A Tikhonov theorem for McKean-Vlasov two-scale systems and a new application to mean field optimal control problems

TL;DR

This paper extends the Tikhonov theorem to two-scale McKean-Vlasov systems, enabling approximation of mean field control solutions without Hamiltonian minimization, and introduces a novel convergence approach for discontinuous limits.

Contribution

It develops a new Tikhonov theorem for two-scale stochastic systems including McKean-Vlasov cases, allowing convergence of discontinuous fast variables and applying it to mean field control problems.

Findings

Established convergence of fast variables to discontinuous limits.

Constructed a two-scale system approximating mean field control solutions.

Provided a method to avoid Hamiltonian minimization in control problems.

Abstract

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 to approximate the solution of the control problem avoiding the usual step of the minimization of the Hamiltonian.