Abstract:A classical result of Strassen asserts that given probabilities $\mu , \nu$ on the real line which are in convex order, there exists a martingale coupling with these marginals, i.e. a random vector $(X_1,X_2)$ such that $X_1\sim \mu , X_2\sim \nu$ and $\mathbb {E}[X_2|X_1]=X_1$. Remarkably, it is a non-trivial problem to construct particular solutions to this problem. Based on the concept of shadow for measures in convex order, we introduce a family of such martingale couplings, each of which admits several characterizations in terms of optimality properties/geometry of the support set/representation through a Skorokhod embedding. As a particular element of this family we recover the (left-)curtain martingale transport, which has recently been studied (see Beiglböck, Henry-Labordère, and Touzi [Stochastic Process. Appl. 127 (2017), pp. 3005–3013]; Beiglböck and Juillet [Ann. Probab. 44 (2016), pp. 42–106]; Campi, Laachir, and Martini [Finance Stoch. 21 (2017), pp. 471–486; Henry-Labordère and Touzi [Finance Stoch. 20 (2016), pp. 635–668]) and which can be viewed as a martingale analogue of the classical monotone rearrangement. As another canonical element of this family we identify a martingale coupling that resembles the usual product coupling and appears as an optimizer in the general transport problem recently introduced by Gozlan et al. In addition, this coupling provides an explicit example of a Lipschitz kernel, shedding new light on Kellerer’s proof of the existence of Markov martingales with specified marginals.

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中文翻译：

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阴影耦合

摘要：Strassen 的一个经典结果断言，给定概率 $\mu , \nu$ 在凸序的实线上，存在与这些边际的鞅耦合，即随机向量 $(X_1,X_2)$ 使得$X_1\sim \mu 、X_2\sim \nu$ 和 $\mathbb {E}[X_2|X_1]=X_1$。值得注意的是，为这个问题构建特定的解决方案是一个不平凡的问题。基于阴影的概念对于凸序的度量，我们引入了一系列这样的鞅耦合，每个耦合都通过 Skorokhod 嵌入在支持集/表示的最优属性/几何方面承认几个特征。作为该家族的一个特殊元素，我们恢复了最近研究过的（左）幕鞅传输（参见 Beiglböck、Henry-Labordère 和 Touzi [Stochastic Process. Appl. 127 (2017), pp. 3005–3013]） ; Beiglböck 和 Juillet [Ann. Probab. 44 (2016), pp. 42–106]; Campi, Laachir, and Martini [Finance Stoch. 21 (2017), pp. 471–486; Henry-Labordère and Touzi [Finance Stoch] . 20 (2016), pp. 635–668]) 并且可以将其视为经典单调重排的鞅类似物。作为这个家族的另一个典型元素，我们确定了一种类似于通常的鞅耦合产品耦合并在 Gozlan 等人最近引入的一般传输问题中作为优化器出现。此外，这种耦合提供了一个明确的 Lipschitz 核示例，为 Kellerer 证明存在具有指定边际的马尔可夫鞅提供了新的线索。

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