Infinite Horizon Linear Quadratic Mean Field Problems with Common Noise and Regime Switching via Conditional McKean-Vlasov FBSDEs
Qingmeng Wei, Yaqi Xu
TL;DR
This work develops a probabilistic framework for infinite-horizon linear-quadratic mean-field control and mean-field game problems with common noise and regime switching, formulated through conditional McKean–Vlasov forward–backward SDEs with Markov switching. It establishes well-posedness via a generalized domination–monotonicity condition and derives necessary and sufficient optimality conditions for open-loop control and mean-field Nash equilibria, using a continuation method to handle full coupling. The infinite-horizon LQ analysis yields explicit or semi-explicit optimal controls via adjoint variables and a Hamiltonian system, including transformations to remove cross terms. Overall, the results provide a rigorous foundation for MF control and MF games in stochastic environments with regime changes and shared randomness, with potential applications to systemic risk and large-scale economic systems.
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.