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In this paper, we consider the portfolio optimization problem in a financialmarket where the underlying stochastic volatility model is driven byn-dimensional Brownian motions. At first, we derive a Hamilton-Jacobi-Bellmanequation including the correlations among the standard Brownian motions. We usean approximation method for the optimization of portfolios. With suchapproximation, the value function is analyzed using the first-order terms ofexpansion of the utility function in the powers of time to the horizon. Theerror of this approximation is controlled using the second-order terms ofexpansion of the utility function. It is also shown that the one-dimensionalversion of this analysis corresponds to a known result in the literature. Wealso generate a close-to-optimal portfolio near the time to horizon using thefirst-order approximation of the utility function. It is shown that the erroris controlled by the square of the time to the horizon. Finally, we provide anapproximation scheme to the value function for all times and generate aclose-to-optimal portfolio.

Analysis of optimal portfolio on finite and small-time horizons for a stochastic volatility model with multiple correlated assets