Monetary risk measures on maximal subspaces of Orlicz classes

Coherent, convex and monetary risk measures were introduced in a setup where uncertain outcomes are modelled by bounded random variables. In this paper, we study such risk measures on maximal subspaces of Orlicz classes. This includes coherent, convex, and monetary risk measures on L-spaces for 1 ≤ p < ∞ and covers a wide range of interesting examples. Moreover, it allows for an elegant duality theory. Our main result shows that every real-valued convex monetary risk measure on such a space admits a representation as maximal penalized expectation with respect to different probability measures. From there we derive Luxemburg-norm-Lipschitz-continuity for coherent risk measures and local Luxemburg-norm-Lipschitz-continuity for convex monetary risk measures. In the second part of the paper we investigate cash-additive hulls of transformed Luxemburg-norms and expected transformed losses. They provide two general classes of coherent and convex monetary risk measures that include many of the currently known examples as special cases. Explicit formulas for their dual representations and the maximizing probability measures are given.