Continuity properties of law - invariant ( quasi - ) convex risk functions on L ∞

Law - invariant risk measures : extension properties and qualitative robustness

We characterize when a convex risk measure associated to a law-invariant acceptance set in L can be extended to L, 1�p<∞, preserving finiteness and continuity. This problem is strongly connected to the statistical robustness of the corresponding risk measures. Special attention is paid to concrete examples including risk measures based on expected utility, max-correlation risk measures, and dis...

Some Properties of Certain Subclasses of Close-to-Convex and Quasi-convex Functions with Respect to 2k-Symmetric Conjugate Points

Optimal capital and risk allocations for law- and cash-invariant convex functions

In this paper we provide the complete solution to the existence and characterisation problem of optimal capital and risk allocations for not necessarily monotone, law-invariant convex risk measures on the model space L, for any p ∈ [1,∞]. Our main result says that the capital and risk allocation problem always admits a solution via contracts whose payoffs are defined as increasing Lipschitz con...

Convexity and Geodesic Metric Spaces

In this paper, we first present a preliminary study on metric segments and geodesics in metric spaces. Then we recall the concept of d-convexity of sets and functions in the sense of Menger and study some properties of d-convex sets and d-convex functions as well as extreme points and faces of d-convex sets in normed spaces. Finally we study the continuity of d-convex functions in geodesic metr...

Convex Risk Measures Beyond Bounded Risks

This work addresses three main issues: Firstly, we study the interplay of risk measures on L∞ and Lp, for p ≥ 1. Our main result is a one-to-one correspondence between law-invariant closed convex risk measures on L∞ and L1. This proves that the canonical model space for the predominant class of law-invariant convex risk measures is L1. Secondly, we provide the solution to the existence and char...