A Tikhonov Theorem for McKean–Vlasov Two-Scale Systems and a New Application to Mean Field Optimal Control Problems

We provide a new version of the Tikhonov theorem for both two-scale forward systems and also two-scale forward-backward systems of stochastic differential equations, which also covers the McKean–Vlasov case. Differently from what is usually done in the literature, we prove a type of convergence for the "fast" variable, which allows the limiting process to be discontinuous. This is relevant for the second part of the paper, where we present a new application of this theory to the approximation of the solution of mean field control problems. Towards this aim, we construct a two-scale system whose "fast" component converges to the optimal control process, while the "slow" component converges to the optimal state process. The interest in such a procedure is that it allows one to approximate the solution of the control problem, avoiding the usual step of the minimization of the Hamiltonian.