  In this paper, we propose and study the stochastic path-dependent
Hamilton-Jacobi-Bellman (SPHJB) equation that arises naturally from the optimal
stochastic control problem of stochastic differential equations with
path-dependence and measurable randomness. Both the notions of viscosity
solution and classical solution are proposed, and the value function of the
optimal stochastic control problem is proved to be the viscosity solution to
the associated SPHJB equation. A uniqueness result about viscosity solutions is
also given for certain superparabolic cases, while the uniqueness of classical
solution is addressed for general cases. In addition, an
It\^o-Kunita-Wentzell-Krylov formula is proved for the compositions of random
fields and stochastic differential equations in the path-dependent setting.
