Stability of martingale optimal transport and weak optimal transport

Understand

Under mild regularity assumptions, the transport problem is stable in the following sense: if a sequence of optimal transport plans $\pi_1, \pi_2, \ldots$ converges weakly to a transport plan $\pi$, then $\pi$ is also optimal (between its marginals).

• Alfonsi, Corbetta and Jourdain asked whether the same property is true for the martingale transport problem.
• This question seems particularly pressing since martingale transport is motivated by robust finance where data is naturally noisy.
• On a technical level, stability in the martingale case appears more intricate than for classical transport since optimal transport plans $\pi$ are not characterized by a `monotonicity'-property of their support.