A remark on the optimal transport between two probability measures sharing the same copula

Aurélien Alfonsi (1, 2) , Benjamin Jourdain (2, 1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MATHRISK - Mathematical Risk handling

We are interested in the Wasserstein distance between two probability measures on $\R^n$ sharing the same copula $C$. The image of the probability measure $dC$ by the vectors of pseudo-inverses of marginal distributions is a natural generalization of the coupling known to be optimal in dimension $n=1$. It turns out that for cost functions $c(x,y)$ equal to the $p$-th power of the $L^q$ norm of $x-y$ in $\R^n$, this coupling is optimal only when $p=q$ i.e. when $c(x,y)$ may be decomposed as the sum of coordinate-wise costs.

Domains: Probability [math.PR]

Aurélien Alfonsi, Benjamin Jourdain. A remark on the optimal transport between two probability measures sharing the same copula. Statistics and Probability Letters, 2014, dx.doi.org/10.1016/j.spl.2013.09.035. ⟨hal-00844906⟩