  In this paper, we consider the portfolio optimization problem in a financial
market where the underlying stochastic volatility model is driven by
n-dimensional Brownian motions. At first, we derive a Hamilton-Jacobi-Bellman
equation including the correlations among the standard Brownian motions. We use
an approximation method for the optimization of portfolios. With such
approximation, the value function is analyzed using the first-order terms of
expansion of the utility function in the powers of time to the horizon. The
error of this approximation is controlled using the second-order terms of
expansion of the utility function. It is also shown that the one-dimensional
version of this analysis corresponds to a known result in the literature. We
also generate a close-to-optimal portfolio near the time to horizon using the
first-order approximation of the utility function. It is shown that the error
is controlled by the square of the time to the horizon. Finally, we provide an
approximation scheme to the value function for all times and generate a
close-to-optimal portfolio.
