May 2022 Martingale Wasserstein inequality for probability measures in the convex order
Benjamin Jourdain, William Margheriti
Author Affiliations +
Bernoulli 28(2): 830-858 (May 2022). DOI: 10.3150/21-BEJ1368


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 |xy| is smaller than twice their W1-distance (Wasserstein distance with index 1). We showed that replacing |xy| and W1 respectively with |xy|ρ and Wρρ does not lead to a finite multiplicative constant. We show here that a finite constant is recovered when replacing Wρρ with the product of Wρ times the centred ρ-th moment of the second marginal to the power ρ1. Then we study the generalisation of this new martingale Wasserstein inequality to higher dimension.

Funding Statement

This research benefited from the support of the “Chaire Risques Financiers”, Fondation du Risque.


We thank Nicolas Juillet for his remarks on a preliminary version of this paper and Nizar Touzi for his suggestion to investigate the martingale Wasserstein inequality for radial probability measures which lead to Proposition 4. We also thank the referees for their remarks that helped us to improve the paper.


Download Citation

Benjamin Jourdain. William Margheriti. "Martingale Wasserstein inequality for probability measures in the convex order." Bernoulli 28 (2) 830 - 858, May 2022.


Received: 1 November 2020; Revised: 1 May 2021; Published: May 2022
First available in Project Euclid: 3 March 2022

MathSciNet: MR4388921
zbMATH: 1510.60017
Digital Object Identifier: 10.3150/21-BEJ1368

Keywords: Convex order , martingale couplings , Martingale optimal transport , Wasserstein distance

Rights: Copyright © 2022 ISI/BS


This article is only available to subscribers.
It is not available for individual sale.

Vol.28 • No. 2 • May 2022
Back to Top