We provide a complete characterisation of the Root solution to the Skorokhod embedding problem (SEP) by means of an optimal stopping formulation. Our methods are purely probabilistic and the analysis relies on a tailored time-reversal argument. This approach allows us to address the long-standing question of a multiple marginals extension of the Root solution of the SEP. Our main result establishes a complete solution to the n-marginal SEP using first hitting times of barrier sets by the time–space process. The barriers are characterised by means of a recursive sequence of optimal stopping problems. Moreover, we prove that our solution enjoys a global optimality property extending the one-marginal Root case. Our results hold for general, one-dimensional, martingale diffusions.

中文翻译：

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多边际嵌入问题的根解：一种最优停止和时间反转方法

我们通过最佳停止公式提供了对 Skorokhod 嵌入问题 (SEP) 的根解的完整表征。我们的方法纯粹是概率性的，分析依赖于量身定制的时间反转参数。这种方法使我们能够解决 SEP 的根解的多重边际扩展这一长期存在的问题。我们的主要结果建立了一个完整的 n-边际 SEP 解决方案，使用时空过程中障碍集的第一次命中时间。障碍的特征在于最优停止问题的递归序列。此外，我们证明我们的解决方案享有扩展单边际根情况的全局最优性。我们的结果适用于一般的、一维的、鞅扩散。