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2020 A new family of one dimensional martingale couplings
B. Jourdain, W. Margheriti
Electron. J. Probab. 25: 1-50 (2020). DOI: 10.1214/20-EJP543


In this paper, we exhibit a new family of martingale couplings between two one-dimensional probability measures $\mu $ and $\nu $ in the convex order. This family is parametrised by two dimensional probability measures on the unit square with respective marginal densities proportional to the positive and negative parts of the difference between the quantile functions of $\mu $ and $\nu $. It contains the inverse transform martingale coupling which is explicit in terms of the quantile functions of these marginal densities. The integral of $|x-y|$ with respect to each of these couplings is smaller than twice the $\mathcal {W}_{1}$ distance between $\mu $ and $\nu $. When the comonotonous coupling between $\mu $ and $\nu $ is given by a map $T$, the elements of the family minimise $\int _{\mathbb {R}}\vert y-T(x)\vert \,M(dx,dy)$ among all martingale couplings between $\mu $ and $\nu $. When $\mu $ and $\nu $ are in the decreasing (resp. increasing) convex order, the construction is generalised to exhibit super (resp. sub) martingale couplings.


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B. Jourdain. W. Margheriti. "A new family of one dimensional martingale couplings." Electron. J. Probab. 25 1 - 50, 2020.


Received: 7 March 2019; Accepted: 22 October 2020; Published: 2020
First available in Project Euclid: 6 November 2020

Digital Object Identifier: 10.1214/20-EJP543

Primary: 60E15 , 60G42 , 91G80

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


Vol.25 • 2020
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