Limit theory for controlled McKean-Vlasov dynamics
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 -state systems, as 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.
Cite
@article{arxiv.1609.08064,
title = {Limit theory for controlled McKean-Vlasov dynamics},
author = {Daniel Lacker},
journal= {arXiv preprint arXiv:1609.08064},
year = {2016}
}