## Bernoulli

• Bernoulli
• Volume 25, Number 3 (2019), 1640-1658.

### Dual attainment for the martingale transport problem

#### Abstract

We investigate existence of dual optimizers in one-dimensional martingale optimal transport problems. While [Ann. Probab. 45 (2017) 3038–3074] established such existence for weak (quasi-sure) duality, [Finance Stoch. 17 (2013) 477–501] showed existence for the natural stronger (pointwise) duality may fail even in regular cases. We establish that (pointwise) dual maximizers exist when $y\mapsto c(x,y)$ is convex, or equivalent to a convex function. It follows that when marginals are compactly supported, the existence holds when the cost $c(x,y)$ is twice continuously differentiable in $y$. Further, this may not be improved as we give examples with $c(x,\cdot)\in C^{2-\varepsilon}$, $\varepsilon>0$, where dual attainment fails. Finally, when measures are compactly supported, we show that dual optimizers are Lipschitz if $c$ is Lipschitz.

#### Article information

Source
Bernoulli, Volume 25, Number 3 (2019), 1640-1658.

Dates
Revised: December 2017
First available in Project Euclid: 12 June 2019

https://projecteuclid.org/euclid.bj/1560326422

Digital Object Identifier
doi:10.3150/17-BEJ1015

Mathematical Reviews number (MathSciNet)
MR3961225

Zentralblatt MATH identifier
07066234

#### Citation

Beiglböck, Mathias; Lim, Tongseok; Obłój, Jan. Dual attainment for the martingale transport problem. Bernoulli 25 (2019), no. 3, 1640--1658. doi:10.3150/17-BEJ1015. https://projecteuclid.org/euclid.bj/1560326422

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