Optimal transport and Skorokhod embedding
Optimal transport and Skorokhod embedding
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DOI：
10.1007/s00222-016-0692-2
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发表时间：
2017-05-01
影响因子：
3.1
通讯作者：
Huesmann, Martin
中科院分区：
文献类型：
--
作者：
Beiglboeck, Mathias;Cox, Alexander M. G.;Huesmann, Martin
Skorokhod嵌入问题是将给定的概率表示为布朗运动在选定停止时间的分布。在过去的50年里，这已经成为概率论中重要的经典问题之一，许多作者已经构造了具有特殊最优性的解。这些构造采用了从偏移理论到势和偏微分理论的各种技术，并已用于纯概率和应用概率的许多不同分支。我们基于最优质量运输的思想和概念开发了一种新的Skorokhod嵌入方法。与Gangbo和McCann关于最优运输几何的著名文章类似，我们建立了具有期望最优性的Skorokhod嵌入的几何表征。这就产生了一种构造最优嵌入的系统方法。它允许我们第一次推导出所有已知的最优Skorokhod嵌入作为一个统一结构的特殊情况，并导致各种新的嵌入。虽然以前的构造通常使用布朗运动的特定性质，但我们的方法适用于所有足够规则的马尔可夫过程。
The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties. These constructions employ a variety of techniques ranging from excursion theory to potential and PDE theory and have been used in many different branches of pure and applied probability. We develop a new approach to Skorokhod embedding based on ideas and concepts from optimal mass transport. In analogy to the celebrated article of Gangbo and McCann on the geometry of optimal transport, we establish a geometric characterization of Skorokhod embeddings with desired optimality properties. This leads to a systematic method to construct optimal embeddings. It allows us, for the first time, to derive all known optimal Skorokhod embeddings as special cases of one unified construction and leads to a variety of new embeddings. While previous constructions typically used particular properties of Brownian motion, our approach applies to all sufficiently regular Markov processes.