Viscosity Solutions of Fully Second-Order HJB Equations in the Wasserstein Space

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, and hence we use the term “fully.” Our argument leverages the construction of smooth approximations from particle systems developed by Cosso et al. [ Trans. Amer. Math. Soc., 377 (2024), pp. 31–83] and the compactness argument via penalization of measure moments in Soner and Yan [ Appl. Math. Optim., 90 (2024), 1]. 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.