Wasserstein distance in terms of the comonotonicity copula

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The aim of this article is to write the p-Wasserstein metric W p with the p-norm, p ∈ [ 1 , ∞ ) , 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 ≠ q , p,q ∈ [ 1 , ∞ ) and that the probability measures µ and ν are sharing the same copula, we also analyze the Wasserstein distance W p , q discussed in [Alfonsi and Jourdain. A remark on the optimal transport between two probability measures sharing the same copula. Statist. Probab. Lett. 84 (2014) 131–134.] 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.