Limit theory for controlled McKean-Vlasov dynamics
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In this paper, the authors rigorously connect the problem of optimal control of McKean-Vlasov dynamics with large systems of interacting controlled state processes, and show that 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).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.read more
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