Sensitivity analysis for expected utility maximization in incomplete brownian market models

We examine the issue of sensitivity with respect to model parameters for the problem of utility maximization from final wealth in an incomplete Samuelson model and mainly for utility functions of power-type. The method consists in moving the parameters through change of measure, which we call a weak perturbation, in particular decoupling the usual wealth equation from the varying parameters. By rewriting the maximization problem in terms of a convex-analytical support function of a weakly-compact set, crucially leveraging on the recent work by Backhoff and Fontbona arXiv:1405.0251, the previous formulation let us prove the Hadamard directional differentiability of the value function w.r.t. the drift and interest rate parameters, as well as for volatility matrices under a stability condition on their Kernel, and derive explicit expressions for the directional derivatives. We contrast our proposed weak perturbations against what we call strong perturbations, whereby the wealth equation is directly influenced by the changing parameters, and find that both points of view generally yield different sensitivities unless e.g. if initial parameters and their perturbations are deterministic.