Dual Representation of Risk Measures on Orlicz Spaces

The objective of this paper is to present a comprehensive study of dual representations of risk measures and convex functionals defined on an Orlicz space L or an Orlicz heart H. The first part of our study is devoted to the Orlicz pair (L, L). In this setting, we present a thorough analysis of the relationship between order closedness of a convex set C in L and the closedness of C with respect to the topology σ(L, L), culminating in the following surprising result: If (and only if) an Orlicz function Φ and its conjugate Ψ both fail the ∆2-condition, then there exists a coherent risk measure on L Φ that has the Fatou property but does not admit a dual representation via L. This result answers an open problem in the representation theory of risk measures. In the second part of our study, we explore the representation problem for the pair (H, H). This part complements the study for the pair (H, L) investigated in [8] and the study for the pair (L, H) investigated in [19]. This paper contains new results and developments on the interplay between topology and order in Orlicz spaces that are of independent interest.