Robust Orlicz spaces: observations and caveats

In this paper, we investigate two different constructions of robust Orlicz spaces as a generalisation of robust $L^p$-spaces. We show that a construction as norm closures of bounded continuous functions typically leads to spaces which are lattice-isomorphic to sublattices of a classical $L^1$-space, thus leading to dominated classes of contingent claims even for nondominated classes of probability measures. We further show that the mathematically very desirable property of $\sigma$ -Dedekind completeness for norm closures of continuous functions ususally aready implies that the considered class of probability measures is dominated. Our second construction, which is top-down, is based on the consideration of the maximal domain of a worst-case Luxemburg norm. From an applied persepective, this approach can be justified by a uniform-boundedness-type result showing that, in typical situations, the worst-case Orlicz space agrees with the intersection of the corresponding individual Orlicz spaces.