Complete duality for martingale optimal transport on the line

Mathias Beiglböck, Marcel Nutz, Nizar Touzi

Research output: Contribution to journalArticlepeer-review


We study the optimal transport between two probability measures on the real line, where the transport plans are laws of one-step martingales. A quasisure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward (cost) functions: absence of a duality gap and existence of dual optimizers. Both properties are shown to fail in the classical formulation. As a consequence of the duality result, we obtain a general principle of cyclical monotonicity describing the geometry of optimal transports.

Original languageEnglish (US)
Pages (from-to)3038-3074
Number of pages37
JournalAnnals of Probability
Issue number5
StatePublished - Sep 1 2017


  • Kantorovich duality
  • Martingale optimal transport

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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