  We consider an optimal investment and consumption problem for a Black-Scholes
financial market with stochastic coefficients driven by a diffusion process. We
assume that an agent makes consumption and investment decisions based on CRRA
utility functions. The dynamical programming approach leads to an investigation
of the Hamilton Jacobi Bellman (HJB) equation which is a highly non linear
partial differential equation (PDE) of the second oder. By using the Feynman -
Kac representation we prove uniqueness and smoothness of the solution.
Moreover, we study the optimal convergence rate of the iterative numerical
schemes for both the value function and the optimal portfolio. We show, that in
this case, the optimal convergence rate is super geometrical, i.e. is more
rapid than any geometrical one. We apply our results to a stochastic volatility
financial market.
