Automatic Continuities of Law-Invariant Risk Measures

Abstract

<p>In this thesis, we investigate the automatic continuity properties of law-invariant risk measures on general model spaces. In Chapter 2, we study automatic order lower semi-continuity of law-invariant risk measures, which is usually termed as the Fatou property in the literature. Let X be a rearrangement-invariant space other than L∞ over a non-atomic probability space. We show that every real-valued, law-invariant, coherent risk measure automatically has the Fatou property at every random variable X ∈ X whose negative tails have vanishing norm (i.e., limn∥X1{X≤−n}∥ = 0) if and only if X satisfies the Almost Order Continuous Equidistributional Average (AOCEA) property, namely, d(CL(X), Xa) = 0 for any nonnegative random variable X ∈ X , where CL(X) is the convex hull of all random variables having the same distribution as X and Xa = { X ∈ X : limn∥X1|X|≥n∥ = 0 } . We also show that the AOCEA property is satisfied by most classical model spaces, including Orlicz spaces. In Chapter 3, we first show that on an r.i. space with the AOCEA property, every real-valued, law-invariant, coherent risk measure is automatically σ(X , X ′ )-lower semicontinuity at every random variable X ∈ X whose negative tails have vanishing norm. Here X ′ is the associated space of X . We also recover a local version of the Fenchel-Moreau Duality and apply it to establish automatic dual representations of risk measures. Finally, in Chapter 4, we apply our results to study when law-invariant bounded linear functionals automatically collapse to the mean, i.e., being scalar multiples of the expectation. We show that on every r.i. space with the AOCEA property, a bounded law-invariant linear functional collapses to the mean. We also construct an r.i. space on which a bounded law-invariant linear functional may fail to collapse to the mean and thus the space fails the AOCEA property. The thesis is based on [10, 11].</p>