Complete duality for martingale optimal transport on the line

Mathias Beiglböck ^[2] ; Marcel Nutz ^[1] ; Nizar Touzi ^[3]
1. [1] Columbia University
  
  Columbia University
  
  Estados Unidos
2. [2] TU Vienna (Austria)
3. [3] Ecole Polytechnique Paris (France)
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Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 45, Nº. 5, 2017, págs. 3038-3074
Idioma: inglés
DOI: 10.1214/16-AOP1131
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Resumen
- We study the optimal transport between two probability measures on the real line, where the transport plans are laws of one-step martingales. A quasi-sure 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.