L\'evy copulas: a probabilistic point of view

TL;DR

This paper establishes a probabilistic framework linking Lévy copulas with proper copulas, enabling new parametric constructions and insights into their structure through max-id distributions and Williamson transforms.

Contribution

It introduces a one-to-one correspondence between Lévy copulas and proper copulas, providing a new probabilistic perspective and methods for defining parametric Lévy copulas as mirror images of proper copulas.

Findings

Lévy copulas correspond uniquely to proper copulas via max-id distributions.

Parametric Lévy copulas can be constructed as mirror images of proper parametric copulas.

Archimedean Lévy copulas are generated by Williamson transforms of transformed radial distribution functions.

Abstract

There is a one-to-one correspondence between L\'{e}vy copulas and proper copulas. The correspondence relies on a relationship between L\'{e}vy copulas sitting on $[0,+\infty]^d$ and max-id distributions. The max-id distributions are defined with respect to a partial order that is compatible with the inclusion of sets bounded away from the origin. An important consequence of the result is the possibility to define parametric L\'{e}vy copulas as mirror images of proper parametric copulas. For example, proper Archimedean copulas are generated by functions that are Williamson $d-$transforms of the cdf of the radial component of random vectors with exchangeable distributions $F_{R}$. In contrast, the generators of Archimedean L\'{e}vy copulas are Williamson $d-$transforms of $-\log(1-F_{R})$.