A remark on smooth solutions to a stochastic control problem with a power terminal cost function and stochastic volatilities

Incomplete financial markets are considered, defined by a multi-dimensional non-homogeneous diffusion process, being the direct sum of an It\^{o} process (the price process), and another non-homogeneous diffusion process (the exogenous process, representing exogenous stochastic sources). The drift and the diffusion matrix of the price process are functions of the time, the price process itself and the exogenous process. In the context of such markets and for power utility functions, it is proved that the stochastic control problem consisting of optimizing the expected utility of the terminal wealth, has a classical solution (i.e. $C^{1,2}$). This result paves the way to a study of the optimal portfolio problem in incomplete forward variance stochastic volatility models, along the lines of Ref: Ekeland et al.