Wasserstein distance in terms of the Comonotonicity Copula

TL;DR

This paper expresses the Wasserstein metric of order p for probability measures on  in terms of the comonotonicity copula, connecting classical results with copula theory and extending previous findings.

Contribution

It explicitly combines copula theory with Wasserstein metrics to represent the Wasserstein distance using the comonotonicity copula, extending known results to higher dimensions.

Findings

Revisits classical 1D Wasserstein results in terms of copulas.

Provides explicit representations of Wasserstein metrics using comonotonicity copula.

Discusses restrictions and extensions to higher dimensions.

Abstract

In this article, we represent the Wasserstein metric of order $p$, where $p\in [1,\infty)$, in terms of the comonotonicity copula, for the case of probability measures on $\R^d$, by revisiting existing results. In 1973, Vallender established the link between the $1$-Wasserstein metric and the corresponding distribution functions for $d=1$. In 1956 Giorgio dall'Aglio showed that the p-Wasserstein metric in $d=1$ could be written in terms of the comonotonicity copula $M$ without being aware of the concept of copulas or Wasserstein metrics. In this article, for the proofs we explicitly combine tools from copula theory and Wasserstein metrics. The extension to general $d\in\N$ has some restriction, as discussed e.g. in \cite{Alfonsi} and \cite{BDS}. Some of the results of \cite{Alfonsi}, \cite{BDS} and \cite{RR} are revisited here in a more explicit form in terms of the comonotonicity copula.