Dual attainment for the martingale transport problem

Journal article

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↦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,⋅)∈C2−ε, ε>0, where dual attainment fails. Finally, when measures are compactly supported, we show that dual optimizers are Lipschitz if c is Lipschitz.

Authors: Beiglboeck, M; Lim, T; Obloj, J

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Bernoulli Society for Mathematical Statistics and Probability
• Journal: Bernoulli
• Volume: 25
• Issue: 3
• Pages: 1640-1658
• Publication date: 2019-06-12
• Acceptance date: 2017-12-15
• DOI: 10.3150/17-BEJ1015
• EISSN: 1573-9759
• ISSN: 1350-7265
• Keywords: robust mathematical finance; martingale optimal transport; Kantorovich duality; dual attainment