Fatou Property, representations, and extensions of law-invariant risk measures on general Orlicz spaces

TL;DR

This paper extends fundamental representation and extension results for law-invariant risk measures from bounded variables to general Orlicz spaces, broadening their applicability in risk theory.

Contribution

It establishes conditions under which classical risk measure representations and properties hold in general Orlicz spaces, including the Fatou property and Kusuoka's representation.

Findings

Delbaen's representation holds under law-invariance in Orlicz spaces.

Characterization of the Fatou property extends to certain Orlicz spaces.

Kusuoka's representation is generalized to Orlicz spaces.

Abstract

We provide a variety of results for (quasi)convex, law-invariant functionals defined on a general Orlicz space, which extend well-known results in the setting of bounded random variables. First, we show that Delbaen's representation of convex functionals with the Fatou property, which fails in a general Orlicz space, can be always achieved under the assumption of law-invariance. Second, we identify the range of Orlicz spaces where the characterization of the Fatou property in terms of norm lower semicontinuity by Jouini, Schachermayer and Touzi continues to hold. Third, we extend Kusuoka's representation to a general Orlicz space. Finally, we prove a version of the extension result by Filipovi\'{c} and Svindland by replacing norm lower semicontinuity with the (generally non-equivalent) Fatou property. Our results have natural applications to the theory of risk measures.