The left-curtain martingale coupling in the presence of atoms

TL;DR

This paper extends the left-curtain martingale coupling to arbitrary initial laws, characterizing it with lower and upper functions, and applies it to derive model-independent bounds for American put options, generalizing previous atom-free results.

Contribution

It introduces a generalized construction of the left-curtain martingale coupling for any starting law, expanding its applicability and theoretical understanding.

Findings

Generalized the left-curtain martingale coupling to arbitrary laws.

Characterized the coupling using lower and upper functions.

Derived model-independent bounds for American put options.

Abstract

Beiglb\"ock and Juillet ("On a problem of optimal transport under marginal martingale constraints") 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 ("Robust bounds for the American Put") on the atom-free case.