ON A PROBLEM OF OPTIMAL TRANSPORT UNDER MARGINAL MARTINGALE CONSTRAINTS
成果类型：
Article
署名作者：
Beiglboeck, Mathias; Juillet, Nicolas
署名单位：
University of Vienna; Universites de Strasbourg Etablissements Associes; Universite de Strasbourg; Centre National de la Recherche Scientifique (CNRS); Universites de Strasbourg Etablissements Associes; Universite de Strasbourg
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/14-AOP966
发表日期：
2016
页码：
42-106
关键词：
摘要：
The basic problem of optimal transportation consists in minimizing the expected costs E[c(X-1, X-2)] by varying the joint distribution (X-1, X-2) where the marginal distributions of the random variables X-1 and X-2 are fixed. Inspired by recent applications in mathematical finance and connections with the peacock problem, we study this problem under the additional condition that (X-i)(i=1,2) is a martingale, that is, E[X-2 vertical bar X-1] = X-1. We establish a variational principle for this problem which enables us to determine optimal martingale transport plans for specific cost functions. In particular, we identify a martingale coupling that resembles the classic monotone quantile coupling in several respects. In analogy with the celebrated theorem of Brenier, the following behavior can be observed: If the initial distribution is continuous, then this monotone martingale is supported by the graphs of two functions T-1, T-2 : R -> R.