  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.
