COMPLETE DUALITY FOR MARTINGALE OPTIMAL TRANSPORT ON THE LINE

• 成果类型: Article
• 署名作者: Beiglbock, Mathias; Nutz, Marcel; Touzi, Nizar
• 署名单位: Technische Universitat Wien; Columbia University; Institut Polytechnique de Paris; ENSTA Paris; Ecole Polytechnique
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1131
• 发表日期: 2017
• 页码: 3038-3074
• 关键词: Discrete-time; constraints; MARGINALS; Couplings; arbitrage; options; THEOREM; version; models; bounds

摘要

We study the optimal transport between two probability measures on the real line, where the transport plans are laws of one-step martingales. A quasi sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward (cost) functions: absence of a duality gap and existence of dual optimizers. Both properties are shown to fail in the classical formulation. As a consequence of the duality result, we obtain a general principle of cyclical monotonicity describing the geometry of optimal transports.