Law-invariant functionals that collapse to the mean
Mathematical Finance
2021-01-21 v2
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.
Keywords
Cite
@article{arxiv.2009.04144,
title = {Law-invariant functionals that collapse to the mean},
author = {Fabio Bellini and Pablo Koch-Medina and Cosimo Munari and Gregor Svindland},
journal= {arXiv preprint arXiv:2009.04144},
year = {2021}
}