Closedness of convex sets in Orlicz spaces with applications to dual representation of risk measures

TL;DR

This paper investigates the conditions under which order closedness and topological closedness coincide for convex sets in Orlicz spaces, with implications for dual representations of risk measures in financial mathematics.

Contribution

It establishes the equivalence of order closedness and $\sigma(L^\Phi,L^\Psi)$-closedness for bounded convex sets in Orlicz spaces and links this to the $\Delta_2$-condition, also revealing cases where dual representations fail.

Findings

Order closedness and $\sigma(L^\Phi,L^\Psi)$-closedness are equivalent for norm bounded convex sets.

Coincidence of these closedness notions depends on the $\Delta_2$-condition for $\Phi$ or $\Psi$.

Existence of coherent risk measures with the Fatou property but lacking Fenchel-Moreau dual representation when both $\Phi$ and $\Psi$ fail the $\Delta_2$-condition.

Abstract

Let $(\Phi,\Psi)$ be a conjugate pair of Orlicz functions. A set in the Orlicz space $L^\Phi$ is said to be order closed if it is closed with respect to dominated convergence of sequences of functions. A well known problem arising from the theory of risk measures in financial mathematics asks whether order closedness of a convex set in $L^\Phi$ characterizes closedness with respect to the topology $\sigma(L^\Phi,L^\Psi)$. (See [26, p.3585].) In this paper, we show that for a norm bounded convex set in $L^\Phi$, order closedness and $\sigma(L^\Phi,L^\Psi)$-closedness are indeed equivalent. In general, however, coincidence of order closedness and $\sigma(L^\Phi,L^\Psi)$-closedness of convex sets in $L^\Phi$ is equivalent to the validity of the Krein-Smulian Theorem for the topology $\sigma(L^\Phi,L^\Psi)$; that is, a convex set is $\sigma(L^\Phi,L^\Psi)$-closed if and only if it is closed with respect to the bounded-$\sigma(L^\Phi,L^\Psi)$ topology. As a result, we show that order closedness and $\sigma(L^\Phi,L^\Psi)$-closedness of convex sets in $L^\Phi$ are equivalent if and only if either $\Phi$ or $\Psi$ satisfies the $\Delta_2$-condition. Using this, we prove the surprising result that: \emph{If (and only if) $\Phi$ and $\Psi$ both fail the $\Delta_2$-condition, then there exists a coherent risk measure on $L^\Phi$ that has the Fatou property but fails the Fenchel-Moreau dual representation with respect to the dual pair $(L^\Phi, L^\Psi)$}. A similar analysis is carried out for the dual pair of Orlicz hearts $(H^\Phi,H^\Psi)$.