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Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics

Abstract : We study the optimal control of general stochastic McKean-Vlasov equation. Such problem is motivated originally from the asymptotic formulation of cooperative equilibrium for a large population of particles (players) in mean-field interaction under common noise. Our first main result is to state a dynamic programming principle for the value function in the Wasserstein space of probability measures, which is proved from a flow property of the conditional law of the controlled state process. Next, by relying on the notion of differentiability with respect to probability measures due to P.L. Lions [32], and Itô's formula along a flow of conditional measures, we derive the dynamic programming Hamilton-Jacobi-Bellman equation, and prove the viscosity property together with a uniqueness result for the value function. Finally, we solve explicitly the linear-quadratic stochastic McKean-Vlasov control problem and give an application to an interbank systemic risk model with common noise.
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https://hal.archives-ouvertes.fr/hal-01302289
Contributor : Huyên Pham <>
Submitted on : Wednesday, January 4, 2017 - 10:50:03 AM
Last modification on : Friday, March 27, 2020 - 2:55:55 AM
Document(s) archivé(s) le : Wednesday, April 5, 2017 - 1:39:34 PM

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  • HAL Id : hal-01302289, version 2
  • ARXIV : 1604.04057

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Huyên Pham, Xiaoli Wei. Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics . 2016. ⟨hal-01302289v2⟩

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