  In this paper we investigate a dynamic stochastic portfolio optimization
problem involving both the expected terminal utility and intertemporal utility
maximization. We solve the problem by means of a solution to a fully nonlinear
evolutionary Hamilton-Jacobi-Bellman (HJB) equation. We propose the so-called
Riccati method for transformation of the fully nonlinear HJB equation into a
quasi-linear parabolic equation with non-local terms involving the
intertemporal utility function. As a numerical method we propose a
semi-implicit scheme in time based on a finite volume approximation in the
spatial variable. By analyzing an explicit traveling wave solution we show that
the numerical method is of the second experimental order of convergence. As a
practical application we compute optimal strategies for a portfolio investment
problem motivated by market financial data of German DAX 30 Index and show the
effect of considering intertemporal utility on optimal portfolio selection.
