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

We consider the expected utility maximisation problem and its response to small changes in the market price of risk in a continuous semimartingale setting. Assuming that the preferences of a rational economic agent are modelled by a general utility function, we obtain a second-order expansion of the value function, a first-order approximation of the terminal wealth, and we construct trading strategies that match the indirect utility function up to the second order. The method, which is presented in an abstract version, relies on a simultaneous expansion with respect to both the state variable and the parameter, and convex duality in the direction of the state variable only (as there is no convexity with respect to the parameter). If a risk-tolerance wealth process exists, using it as numéraire and under an appropriate change of measure, we reduce the approximation problem to a Kunita–Watanabe decomposition.