Martingale Wasserstein inequality for probability measures in the convex order (Q2136998)

The authors show that two one-dimensional probability measures in the convex order admit a martingale coupling with respect to which the integral of \(|x-y|\) is smaller than twice their \(W_1\)-distance (Wasserstein distance with index \(1\)). It is also shown that replacing \(|x-y|\) and \(W_1\) respectively with \(|x- y|^p\) and \(W_\rho^p\). does not lead to a finite multiplicative constant. The authors show that a finite constant is recovered when replacing \(W_\rho^p\) with the product of \(W_\rho\) times the centered \(\rho\)-th moment of the second marginal to the power \(\rho-1\). Then the authors study the generalisation of this new martingale Wasserstein inequality to higher dimension.