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This paper introduces a new type of second order stochastic backwardHamilton-Jacobi-Bellman (HJB) equations for optimal stochastic control problemswith a currently observable but non-predicable parameter process, in additionto the driving Brownian motion. The main feature of this HJB equation is thatit excludes specifications of the parameter process which dynamics can beunspecified or unknown. This allows to reduce the dimension of the state space.The paper considers the case of control dependent diffusion coefficients andfully nonlinear HJB equations under so-called Cordes conditions.

Stochastic control problems and HJB equations with excluded parameters of random inputs