Irreducible convex paving for decomposition of multidimensional martingale transport plans

Hadrien De March, Nizar Touzi

Research output: Contribution to journalArticlepeer-review


Martingale transport plans on the line are known from Beiglböck and Juillet (Ann. Probab. 44 (2016) 42-106) to have an irreducible decomposition on a (at most) countable union of intervals. We provide an extension of this decomposition for martingale transport plans in ℝ d , d ≥ 1. Our decomposition is a partition of ℝ d consisting of a possibly uncountable family of relatively open convex components, with the required measurability so that the disintegration is well defined. We justify the relevance of our decomposition by proving the existence of a martingale transport plan filling these components. We also deduce from this decomposition a characterization of the structure of polar sets with respect to all martingale transport plans.

Original languageEnglish (US)
Pages (from-to)1726-1774
Number of pages49
JournalAnnals of Probability
Issue number3
StatePublished - May 1 2019


  • Irreducible decomposition
  • Martingale optimal transport
  • Polar sets

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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