Wasserstein distance in terms of the comonotonicity Copula

TL;DR

This paper expresses the p-Wasserstein metric using copulas, especially the comonotonicity copula, and explores bounds for Wasserstein distances between distributions sharing the same copula, extending to multivariate cases.

Contribution

It provides a copula-based formulation of the Wasserstein metric and derives bounds for Wasserstein distances under shared dependence structures, including multivariate cases.

Findings

Wasserstein distance expressed via comonotonicity copula.

Bounds for Wasserstein distances when distributions share the same copula.

Extension of results to multivariate distributions with same dependence structure.

Abstract

The aim of this article is to write the $p$-Wasserstein metric $W_p$ with the $p$-norm, $p\in [1,\infty)$, on $\R^d$ in terms of copula. In particular for the case of one-dimensional distributions, we get that the copula employed to get the optimal coupling of the Wasserstein distances is the comotonicity copula. We obtain the equivalent result also for $d$-dimensional distributions under the sufficient and necessary condition that these have the same dependence structure of their one-dimensional marginals, i.e that the $d$-dimensional distributions share the same copula. Assuming $p\neq q$, $p,q$ $\in [1,\infty)$ and that the probability measures $\mu$ and $\nu$ are sharing the same copula, we also analyze the Wasserstein distance $W_{p,q}$ discussed in \cite{Alfonsi} and get an upper and lower bounds of $W_{p,q}$ in terms of $W_p$, written in terms of comonotonicity copula. We show that as a consequence the lower and upper bound of $W_{p,q}$ can be written in terms of generalized inverse functions.