Martingale Wasserstein inequality for probability measures in the convex order

Abstract

It was shown by the authors that two one-dimensional probability measures in the convex order admit a martingale coupling with respect to which the integral of $|\mathit{x}-\mathit{y}|$ is smaller than twice their ${\mathcal{W}}_1$-distance (Wasserstein distance with index 1). We showed that replacing $|\mathit{x}-\mathit{y}|$ and ${\mathcal{W}}_1$ respectively with $|\mathit{x}-\mathit{y}{|}^{\mathit{\rho }}$ and ${\mathcal{W}}_{\mathit{\rho }}^{\mathit{\rho }}$ does not lead to a finite multiplicative constant. We show here that a finite constant is recovered when replacing ${\mathcal{W}}_{\mathit{\rho }}^{\mathit{\rho }}$ with the product of ${\mathcal{W}}_{\mathit{\rho }}$ times the centred ρ-th moment of the second marginal to the power $\mathit{\rho }-1$. Then we study the generalisation of this new martingale Wasserstein inequality to higher dimension.