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March 2021 A solution to the Monge transport problem for Brownian martingales
Nassif Ghoussoub, Young-Heon Kim, Aaron Zeff Palmer
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Ann. Probab. 49(2): 877-907 (March 2021). DOI: 10.1214/20-AOP1462

Abstract

We provide a solution to the problem of optimal transport by Brownian martingales in general dimensions whenever the transport cost satisfies certain subharmonic properties in the target variable as well as a stochastic version of the standard “twist condition” frequently used in deterministic Monge transport theory. This setting includes, in particular, the case of the distance cost c(x,y)=|xy|. We prove existence and uniqueness of the solution and characterize it as the first time Brownian motion hits a barrier that is determined by solutions to a corresponding dual problem.

Acknowledgments

The first two authors are partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Citation

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Nassif Ghoussoub. Young-Heon Kim. Aaron Zeff Palmer. "A solution to the Monge transport problem for Brownian martingales." Ann. Probab. 49 (2) 877 - 907, March 2021. https://doi.org/10.1214/20-AOP1462

Information

Received: 1 July 2019; Revised: 1 April 2020; Published: March 2021
First available in Project Euclid: 17 March 2021

Digital Object Identifier: 10.1214/20-AOP1462

Subjects:
Primary: 49 , 60
Secondary: 52

Keywords: Optimal transport , Skorokhod embedding , variational inequality

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 2 • March 2021
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