  The aim of this paper is to study the fast computation of the lower and upper
bounds on the value function for utility maximization under the Heston
stochastic volatility model with general utility functions. It is well known
there is a closed form solution of the HJB equation for power utility due to
its homothetic property. It is not possible to get closed form solution for
general utilities and there is little literature on the numerical scheme to
solve the HJB equation for the Heston model. In this paper we propose an
efficient dual control Monte Carlo method for computing tight lower and upper
bounds of the value function. We identify a particular form of the dual control
which leads to the closed form upper bound for a class of utility functions,
including power, non-HARA and Yarri utilities. Finally, we perform some
numerical tests to see the efficiency, accuracy, and robustness of the method.
The numerical results support strongly our proposed scheme.
