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^\Phi or an Orlicz heart H^\Phi. The first part of our study is devoted to the Orlicz pair (L^\Phi,L^\Psi). In this setting, we present a thorough analysis of the relationship between order closedness of a convex set \mathcal{C ] in L^\Phi and the closedness of \mathcal{C topology \sigma(L^\Phi,L^\Psi) , culminating in the following surprising result:If (and only if) an Orlicz function \Phi and its conjugate \Psi both fail the \Delta_2-condition, then there exists a coherent risk measure on L^\Phi that has the Fatou property but does not admit a dual representation via L^{\Psi . 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^\Phi,H^\Psi) . This part complements the study for the pair (H ^\Phi,L^\Psi) investigated in 8%DIFDELCMD < ] %%% and the study for the pair (L^{\Phi,H^{\Psi})investigated in }%DIFDELCMD < [%%% 19 . This paper contains new results and developments on the interplay between topology and order in Orlicz spaces that are of independent interest\emph{  .
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) %DIFDELCMD < ] %%% ,H^{\Psi})investigated in }%DIFDELCMD < [%%% -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) .
R
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^\Phi or an Orlicz heart H^\Phi. The first part of our study is devoted to the Orlicz pair
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
R
In this setting, we present a thorough analysis of the relationship between order closedness of a convex set \mathcal{C
(See
A
26, p.3585
A
.) 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
R
and the closedness of \mathcal{C
is equivalent to the validity of the Krein-Smulian Theorem for the
D
, culminating in the following surprising result:
D
If (and only if) an Orlicz function \Phi and its conjugate \Psi both fail the \Delta_2-condition, then there exists a coherent risk measure on L^\Phi that has the Fatou property but does not admit a dual representation via L^{\Psi
R
. 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^\Phi,H^\Psi) . This part complements the study for the pair (H
; 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
D
investigated in
D
8
D
and the study for the pair (L^{\Phi
D
19
R
. This paper contains new results and developments on the interplay between topology and order in Orlicz spaces that are of independent interest
-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:
A
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
. A similar analysis is carried out for the dual pair of Orlicz hearts (H^\Phi,H^\Psi)