimal control in the control problem and the mean-field Nash equilibria in the game problem.