Open Access
August 2019 Dual attainment for the martingale transport problem
Mathias Beiglböck, Tongseok Lim, Jan Obłój
Bernoulli 25(3): 1640-1658 (August 2019). DOI: 10.3150/17-BEJ1015


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.


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Mathias Beiglböck. Tongseok Lim. Jan Obłój. "Dual attainment for the martingale transport problem." Bernoulli 25 (3) 1640 - 1658, August 2019.


Received: 1 May 2017; Revised: 1 December 2017; Published: August 2019
First available in Project Euclid: 12 June 2019

zbMATH: 07066234
MathSciNet: MR3961225
Digital Object Identifier: 10.3150/17-BEJ1015

Keywords: dual attainment , Kantorovich duality , Martingale optimal transport , robust mathematical finance

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 3 • August 2019
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