We provide a variety of results for (quasi)convex, law-invariant functionals defined on a general Orlicz space, which extend well-known results in the setting of bounded random variables. First, we show that Delbaen's dual characterization of the Fatou property, which no longer holds in a general Orlicz space, continues to hold under the assumption of law-invariance. Second, we identify the range of Orlicz spaces where the characterization of the Fatou property in terms of norm lower semicontinuity by Jouini, Schachermayer and Touzi still holds . Third, we extend Kusuoka's dual representation to a general Orlicz space. Finally, we prove a version of the extension result by Filipovi\'{c} and Svindland by replacing norm lower semicontinuity with the (generally non-equivalent) Fatou property. Our results have natural applications to the theory of risk measures.
We provide a variety of results for (quasi)convex, law-invariant functionals defined on a general Orlicz space, which extend well-known results in the setting of bounded random variables. First, we show that Delbaen's representation of convex functionals with the Fatou property, which fails in a general Orlicz space, can be always achieved under the assumption of law-invariance. Second, we identify the range of Orlicz spaces where the characterization of the Fatou property in terms of norm lower semicontinuity by Jouini, Schachermayer and Touzi continues to hold . Third, we extend Kusuoka's representation to a general Orlicz space. Finally, we prove a version of the extension result by Filipovi\'{c} and Svindland by replacing norm lower semicontinuity with the (generally non-equivalent) Fatou property. Our results have natural applications to the theory of risk measures.
R
dual characterization of
representation of convex functionals with
R
no longer holds
fails
R
continues to hold
can be always achieved
R
still holds
continues to hold
D
dual