Ergodic control of McKean–Vlasov SDEs and associated Bellman equation

Abstract

We consider the ergodic control problem for McKean-Vlasov stochastic differential equations and prove the existence and uniqueness of the viscosity solution to the associated fully nonlinear HJB equation in a lifted sense. Furthermore, as the time horizon goes to infinity, we show that the solutions of finite-horizon time-averaging optimal control problems converge to that of the ergodic control problem. Our results require dissipativity conditions and dissipativity-like conditions on distribution variables of both drift and diffusion coefficients.