Robust utility maximization without model compactness

Julio Backhoff, Joaquín Fontbona

Published 2014-05-01, updated 2014-09-09Version 3

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. Relevant examples of such a situation typically arise when the uncertainty set is determined through moment constraints, the treatment of which is beyond the reach of the existing literature. Our approach is based on identifying functional spaces naturally associated with the elements of each problem. For general markets these are modular spaces, through which we can prove a minimax equality and the existence of optimal strategies by exploiting the compactness, which we establish, of the image by the utility function of the set of attainable wealths. This line of argumentation starkly differs from the usual ones in the literature, in particular by not requiring a worst-case measure to exist. In complete markets the relevant space is an Orlicz space, and upon granting its reflexivity under verifiable conditions on the utility function, we obtain additionally the existence of a worst-case measure. Combining our ideas with abstract entropy minimization techniques, we moreover provide in that case a novel methodology to characterize such measure. This is done in terms of the solution to a certain "bi-dual" problem, which can in practical cases be simpler to solve.