Viscosity Solutions of Stochastic Hamilton--Jacobi--Bellman Equations

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.