Sensitivity analysis of the utility maximization problem with respect to model perturbations

We study the sensitivity of the expected utility maximization problem in a continuous semi-martingale market with respect to small changes in the market price of risk. Assuming that the preferences of a rational economic agent are modeled with a general utility function, we obtain a second-order expansion of the value function, a first-order approximation of the terminal wealth, and construct trading strategies that match the indirect utility function up to the second order. If a risk-tolerance wealth process exists, using it as a num\'eraire and under an appropriate change of measure, we reduce the approximation problem to a Kunita-Watanabe decomposition.