Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics

TL;DR

This paper develops a dynamic programming framework for optimal control of stochastic McKean-Vlasov equations, deriving the Hamilton-Jacobi-Bellman equation and solving a linear-quadratic case with applications to systemic risk.

Contribution

It introduces a dynamic programming principle in Wasserstein space and derives the HJB equation for McKean-Vlasov control problems, including explicit solutions and applications.

Findings

Established a dynamic programming principle in Wasserstein space

Derived the HJB equation and proved viscosity and uniqueness

Explicitly solved a linear-quadratic McKean-Vlasov control problem

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 It{\^o}'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.