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

Abstract

This paper introduces a new type of second order stochastic backward Hamilton-Jacobi-Bellman (HJB) equations for optimal stochastic control problems with a currently observable but non-predicable parameter process, in addition to the driving Brownian motion. The main feature of this HJB equation is that it excludes specifications of the parameter process which dynamics can be unspecified or unknown. This allows to reduce the dimension of the state space. The paper considers the case of control dependent diffusion coefficients and fully nonlinear HJB equations under so-called Cordes conditions.