Wasserstein distance in terms of the comonotonicity Copula

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 $μ$ and $ν$ 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.

Theorems & Definitions (26)

• Theorem 1.1: Existence of optimal coupling, Villani
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Sklar's theorem, Nelsen
• Theorem 2.4: Fréchet-Hoeffding bounds in $d$-dimensions, Nelsen
• Theorem 2.5: Equivalent conditions comonotonicity, DDGKV2001
• Corollary 2.6
• Theorem 3.1: Wasserstein distance in terms of copula in $\mathbb{R}$
• proof : Proof of Theorem \ref{['main theorem']}, $p=1$