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2022 Instability of martingale optimal transport in dimension d2
Martin Brückerhoff, Nicolas Juillet
Author Affiliations +
Electron. Commun. Probab. 27: 1-10 (2022). DOI: 10.1214/22-ECP463

Abstract

Stability of the value function and the set of minimizers w.r.t. the given data is a desirable feature of optimal transport problems. For the classical Kantorovich transport problem, stability is satisfied under mild assumptions and in general frameworks such as the one of Polish spaces. However, for the martingale transport problem several works based on different strategies established stability results for Ronly. We show that the restriction to dimension d=1 is not accidental by presenting a sequence of marginal distributions on R2 for which the martingale optimal transport problem is neither stable w.r.t. the value nor the set of minimizers. Our construction adapts to any dimension d2. For d2 it also provides a contradiction to the martingale Wasserstein inequality established by Jourdain and Margheriti in d=1.

Funding Statement

MB is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044 –390685587, Mathematics Münster: Dynamics–Geometry–Structure.

Acknowledgments

We thank the anonymous referees for interesting comments and the idea to discuss the impact of irreducible components in Remark 3.2 (iii). Another suggestion of the referees is that an additional condition may restore the stability in dimension 2. We are looking forward for such an evolution in the topic.

Citation

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Martin Brückerhoff. Nicolas Juillet. "Instability of martingale optimal transport in dimension d2." Electron. Commun. Probab. 27 1 - 10, 2022. https://doi.org/10.1214/22-ECP463

Information

Received: 19 March 2021; Accepted: 22 March 2022; Published: 2022
First available in Project Euclid: 12 April 2022

arXiv: 2101.06964
Digital Object Identifier: 10.1214/22-ECP463

Subjects:
Primary: 49Q22 , 60B10 , 60E15 , 60G42

Keywords: Convex order , martingale couplings , Martingale optimal transport , Monge-Kantorovich transport problems , stability

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