Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics

Huyên Pham (1, 2) , Xiaoli Wei (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 CREST - Centre de Recherche en Économie et Statistique

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.

Mots clés: Stochastic McKean-Vlasov SDEs; dynamic programming principle; Bellman equation; Wasserstein space; viscosity solutions

Domaines: Probabilités [math.PR]; Optimisation et contrôle [math.OC]

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