Wasserstein distance in terms of the comonotonicity copula

The aim of this article is to write the p-Wasserstein metric 𝑊𝑝 with the p-norm, 𝑝∈[1,∞), on ℝ𝑑 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 𝑊𝑝,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 𝑊𝑝,𝑞 in terms of 𝑊𝑝, written in terms of comonotonicity copula. We show that as a consequence the lower and upper bound of 𝑊𝑝,𝑞 can be written in terms of generalized inverse functions.