  In this paper we study the fully nonlinear stochastic Hamilton-Jacobi-Bellman
(HJB) equation for the optimal stochastic control problem of stochastic
differential equations with random coefficients. The notion of viscosity
solution is introduced, and we prove that the value function of the optimal
stochastic control problem is the maximal viscosity solution of the associated
stochastic HJB equation. For the superparabolic cases when the diffusion
coefficients are deterministic functions of time, states and controls, the
uniqueness is addressed as well.
