The potential of the shadow measure
The potential of the shadow measure
复制标题
影子措施的潜力
DOI：
10.1214/22-ecp457
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发表时间：
2020
影响因子：
0.5
通讯作者：
Dominykas Norgilas
中科院分区：
文献类型：
--
作者：
Mathias Beiglbock;D. Hobson;Dominykas Norgilas
众所周知，给定$\mathbb{R}$上凸序的两个概率测度$\mu$和$\nu$，存在具有这些边缘的离散时间鞅。已知有几种解（例如来自布朗运动中的Skorokhod嵌入问题的文献）。但是，如果我们增加一个要求，即鞅应该最小化其开始和结束位置的某些泛函的期望值，那么问题就变得更加困难。Beiglb\"{o}ck and Juillet（Ann. Probab. 44（2016）42-106）引入了引入鞅耦合族的阴影测度，并解决了一类二元目标函数的最优鞅运输问题。在这篇文章中，我们推广了他们的（存在性和唯一性）结果，提供了一个明确的结构的阴影措施，并作为应用，给出了一个简单的证明其结合性。
It is well known that given two probability measures $\mu$ and $\nu$ on $\mathbb{R}$ in convex order there exists a discrete-time martingale with these marginals. Several solutions are known (for example from the literature on the Skorokhod embedding problem in Brownian motion). But, if we add a requirement that the martingale should minimise the expected value of some functional of its starting and finishing positions then the problem becomes more difficult. Beiglb\"{o}ck and Juillet (Ann. Probab. 44 (2016) 42-106) introduced the shadow measure which induces a family of martingale couplings, and solves the optimal martingale transport problem for a class of bivariate objective functions. In this article we extend their (existence and uniqueness) results by providing an explicit construction of the shadow measure and, as an application, give a simple proof of its associativity.