Law-invariant functionals that collapse to the mean: Beyond convexity

Abstract

We establish general "collapse to the mean" principles that provide conditions under which a law-invariant functional reduces to an expectation. In the convex setting, we retrieve and sharpen known results from the literature. However, our results also apply beyond the convex setting. We illustrate this by providing a complete account of the "collapse to the mean" for quasiconvex functionals. In the special cases of consistent risk measures and Choquet integrals, we can even dispense with quasiconvexity. In addition, we relate the "collapse to the mean" to the study of solutions of a broad class of optimisation problems with law-invariant objectives that appear in mathematical finance, insurance, and economics. We show that the corresponding quantile formulations studied in the literature are sometimes illegitimate and require further analysis.

A Framework for Measures of Risk under Uncertainty

“…For instance, additivity of the core may be sensible in our framework (Theorem 3) and it nicely connects to the scenario-based margin calculation used by CME. However, such a property is not desirable for traditional risk measures, as it essentially forces the risk measure to collapse to the mean; see e.g., Liebrich and Munari (2021) and Chen et al (2021) .…”

Risk sharing under heterogeneous beliefs without convexity

“…In the theory of risk measures, these go back at least to [23]. We refer to the more detailed discussion in [30].…”

Do law-invariant linear functionals collapse to the mean?

“…The functionals considered in [3,11] are assumed to be σ(X , X ) lower semicontinuous. In general, weak continuities are rather restrictive assumptions to impose.…”

“…In the past few years, law-invariant risk measures have been of intense research interest in Financial Mathematics; see, e.g., [2,3,5,7,8,10,12,13,14,15,17,19,20,21,22]. Of particular interest to us are two recent papers [3,11] that investigate when a law-invariant risk measure collapses to the mean, i.e., being a scalar multiple of expectation. We are motivated to study the problem for linear functionals.…”

“…We are motivated to study the problem for linear functionals. Precisely, we investigate whether law-invariant bounded linear functionals on rearrangementinvariant spaces automatically collapse to the mean, without the rather restrictive assumption imposed in [3,11] of lower semicontinuity with respect to the weak topology generated by the associated space (i.e., order continuous dual). For linear functionals, this assumption is strictly stronger than norm boundedness, except on spaces with order continuous norm, and together with law-invariance, it implies collapsing to the mean in a straightforward way (see Lemma 2.1 below).…”