Open Access
2010 Measurability of optimal transportation and strong coupling of martingale measures
Joaquin Fontbona, Hélène Guérin, Sylvie Méléard
Author Affiliations +
Electron. Commun. Probab. 15: 124-133 (2010). DOI: 10.1214/ECP.v15-1534


We consider the optimal mass transportation problem in $\mathbb{R}^d$ with measurably parameterized marginals under conditions ensuring the existence of a unique optimal transport map. We prove a joint measurability result for this map, with respect to the space variable and to the parameter. The proof needs to establish the measurability of some set-valued mappings, related to the support of the optimal transference plans, which we use to perform a suitable discrete approximation procedure. A motivation is the construction of a strong coupling between orthogonal martingale measures. By this we mean that, given a martingale measure, we construct in the same probability space a second one with a specified covariance measure process. This is done by pushing forward the first martingale measure through a predictable version of the optimal transport map between the covariance measures. This coupling allows us to obtain quantitative estimates in terms of the Wasserstein distance between those covariance measures.


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Joaquin Fontbona. Hélène Guérin. Sylvie Méléard. "Measurability of optimal transportation and strong coupling of martingale measures." Electron. Commun. Probab. 15 124 - 133, 2010.


Accepted: 26 April 2010; Published: 2010
First available in Project Euclid: 6 June 2016

zbMATH: 1232.49052
MathSciNet: MR2643592
Digital Object Identifier: 10.1214/ECP.v15-1534

Primary: 49Q20
Secondary: 60G57

Keywords: coupling between orthogonal martingale measures , measurability of optimal transport , predictable transport process

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