Automatic Fatou Property of Law-invariant Risk Measures

Abstract

In this paper, we show that, on classical model spaces including Orlicz spaces, every real-valued, law-invariant, coherent risk measure automatically has the Fatou property at every point whose negative part has a thin tail.

Do law-invariant linear functionals collapse to the mean?

“…The following property was shown by the authors in [6] to play a pivotal role in automatic continuity of law-invariant convex functionals: d CL(X), X a = 0 for any X ∈ X + , where CL(X) = co({Y : Y ∼ X}). (3.1) It was termed there as the Almost Order Continuous Equidistributional Average (AO-CEA) property.…”

“…An alternative proof of this proposition in the spirit of [6] is included in the appendix. It was also proved in [6] that when X = L ∞ , (3.1) is equivalent to the following condition:…”

“…Remark 3.5. It was shown in [6] that the AOCEA property (3.1) is satisfied by nearly all classical r.i. spaces other than L ∞ , including Lebesgue spaces L p (1 ≤ p < ∞), Lorentz spaces L p,q (1 < p < ∞, 1 ≤ q ≤ ∞) and Orlicz spaces. Proposition 3.2 gives another class of r.i. spaces with the AOCEA property: Let X be an r.i. space over a non-atomic probability space other than L ∞ .…”