A Viscosity Solution Theory of Stochastic Hamilton-Jacobi-Bellman equations in the Wasserstein Space

This paper is devoted to a viscosity solution theory of the stochastic Hamilton-Jacobi-Bellman equation in the Wasserstein spaces for the mean-field type control problem which allows for random coefficients and may thus be non-Markovian. The value function of the control problem is proven to be the unique viscosity solution. The major challenge lies in the mixture of the lack of local compactness of the Wasserstein spaces and the non-Markovian setting with random coefficients and various techniques are used, including Ito processes parameterized by random measures, the conditional law invariance of the value function, a novel tailor-made compact subset of measure-valued processes, finite dimensional approximations via stochastic n-player differential games with common noises, and so on.