We study the two-times differentiability of the value functions of the primaland dual optimization problems that appear in the setting of expected utilitymaximization in incomplete markets. We also study the differentiability of thesolutions to these problems with respect to their initial values. We show thatthe key conditions for the results to hold true are that the relative riskaversion coefficient of the utility function is uniformly bounded away fromzero and infinity, and that the prices of traded securities are sigma-boundedunder the num\'{e}raire given by the optimal wealth process.
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