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

Dual control Monte Carlo method for tight bounds of value function under Heston stochastic volatility model