MARTINGALE TRANSPORTS AND MONGE MAPS

• 成果类型: Article
• 署名作者: Nutz, Marcel; Wang, Ruodu; Zhang, Zhenyuan
• 署名单位: Columbia University; University of Waterloo; Stanford University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/24-AAP2099
• 发表日期: 2024
• 页码: 5556-5577
• 关键词: duality; brenier; bounds

摘要

It is well known that martingale transport plans between marginals mu not equal nu are never given by Monge maps-with the understanding that the map is over the first marginal mu, or forward in time. Here, we change the perspective, with surprising results. We show that any distributions mu, nu in convex order with nu atomless admit a martingale coupling given by a Monge map over the second marginal nu. Namely, we construct a particular coupling called the barcode tingale transports are dense in the set of all martingale couplings, paralleling the classical denseness result for Monge transports in the Kantorovich formulation of optimal transport. Various properties and applications are presented, including a refined version of Strassen's theorem and a mimicking theorem where the marginals of a given martingale are reproduced by a backward deterministic martingale, a remarkable type of process whose current state encodes its whole history.