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We consider a utility maximization problem for an investment-consumptionportfolio when the current utility depends also on the wealth process. Suchkind of problems arise, e.g., in portfolio optimization with random horizon orwith random trading times. To overcome the difficulties of the problem we usethe dual approach. We define a dual problem and treat it by means of dynamicprogramming, showing that the viscosity solutions of the associatedHamilton-Jacobi-Bellman equation belong to a suitable class of smoothfunctions. This allows to define a smooth solution of the primalHamilton-Jacobi-Bellman equation, proving that this solution is indeed uniquein a suitable class and coincides with the value function of the primalproblem. Some financial applications of the results are provided.

Utility maximization with current utility on the wealth: regularity of solutions to the HJB equation