  In this paper, we consider the portfolio optimization problem in a financial
market under a general utility function. Empirical results suggest that if a
significant market fluctuation occurs, invested wealth tends to have a notable
change from its current value. We consider an incomplete stochastic volatility
market model, that is driven by both a Brownian motion and a jump process. At
first, we obtain a closed-form formula for an approximation to the optimal
portfolio in a small-time horizon. This is obtained by finding the associated
Hamilton-Jacobi-Bellman integro-differential equation and then approximating
the value function by constructing appropriate super-solution and sub-solution.
It is shown that the true value function can be obtained by sandwiching the
constructed super-solution and sub-solution. We also prove the accuracy of the
approximation formulas. Finally, we provide a procedure for generating a
close-to-optimal portfolio for a finite time horizon.
