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In this paper, we propose and study the stochastic path-dependentHamilton-Jacobi-Bellman (SPHJB) equation that arises naturally from the optimalstochastic control problem of stochastic differential equations withpath-dependence and measurable randomness. Both the notions of viscositysolution and classical solution are proposed, and the value function of theoptimal stochastic control problem is proved to be the viscosity solution tothe associated SPHJB equation. A uniqueness result about viscosity solutions isalso given for certain superparabolic cases, while the uniqueness of classicalsolution is addressed for general cases. In addition, anIt\^o-Kunita-Wentzell-Krylov formula is proved for the compositions of randomfields and stochastic differential equations in the path-dependent setting.

Stochastic Path-Dependent Hamilton-Jacobi-Bellman Equations and Controlled Stochastic Differential Equations with Random Path-Dependent Coefficients