Martingale Wasserstein inequality for probability measures in the convex order

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 $\mathrm{\mid }x-y\mathrm{\mid }$ is smaller than twice their ${\mathcal{W}}_1$-distance (Wasserstein distance with index $1$). We showed that replacing $\vert x-y\vert$and$\mathcal W_1$respectivelywith$\vert x-y\vert^\rho$and$\mathcal W_\rho^\rho$ does not lead to a finite multiplicative constant. We show here that a finite constant is recovered when replacing $\mathcal W_\rho^\rho$withtheproductof$\mathcal W_\rho$timesthecentred$\rho$-th moment of the second marginal to the power $\rho -1$. Then we study the generalisation of this new stability inequality to higher dimension.