Exponential utility maximization under model uncertainty for unbounded endowments

TL;DR

This paper addresses robust exponential utility maximization under model uncertainty in discrete time, establishing existence of optimal strategies, dual representations, and convergence to superhedging prices, applicable to unbounded endowments.

Contribution

It provides a comprehensive framework for utility maximization under nondominated models, including existence results, duality, dynamic programming, and asymptotic analysis.

Findings

Optimal strategies exist for all measurable endowments.

Dual representation involves calibrated martingale measures.

Utility maximization value converges to superhedging price as risk aversion increases.

Abstract

We consider the robust exponential utility maximization problem in discrete time: An investor maximizes the worst case expected exponential utility with respect to a family of nondominated probabilistic models of her endowment by dynamically investing in a financial market, and statically in available options. We show that, for any measurable random endowment (regardless of whether the problem is finite or not) an optimal strategy exists, a dual representation in terms of (calibrated) martingale measures holds true, and that the problem satisfies the dynamic programming principle (in case of no options). Further it is shown that the value of the utility maximization problem converges to the robust superhedging price as the risk aversion parameter gets large, and examples of nondominated probabilistic models are discussed.