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

Let $(\mathrm{\Phi },\mathrm{\Psi })$ be a conjugate pair of Orlicz functions. A set in the Orlicz space $L^{\mathrm{\Phi }}$ 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 $L^{\mathrm{\Phi }}$ characterizes closedness with respect to the topology $\sigma (L^{\mathrm{\Phi }},L^{\mathrm{\Psi }})$. (See [26, p.3585].) In this paper, we show that for a norm bounded convex set in $L^{\mathrm{\Phi }}$, order closedness and $\sigma (L^{\mathrm{\Phi }},L^{\mathrm{\Psi }})$-closedness are indeed equivalent. In general, however, coincidence of order closedness and $\sigma (L^{\mathrm{\Phi }},L^{\mathrm{\Psi }})$-closedness of convex sets in $L^{\mathrm{\Phi }}$ is equivalent to the validity of the Krein-Smulian Theorem for the topology $\sigma (L^{\mathrm{\Phi }},L^{\mathrm{\Psi }})$; that is, a convex set is $\sigma (L^{\mathrm{\Phi }},L^{\mathrm{\Psi }})$-closed if and only if it is closed with respect to the bounded-$\sigma (L^{\mathrm{\Phi }},L^{\mathrm{\Psi }})$ topology. As a result, we show that order closedness and $\sigma (L^{\mathrm{\Phi }},L^{\mathrm{\Psi }})$-closedness of convex sets in $L^{\mathrm{\Phi }}$ are equivalent if and only if either $\mathrm{\Phi }$ or $\mathrm{\Psi }$ satisfies the ${\mathrm{\Delta }}_2$-condition. Using this, we prove the surprising result that: \emph{If (and only if) $\mathrm{\Phi }$ and $\mathrm{\Psi }$ both fail the ${\mathrm{\Delta }}_2$-condition, then there exists a coherent risk measure on $L^{\mathrm{\Phi }}$ that has the Fatou property but fails the Fenchel-Moreau dual representation with respect to the dual pair $(L^{\mathrm{\Phi }},L^{\mathrm{\Psi }})$}. A similar analysis is carried out for the dual pair of Orlicz hearts $(H^{\mathrm{\Phi }},H^{\mathrm{\Psi }})$.