Separability vs. robustness of Orlicz spaces: financial and economic perspectives

TL;DR

This paper explores the structure and properties of robust Orlicz spaces, highlighting their implications for robust finance, especially regarding separability, dominatedness, and the limitations of options under uncertainty.

Contribution

It introduces two constructions of robust Orlicz spaces, analyzes their separability and order properties, and discusses their significance in robust financial modeling.

Findings

Robust Orlicz spaces' separability implies strong dominatedness conditions.

Spaces considered are lattice isomorphic to sublattices of classical L^1 but lack order completeness.

Options have limited topological spanning power under nondominated uncertainty.

Abstract

We investigate robust Orlicz spaces as a generalisation of robust $L^p$-spaces. Two constructions of such spaces are distinguished, a top-down approach and a bottom-up approach. We show that separability of robust Orlicz spaces or their subspaces has very strong implications in terms of the dominatedness of the set of priors and the lack of order completeness. Our results have subtle implications for the field of robust finance. For instance, norm closures of bounded continuous functions with respect to the worst-case $L^p$-norm, as considered in the $G$-framework, lead to spaces which are lattice isomorphic to a sublattice of a classical $L^1$-space lacking, however, any form of order completeness. We further show that the topological spanning power of options is always limited under nondominated uncertainty.