## Bernoulli

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

### Dual attainment for the martingale transport problem

Mathias Beiglböck, Tongseok Lim, and Jan Obłój

#### 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**

Received: May 2017

Revised: December 2017

First available in Project Euclid: 12 June 2019

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.3150/17-BEJ1015

**Mathematical Reviews number (MathSciNet)**

MR3961225

**Zentralblatt MATH identifier**

07066234

**Keywords**

dual attainment Kantorovich duality martingale optimal transport robust mathematical finance

#### 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