当前位置: X-MOL 学术Probab Theory Relat Fields › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
The geometry of multi-marginal Skorokhod Embedding
Probability Theory and Related Fields ( IF 2 ) Pub Date : 2019-08-01 , DOI: 10.1007/s00440-019-00935-z
Mathias Beiglböck 1 , Alexander M G Cox 2 , Martin Huesmann 1
Affiliation  

The Skorokhod Embedding Problem is one of the classical problems in the theory of stochastic processes, with applications in many different fields [cf. the surveys (Hobson in: Paris-Princeton lectures on mathematical finance 2010, Volume 2003 of Lecture Notes in Mathematics, Springer, Berlin, 2011 ; Obłój in: Probab Surv 1:321–390, 2004 )]. Many of these applications have natural multi-marginal extensions leading to the (optimal) multi-marginal Skorokhod problem . Some of the first papers to consider this problem are Brown et al. (Probab Theory Relat Fields 119(4):558–578, 2001 ), Hobson (Séminaire de Probabilités, XXXII, Volume 1686 of Lecture Notes in Mathematics, Springer, Berlin, 1998 ), Madan and Yor (Bernoulli 8(4):509–536, 2002 ). However, this turns out to be difficult using existing techniques: only recently a complete solution was be obtained in Cox et al. (Probab Theory Relat Fields 173:211–259, 2018 ) establishing an extension of the Root construction, while other instances are only partially answered or remain wide open. In this paper, we extend the theory developed in Beiglböck et al. (Invent Math 208(2):327–400, 2017 ) to the multi-marginal setup which is comparable to the extension of the optimal transport problem to the multi-marginal optimal transport problem. As for the one-marginal case, this viewpoint turns out to be very powerful. In particular, we are able to show that all classical optimal embeddings have natural multi-marginal counterparts. Notably these different constructions are linked through a joint geometric structure and the classical solutions are recovered as particular cases. Moreover, our results also have consequences for the study of the martingale transport problem as well as the peacock problem.

中文翻译:

多边缘 Skorokhod 嵌入的几何

Skorokhod 嵌入问题是随机过程理论中的经典问题之一,在许多不同领域都有应用[cf. 调查(霍布森在:2010 年巴黎-普林斯顿数学金融讲座,数学讲义第 2003 卷,施普林格,柏林,2011 年;Obłój 在:Probab Surv 1:321–390, 2004)]。其中许多应用程序具有自然的多边际扩展,从而导致(最佳)多边际 Skorokhod 问题。一些最早考虑这个问题的论文是 Brown 等人。(Probab Theory Relat Fields 119(4):558–578, 2001), Hobson (Séminaire de Probabilités, XXXII, Volume 1686 of Lecture Notes in Mathematics, Springer, Berlin, 1998), Madan and Yor (Bernoulli 8(4): 509–536, 2002). 然而,事实证明这很难使用现有技术:直到最近,Cox 等人才获得了完整的解决方案。(Probab Theory Relat Fields 173:211–259, 2018) 建立根结构的扩展,而其他实例仅得到部分回答或保持开放。在本文中,我们扩展了 Beiglböck 等人提出的理论。(Invent Math 208(2):327–400, 2017) 到多边际设置,这相当于将最优运输问题扩展到多边际最优运输问题。对于单边情况,这个观点被证明是非常有说服力的。特别是,我们能够证明所有经典的最优嵌入都有自然的多边际对应物。值得注意的是,这些不同的结构通过一个联合几何结构联系起来,经典的解决方案被恢复为特殊情况。而且,
更新日期:2019-08-01
down
wechat
bug