The left-curtain martingale coupling in the presence of atoms

Beiglböck and Juillet (Ann. Probab. 44 (2016) 42.106) introduced the left-curtain martingale coupling of probability measures . and ., and proved that, when the initial law .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 . 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.