Beiglböck and Juillet (Ann. Probab. 44 (2016) 42–106) introduced the left-curtain martingale coupling of probability measures $\mu$ and $\nu$, and proved that, when the initial law $\mu$ is continuous, it is supported by the graphs of two functions. We extend the later result by constructing the generalised left-curtain martingale coupling and show that for an arbitrary starting law $\mu$ it is characterised by two appropriately defined lower and upper functions.
As an application of this result, we derive the model-independent upper bound of an American put option. This extends recent results of Hobson and Norgilas (2017) on the atom-free case.
"The left-curtain martingale coupling in the presence of atoms." Ann. Appl. Probab. 29 (3) 1904 - 1928, June 2019. https://doi.org/10.1214/18-AAP1450