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The geometry of multi-marginal Skorokhod Embedding

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  • Mathias Beiglboeck
  • Alexander Cox
  • Martin Huesmann

Abstract

The Skorokhod Embedding Problem (SEP) is one of the classical problems in the study of stochastic processes, with applications in many different fields (cf.~ the surveys \cite{Ob04,Ho11}). Many of these applications have natural multi-marginal extensions leading to the \emph{(optimal) multi-marginal Skorokhod problem} (MSEP). Some of the first papers to consider this problem are \cite{Ho98b, BrHoRo01b, MaYo02}. However, this turns out to be difficult using existing techniques: only recently a complete solution was be obtained in \cite{CoObTo15} establishing an extension of the Root construction, while other instances are only partially answered or remain wide open. In this paper, we extend the theory developed in \cite{BeCoHu14} to the multi-marginal setup which is comparable to the extension of the optimal transport problem to the multi-marginal optimal transport problem. As for the one-marginal case, this viewpoint turns out to be very powerful. In particular, we are able to show that all classical optimal embeddings have natural multi-marginal counterparts. Notably these different constructions are linked through a joint geometric structure and the classical solutions are recovered as particular cases. Moreover, our results also have consequences for the study of the martingale transport problem as well as the peacock problem.

Suggested Citation

  • Mathias Beiglboeck & Alexander Cox & Martin Huesmann, 2017. "The geometry of multi-marginal Skorokhod Embedding," Papers 1705.09505, arXiv.org.
    • Handle: RePEc:arx:papers:1705.09505
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    References listed on IDEAS

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    Cited by:

    1. Nutz, Marcel & Stebegg, Florian & Tan, Xiaowei, 2020. "Multiperiod martingale transport," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1568-1615.
    2. Mathias Beiglbock & Marcel Nutz & Florian Stebegg, 2019. "Fine Properties of the Optimal Skorokhod Embedding Problem," Papers 1903.03887, arXiv.org, revised Apr 2020.
    3. Erhan Bayraktar & Thomas Bernhardt, 2020. "On the Continuity of the Root Barrier," Papers 2010.14695, arXiv.org, revised Jul 2021.

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