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