On the extension of the Namioka-Klee theorem and on the Fatou property for Risk Measures

This paper has been motivated by general considerations on the topic of Risk Measures, which essentially are convex monotone maps de…ned on spaces of random variables, possibly with the so-called Fatou property. We show …rst that the celebrated Namioka-Klee theorem for linear, positive functionals holds also for convex monotone maps on Frechet lattices. It is well-known among the specialists that the Fatou property for risk measures on L1 enables a simpli…ed dual representation, via probability measures only. The Fatou property in a general framework of lattices is nothing but the lower order semicontinuity property for . Our second goal is thus to prove that a similar simpli…ed dual representation holds also for order lower semicontinuous, convex and monotone functionals de…ned on more general spaces X (locally convex Frechet lattices). To this end, we identify a link between the topology and the order structure in X the C-property that enables the simpli…ed representation. One main application of these results leads to the study of convex risk measures de…ned on Orlicz spaces and of their dual representation. Acknowledgements The …rst author would like to thank B. Rudlo¤, P. Cheridito and A. Hamel for some discussions while she was visiting the ORFE Department at Princeton University. The second author would like to thank Marco Maggis, PhD student at Milano University for helpful discussion on this subject.