Lebesgue property for convex risk measures on Orlicz spaces

Abstract

We present a robust representation theorem for monetary convex risk measures \({\rho : \mathcal{X} \rightarrow \mathbb{R}}\) such that $$\lim_n\rho(X_n) = \rho(X)\,{\rm whenever}\,(X_n)\,{\rm almost\,surely\,converges\,to}\,X,$$ \({|Xn| \leq Z \in \mathcal{X}, {\rm for\,all}\,n \in \mathbb{N}}\) and \({\mathcal{X}}\) is an arbitrary Orlicz space. The separable \({\langle\mathbb{L}^{1}, \mathbb{L}^{\infty}\rangle}\) case of Jouini et al. (Adv Math Econ 9:49–71 2006), as well as the non-separable version of Delbaen [7], are contained as a particular case here. We answer a natural question posed by Biagini and Fritelli [2]. Our approach is based on the study for unbounded sets, as the epigraph of a given penalty function associated with ρ, of the celebrated weak compactness Theorem due to James (Isr J Math 13:289–300 1972).