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

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

We study continuity properties of law-invariant (quasi-)convex functions f : L∞( ,F,P) → (−∞,∞] over a non-atomic probability space ( ,F,P). This is a supplementary note to Jouini et al. (Adv Math Econ 9:49–71, 2006).

Subgradients of Law-Invariant Convex Risk Measures on L

We introduce a generalised subgradient for law-invariant closed convex risk measures on L and establish its relationship with optimal risk allocations and equilibria. Our main result gives sufficient conditions ensuring a non-empty generalised subgradient.

Law invariant risk measures on L∞(Rd)

Kusuoka (2001) has obtained explicit representation theorems for comonotone risk measures and, more generally, for law invariant risk measures. These theorems pertain, like most of the previous literature, to the case of scalar-valued risks. Jouini-Meddeb-Touzi (2004) and Burgert-Rüschendorf (2006) extended the notion of risk measures to the vector-valued case. Recently Ekeland-Galichon-Henry (...

Subgradients of Law-Invariant Convex Risk Measures on L1∗

We introduce a generalised subgradient for law-invariant closed convex risk measures on L and establish its relationship with optimal risk allocations and equilibria. Our main result gives sufficient conditions ensuring a non-empty generalised subgradient.

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