Risk Measures on Orlicz Hearts

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 Orlicz hearts. 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. We prove that every coherent or convex monetary risk measure on an Orlicz heart which is real-valued on a set with non-empty algebraic interior is real-valued on the whole space and admits a robust representation as maximal penalized expectation with respect to different probability measures. We also show that penalty functions of such risk measures have to satisfy a certain growth condition and that our risk measures are Luxemburg-norm Lipschitz-continuous in the coherent case and locally Luxemburg-norm Lipschitz-continuous in the convex monetary case. 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 robust representations and the maximizing probability measures are given.