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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 |
ISSN | 0091-1798 |
卷号 | 47期号:1页码:109-164 |
发表状态 | 已发表 |
DOI | 10.1214/18-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 |
推荐引用方式 GB/T 7714 | 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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