There are many transformations mapping martingales into martingales,
e.g.\ $M_t\mapsto M_t^2-\langle M\rangle_t$. However, looking at
solutions to the martingale optimal transport problem it turns out that
there are only very few transformations that map optimal martingale
couplings into optimal martingale couplings.
After a short introduction to and review of the state of the art of
martingale optimal transport we will give a characterization of all
these monotonicity preserving transformations. In particular, these
transformation reveal certain symmetries between different solutions to
the martingale optimal transport problem. Moreover, we will show that
the transport approach to Skorokhod embedding is a powerful tool to
study martingale optimal transport and, in turn, that the new symmetries
for martingale optimal transport disclose symmetries for solutions to
the Skorokhod embedding problem. (based on joint work with Florian Stebegg)http://cvgmt.sns.it/seminar/538/
