影子措施的潜力
The potential of the shadow measure
摘要：
众所周知，给定$\mathbb{R}$上凸序的两个概率测度$μ$和$ν$，存在具有这些边缘的离散鞅.已知几种解（例如来自关于布朗运动中的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。Beiglböck 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。
英文摘要：
It is well known that given two probability measures $μ$ and $ν$ 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. Beiglböck 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.
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