A SOLUTION TO THE MONGE TRANSPORT PROBLEM FOR BROWNIAN MARTINGALES

• 成果类型: Article
• 署名作者: Ghoussoub, Nassif; Kim, Young-Heon; Palmer, Aaron Zeff
• 署名单位: University of British Columbia
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1462
• 发表日期: 2021
• 页码: 877-907
• 关键词: viscosity solutions; MONOTONICITY; EXISTENCE; EQUATIONS; MAPS

摘要

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) = | x - y|. 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.