Journal
ANNALS OF PROBABILITY
Volume 43, Issue 5, Pages 2647-2700Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/14-AOP946
Keywords
Stochastic control; McKean-Vlasov diffusion; stochastic Pontryagin principle; mean-field interaction; mean-field forward-backward stochastic differential equation
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Funding
- NSF [DMS-08-06591]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1211928] Funding Source: National Science Foundation
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The purpose of this paper is to provide a detailed probabilistic analysis of the optimal control of nonlinear stochastic dynamical systems of McKean Vlasov type. Motivated by the recent interest in mean-field games, we highlight the connection and the differences between the two sets of problems. We prove a new version of the stochastic maximum principle and give sufficient conditions for existence of an optimal control. We also provide examples for which our sufficient conditions for existence of an optimal solution are satisfied. Finally we show that our solution to the control problem provides approximate equilibria for large stochastic controlled systems with mean-field interactions when subject to a common policy.
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