MARTINGALE BENAMOU-BRENIER: A PROBABILISTIC PERSPECTIVE

• 成果类型: Article
• 署名作者: Backhoff-Veraguas, Julio; Beiglboeck, Mathias; Huesmann, Martin; Kallblad, Sigrid
• 署名单位: University of Twente; University of Vienna; Royal Institute of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1422
• 发表日期: 2020
• 页码: 2258-2289
• 关键词: optimal transport; DISCRETE-TIME; Duality; DECOMPOSITION; rearrangement; MONOTONICITY; arbitrage; MARGINALS; geometry

摘要

In classical optimal transport, the contributions of Benamou-Brenier and McCann regarding the time-dependent version of the problem are corner-stones of the field and form the basis for a variety of applications in other mathematical areas. We suggest a Benamou-Brenier type formulation of the martingale transport problem for given d-dimensional distributions mu, nu in convex order. The unique solution M* = (M-t*)(t is an element of[0,1]) of this problem turns out to be a Markov-martingale which has several notable properties: In a specific sense it mimics the movement of a Brownian particle as closely as possible subject to the conditions M-0*similar to mu, M-1*similar to nu. Similar to McCann's displacement-interpolation, M* provides a time-consistent interpolation between mu and nu. For particular choices of the initial and terminal law, M* recovers archetypical martingales such as Brownian motion, geometric Brownian motion, and the Bass martingale. Furthermore, it yields a natural approximation to the local vol model and a new approach to Kellerer's theorem. This article is parallel to the work of Huesmann-Trevisan, who consider a related class of problems from a PDE-oriented perspective.