  We consider a semilinear equation linked to the finite horizon consumption -
investment problem under the stochastic factor framework and we prove it admits
a classical solution and provide all obligatory estimates to successfully apply
a verification reasoning. The paper covers the standard time additive utility,
as well as the recursive utility framework. We extend existing results by
considering more general factor dynamics including a non-trivial diffusion part
and a stochastic correlation between assets and factors. In addition, this is
the first paper which compromises many other optimization problems in finance,
for example those related to the indifference pricing or the quadratic hedging
problem. The extension of the result to the stochastic differential utility and
robust portfolio optimization is provided as well. The essence of our paper
lays in using improved stochastic methods to prove gradient estimates for
suitable HJB equations with restricted control space.
