Before revision:  The existence and uniqueness theorem for the Hamilton-Jacobi-Bellman (HJB) equation is shown. We construct an iterative sequence of functions converging to the solutionof this equation. An optimal convergence rate for this sequence is found and sharp computable upper bounds for the approximation accuracy of the optimal consumption - investment strategies are obtained.
After revision:  The existence and uniqueness theorem for 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 for this sequence is found and sharp computable upper bounds for the approximation accuracy of the optimal consumption - investment strategies are obtained.