## The Annals of Probability

### Complete duality for martingale optimal transport on the line

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 45, Number 5 (2017), 3038-3074.

Dates
Revised: May 2016
First available in Project Euclid: 23 September 2017

https://projecteuclid.org/euclid.aop/1506132033

Digital Object Identifier
doi:10.1214/16-AOP1131

Mathematical Reviews number (MathSciNet)
MR3706738

Zentralblatt MATH identifier
06812200

#### Citation

Beiglböck, Mathias; Nutz, Marcel; Touzi, Nizar. Complete duality for martingale optimal transport on the line. Ann. Probab. 45 (2017), no. 5, 3038--3074. doi:10.1214/16-AOP1131. https://projecteuclid.org/euclid.aop/1506132033

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