Ergodic control of McKean-Vlasov SDEs and associated Bellman equation

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.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

We study the ergodic control problem for McKean-Vlasov stochastic differential equations and establish the existence and uniqueness of the viscosity solution to the associated fully nonlinear HJB equation in a lifted sense. Moreover, we demonstrate the convergence of the solutions of finite-horizon time-averaging optimal control problems to that of the ergodic control problem as the time horizon tends to infinity. Our results rely on dissipativity conditions and dissipativity-like conditions on the distribution variables of both drift and diffusion coefficients. (c) 2023 Elsevier Inc. All rights reserved.