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 $|x-y|$ is smaller than twice their ${\mathcal{W}}_1$-distance (Wasserstein distance with index $1$). We showed that replacing $|x-y|$ and ${\mathcal{W}}_1$ respectively with $|x-y{|}^{\rho }$ and ${\mathcal{W}}_{\rho }^{\rho }$ does not lead to a finite multiplicative constant. We show here that a finite constant is recovered when replacing ${\mathcal{W}}_{\rho }^{\rho }$ with the product of ${\mathcal{W}}_{\rho }$ times the centred $\rho$-th moment of the second marginal to the power $\rho -1$. Then we study the generalisation of this new stability inequality to higher dimension.