Dual attainment for the martingale transport problem

TL;DR

This paper studies the existence of dual optimizers in one-dimensional martingale optimal transport problems, identifying conditions under which dual maximizers exist or fail, with implications for regularity and cost functions.

Contribution

It establishes the existence of dual maximizers under convexity conditions on the cost function and explores regularity and failure cases, advancing understanding of duality in martingale transport.

Findings

Dual maximizers exist when $y\mapsto c(x,y)$ is convex.

Existence holds for twice continuously differentiable costs with compactly supported marginals.

Dual optimizers are Lipschitz if the cost function is Lipschitz.

Abstract

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 $y$. Further, this may not be improved as we give examples with $c(x,\cdot)\in C^{2-\epsilon}$, $\epsilon >0$, where dual attainment fails. Finally, when measures are compactly supported, we show that dual optimizers are Lipschitz if $c$ is Lipschitz.