Robust expected utility maximization with medial limits

In this paper we study a robust expected utility maximization problem with random endowment in discrete time. We give conditions under which an optimal strategy exists and derive a dual representation for the optimal utility. Our approach is based on a general representation result for monotone convex functionals, a functional version of Choquet's capacitability theorem and medial limits. The novelty is that it works under nondominated model uncertainty without any assumptions of time-consistency. As applications, we discuss robust utility maximization problems with moment constraints, Wasserstein constraints and Wasserstein penalties.