Viscosity Solutions of Fully second-order HJB Equations in the Wasserstein Space

TL;DR

This paper establishes that the value functions of mean field control problems with common noise are the unique viscosity solutions to fully second-order Hamilton-Jacobi-Bellman equations in the Wasserstein space, addressing unbounded dynamics and state-dependent noise.

Contribution

It introduces a novel framework for viscosity solutions of second-order HJB equations in Wasserstein space with state-dependent derivatives, extending prior methods to more complex dynamics.

Findings

Proves uniqueness of viscosity solutions in this setting

Handles unbounded dynamics and state-dependent noise volatility

Develops approximation techniques from particle systems

Abstract

In this paper, we show that the value functions of mean field control problems with common noise are the unique viscosity solutions to fully second-order Hamilton-Jacobi-Bellman equations, in a Crandall-Lions-like framework. We allow the second-order derivative in measure to be state-dependent and thus infinite-dimensional, rather than derived from a finite-dimensional operator, hence the term ''fully''. Our argument leverages the construction of smooth approximations from particle systems developed by Cosso, Gozzi, Kharroubi, Pham, and Rosestolato [Trans. Amer. Math. Soc., 2023], and the compactness argument via penalization of measure moments in Soner and Yan [Appl. Math. Optim., 2024]. Our work addresses unbounded dynamics and state-dependent common noise volatility, and to our knowledge, this is the first result of its kind in the literature.