Robust Utility Maximization without Model Compactness

We formulate conditions for the solvability of the problem of robust utility maximization from final wealth in continuous time financial markets, without assuming weak compactness of the densities of the uncertainty set, as customarily done in the literature. Relevant examples of such a situation typically arise when the uncertainty set is determined through moment constraints. Our approach is based on identifying functional spaces naturally associated with the elements of each problem. For general markets these are convex modular spaces, which we can use to prove a minimax equality and the existence of optimal strategies, by establishing and exploiting the compactness of the image through the utility function of the set of attainable wealths. In complete markets we additionally obtain the existence of a worst-case measure, and, combining our ideas with abstract entropy minimization techniques, we moreover provide a novel and general methodology to characterize it.