Dual attainment for the martingale transport problem

Understand

We investigate existence of dual optimizers in one-dimensional martingale optimal transport problems.

• While [BNT16] established such existence for weak (quasi-sure) duality, [BHP13] 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 $y$.