Dual attainment for the martingale transport problem

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 (sic) 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, center dot) epsilon C2-epsilon, epsilon > 0, where dual attainment fails. Finally, when measures are compactly supported, we show that dual optimizers are Lipschitz if c is Lipschitz.