Stability of the exponential utility maximization problem with respect to preferences

TL;DR

This paper investigates how small changes in an agent's utility function affect the solutions to the exponential utility maximization problem, demonstrating convergence of strategies and prices under various conditions.

Contribution

It provides new stability results for exponential utility maximization under utility variations, extending previous convergence results to broader settings.

Findings

Convergence of value functions, payoffs, and strategies under utility perturbations.

Stability of utility-based pricing with utility variations.

Convergence of optimal strategies for scaled utilities to the exponential hedging strategy.

Abstract

This paper studies stability of the exponential utility maximization when there are small variations on agent's utility function. Two settings are considered. First, in a general semimartingale model where random endowments are present, a sequence of utilities defined on R converges to the exponential utility. Under a uniform condition on their marginal utilities, convergence of value functions, optimal payoffs and optimal investment strategies are obtained, their rate of convergence are also determined. Stability of utility-based pricing is studied as an application. Second, a sequence of utilities defined on R_+ converges to the exponential utility after shifting and scaling. Their associated optimal strategies, after appropriate scaling, converge to the optimal strategy for the exponential hedging problem. This complements Theorem 3.2 in \textit{M. Nutz, Probab. Theory Relat. Fields, 152, 2012}, which establishes the convergence for a sequence of power utilities.