Near Optimality of Stochastic Control for Singularly Perturbed McKean--Vlasov Systems

This is the authors' abstract. We don't add key points for this paper.

In this paper, we are concerned with the optimal control problems for a class of systems with fast-slow processes. The problem under consideration is to minimize a functional subject to a system described by a two-time scaled McKean--Vlasov stochastic differential equation whose coefficients depend on state components and their probability distributions. Firstly, we establish the existence and uniqueness of the invariant probability measure for the fast process. Then, by using the relaxed control representation and the martingale method, we prove the weak convergence of the slow process and control process in the original problem, and we obtain an associated limit problem in which the coefficients are determined by the average of those of the original problem with respect to the invariant probability measure. Finally, by establishing the nearly optimal control of the limit problem, we obtain the near optimality of the original problem.