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 . 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
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
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