Limit theory for controlled McKean-Vlasov dynamics

TL;DR

This paper establishes a rigorous connection between the optimal control of McKean-Vlasov dynamics and large systems of interacting controlled processes, showing convergence of empirical distributions to solutions of the McKean-Vlasov problem.

Contribution

It proves that empirical distributions of near-optimal controls in large systems converge to the McKean-Vlasov control solutions, and any such solution can be approximated by large systems.

Findings

Empirical distributions converge to McKean-Vlasov control solutions as system size grows.

Any McKean-Vlasov control solution can be realized as a limit of large system controls.

Existence of optimal Markovian controls for a broad class of McKean-Vlasov problems.

Abstract

This paper rigorously connects the problem of optimal control of McKean-Vlasov dynamics with large systems of interacting controlled state processes. Precisely, the empirical distributions of near-optimal control-state pairs for the $n$-state systems, as $n$ tends to infinity, admit limit points in distribution (if the objective functions are suitably coercive), and every such limit is supported on the set of optimal control-state pairs for the McKean-Vlasov problem. Conversely, any distribution on the set of optimal control-state pairs for the McKean-Vlasov problem can be realized as a limit in this manner. Arguments are based on controlled martingale problems, which lend themselves naturally to existence proofs; along the way it is shown that a large class of McKean-Vlasov control problems admit optimal Markovian controls.