Law-invariant functionals that collapse to the mean

TL;DR

This paper characterizes when law-invariant convex functionals essentially reduce to the expectation, highlighting conditions under which they behave linearly along certain directions, with implications for pricing and risk measures.

Contribution

It extends existing results by identifying conditions where law-invariant convex functionals collapse to the mean in broader spaces and under milder assumptions.

Findings

Expectation is the unique law-invariant convex functional linear along nonconstant directions.

Results apply to a wide class of spaces of random variables.

Implications for pricing rules and risk measures are discussed.

Abstract

We discuss when law-invariant convex functionals "collapse to the mean". More precisely, we show that, in a large class of spaces of random variables and under mild semicontinuity assumptions, the expectation functional is, up to an affine transformation, the only law-invariant convex functional that is linear along the direction of a nonconstant random variable with nonzero expectation. This extends results obtained in the literature in a bounded setting and under additional assumptions on the functionals. We illustrate the implications of our general results for pricing rules and risk measures.