Sensitivity analysis for expected utility maximization in incomplete Brownian market models

摘要

We examine the issue of sensitivity with respect to model parameters for theproblem of utility maximization from final wealth in an incomplete Samuelsonmodel and mainly, but not exclusively, for utility functions of positivepower-type. The method consists in moving the parameters through change ofmeasure, which we call a weak perturbation, decoupling the usual wealthequation from the varying parameters. By rewriting the maximization problem interms of a convex-analytical support function of a weakly-compact set,crucially leveraging on the work of Backhoff and Fontbona (SIFIN 2016), theprevious formulation let us prove the Hadamard directional differentiability ofthe value function w.r.t. the drift and interest rate parameters, as well asfor volatility matrices under a stability condition on their Kernel, and deriveexplicit expressions for the directional derivatives. We contrast our proposedweak perturbations against what we call strong perturbations, where the wealthequation is directly influenced by the changing parameters. Contrary toconventional wisdom, we find that both points of view generally yield differentsensitivities unless e.g. if initial parameters and their perturbations aredeterministic.