Characterizing and constructing the solutions to stochastic optimization problems of optimal portfolio choice is a long standing problem. In this talk, I will discuss a new method based on a splitting scheme for the associated Hamilton-Jacobi-Bellman equation in a two-factor stochastic volatility model for the stock price. The scheme converges to a solution of the corresponding PDE, and yields an explicit uniform approximation of the optimal investment strategy. This solution approach offers, among others, insightful observations on how market incompleteness is processed and how it affects the 'infinitesimal' investment preferences. This is joint work with Thaleia Zariphopoulou.