RISK MEASURES ON ORLICZ HEARTS

TLDR

Risk measures such as coherent, convex, and monetary were originally defined for bounded random variables. This work investigates these risk measures within Orlicz hearts. The authors analyze risk measures on Lp‑spaces and study cash‑additive hulls of transformed Luxemburg norms and expected transformed losses. They show that any coherent or convex monetary risk measure on an Orlicz heart that is real‑valued on a set with non‑empty algebraic interior is real‑valued on the whole space, admits a robust representation as a maximal penalized expectation, satisfies a growth condition on its penalty function, and is Lipschitz‑continuous in the Luxemburg norm, while also providing explicit robust representations and maximizing measures for two general classes of coherent and convex monetary risk measures.

Abstract

Coherent, convex, and monetary risk measures were introduced in a setup where uncertain outcomes are modeled 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 p ‐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.