STRUCTURE OF OPTIMAL MARTINGALE TRANSPORT PLANS IN GENERAL DIMENSIONS
Ghoussoub, Nassif1; Kim, Young-Heon1,2; Lim, Tongseok3
2019-01
发表期刊ANNALS OF PROBABILITY
ISSN0091-1798
卷号47期号:1页码:109-164
发表状态已发表
DOI10.1214/18-AOP1258
摘要Given two probability measures mu and nu in "convex order" on R-d, we study the profile of one-step martingale plans pi on R-d x R-d that optimize the expected value of the modulus of their increment among all martingales having mu and nu as marginals. While there is a great deal of results for the real line (i.e., when d = 1), much less is known in the richer and more delicate higher-dimensional case that we tackle in this paper. We show that many structural results can be obtained, provided the initial measure mu is absolutely continuous with respect to the Lebesgue measure. One such a property is that mu-almost every x in R-d is transported by the optimal martingale plan into a probability measure pi(x) concentrated on the extreme points of the closed convex hull of its support. This will be established for the distance cost c(x, y) = vertical bar x - y vertical bar in the two-dimensional case, and also for any d >= 3 as long as the marginals are in "subharmonic order." In some cases, pi(x) is supported on the vertices of a k(x)-dimensional polytope, such as when the target measure is discrete. Duality plays a crucial role in our approach, even though, in contrast to standard optimal transports, the dual extremal problem may not be attained in general. We show however that "martingale supporting" Borel subsets of R-d x R-d can be decomposed into a collection of mutually disjoint components by means of a "convex paving" of the source space, in such a way that when the martingale is optimal for a general cost function, each of the components then supports a restricted optimal martingale transport whose dual problem is attained. This decomposition is used to obtain structural results in cases where global duality is not attained. On the other hand, it shows that certain "optimal martingale supporting" Borel sets can be viewed as higher-dimensional versions of Nikodym-type sets. The paper focuses on the distance cost, but much of the results hold for general Lipschitz cost functions.
关键词Optimal transport martingale Choquet boundary duality convex paving
收录类别SCI ; SCIE
语种英语
资助项目European Research Council under European Union[335421]
WOS研究方向Mathematics
WOS类目Statistics & Probability
WOS记录号WOS:000453000100003
出版者INST MATHEMATICAL STATISTICS
WOS关键词EXISTENCE ; BOUNDS
原始文献类型Article
引用统计
文献类型期刊论文
条目标识符https://kms.shanghaitech.edu.cn/handle/2MSLDSTB/29949
专题数学科学研究所_PI研究组(P)_Tongseok Lim组
数学科学研究所
通讯作者Ghoussoub, Nassif
作者单位1.Univ British Columbia, Dept Math, Vancouver, BC V6T 1Z2, Canada
2.Korea Inst Adv Study, Ctr Math Challenges, 85 Hoegiro, Seoul, South Korea
3.ShanghaiTech univ, Inst Math Sci, 393 Middle Huaxia Rd, Shanghai 201210, Peoples R China
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Ghoussoub, Nassif,Kim, Young-Heon,Lim, Tongseok. STRUCTURE OF OPTIMAL MARTINGALE TRANSPORT PLANS IN GENERAL DIMENSIONS[J]. ANNALS OF PROBABILITY,2019,47(1):109-164.
APA Ghoussoub, Nassif,Kim, Young-Heon,&Lim, Tongseok.(2019).STRUCTURE OF OPTIMAL MARTINGALE TRANSPORT PLANS IN GENERAL DIMENSIONS.ANNALS OF PROBABILITY,47(1),109-164.
MLA Ghoussoub, Nassif,et al."STRUCTURE OF OPTIMAL MARTINGALE TRANSPORT PLANS IN GENERAL DIMENSIONS".ANNALS OF PROBABILITY 47.1(2019):109-164.
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