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Multiperiod martingale transport

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  • Nutz, Marcel
  • Stebegg, Florian
  • Tan, Xiaowei

Abstract

Consider a multiperiod optimal transport problem where distributions μ0,…,μn are prescribed and a transport corresponds to a scalar martingale X with marginals Xt∼μt. We introduce particular couplings called left-monotone transports; they are characterized equivalently by a no-crossing property of their support, as simultaneous optimizers for a class of bivariate transport cost functions with a Spence–Mirrlees property, and by an order-theoretic minimality property. Left-monotone transports are unique if μ0 is atomless, but not in general. In the one-period case n=1, these transports reduce to the Left-Curtain coupling of Beiglböck and Juillet. In the multiperiod case, the bivariate marginals for dates (0,t) are of Left-Curtain type, if and only if μ0,…,μn have a specific order property. The general analysis of the transport problem also gives rise to a strong duality result and a description of its polar sets. Finally, we study a variant where the intermediate marginals μ1,…,μn−1 are not prescribed.

Suggested Citation

  • Nutz, Marcel & Stebegg, Florian & Tan, Xiaowei, 2020. "Multiperiod martingale transport," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1568-1615.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:3:p:1568-1615
    DOI: 10.1016/j.spa.2019.05.010
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    References listed on IDEAS

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    Cited by:

    1. Zhengqing Zhou & Jose Blanchet & Peter W. Glynn, 2021. "Distributionally Robust Martingale Optimal Transport," Papers 2106.07191, arXiv.org, revised Nov 2021.
    2. Erhan Bayraktar & Shuoqing Deng & Dominykas Norgilas, 2023. "Supermartingale Brenier’s Theorem with Full-Marginal Constraint," World Scientific Book Chapters, in: Robert A Jarrow & Dilip B Madan (ed.), Peter Carr Gedenkschrift Research Advances in Mathematical Finance, chapter 17, pages 569-636, World Scientific Publishing Co. Pte. Ltd..

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