  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.
