Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions

TL;DR

This paper establishes the first uniqueness result for viscosity solutions of the Master Bellman equation in the Wasserstein space, crucial for mean field control problems, using novel approximation and regularization techniques.

Contribution

It introduces a new comparison principle for viscosity solutions in Wasserstein space, proving uniqueness for the Master Bellman equation in a second-order setting.

Findings

Proves the uniqueness of viscosity solutions in Wasserstein space.

Develops finite-dimensional approximation methods.

Constructs a smooth gauge-type function for analysis.

Abstract

We study the Bellman equation in the Wasserstein space arising in the study of mean field control problems, namely stochastic optimal control problems for McKean-Vlasov diffusion processes.Using the standard notion of viscosity solution \`a la Crandall-Lions extended to our Wasserstein setting,we prove a comparison result under general conditions on the drift and reward coefficients, whichcoupled with the dynamic programming principle, implies that the value function is the unique viscosity solution of the Master Bellman equation.This is the first uniqueness result in such a second-order context. The classical arguments used in the standard cases of equations in finite-dimensional spaces or in infinite-dimensional separable Hilbert spaces do not extend to the present framework, due to the awkward nature of the underlying Wasserstein space. The adopted strategy is based on finite-dimensional approximations of the value function obtained in terms of the related cooperative $n$-player game, and on the construction of a smooth gauge-type function, built starting from a regularization of a sharp estimate of the Wasserstein metric; such a gauge-type function is used to generate maxima/minima through a suitable extension of the Borwein-Preiss generalization of Ekeland's variational principle on the Wasserstein space.