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

摘要

Let $(Phi,Psi)$ be a conjugate pair of Orlicz functions. A set in theOrlicz space $L^Phi$ is said to be order closed if it is closed with respectto dominated convergence of sequences of functions. A well known problemarising from the theory of risk measures in financial mathematics asks whetherorder closedness of a convex set in $L^Phi$ characterizes closedness withrespect to the topology $sigma(L^Phi,L^Psi)$. (See [26, p.3585].) In thispaper, we show that for a norm bounded convex set in $L^Phi$, order closednessand $sigma(L^Phi,L^Psi)$-closedness are indeed equivalent. In general,however, coincidence of order closedness and $sigma(L^Phi,L^Psi)$-closednessof convex sets in $L^Phi$ is equivalent to the validity of the Krein-SmulianTheorem 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 thebounded-$sigma(L^Phi,L^Psi)$ topology. As a result, we show that orderclosedness 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 thereexists a coherent risk measure on $L^Phi$ that has the Fatou property butfails 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 ofOrlicz hearts $(H^Phi,H^Psi)$.