Change of numeraire for weak martingale transport

TL;DR

This paper extends the classical change of numeraire technique to weak martingale transport, applying it to shadow couplings and continuous-time martingale problems, and establishing a link with stretched Brownian motion.

Contribution

It generalizes the change of numeraire to weak martingale transport and connects it with geometric stretched Brownian motion, providing a new perspective in the framework.

Findings

Extension of change of numeraire to weak martingale transport.

Application to shadow couplings and continuous-time martingale problems.

Establishment of the correspondence between stretched Brownian motion and its geometric counterpart.

Abstract

Change of numeraire is a classical tool in mathematical finance. Campi-Laachir-Martini established its applicability to martingale optimal transport. We note that the results of Campi-Laachir-Martini extend to the case of weak martingale transport. We apply this to shadow couplings, continuous time martingale transport problems in the framework of Huesmann-Trevisan and in particular to establish the correspondence between stretched Brownian motion with its geometric counterpart. Note: We emphasize that we learned about the geometric stretched Brownian motion gSBM (defined in PDE terms) in a presentation of Loeper \cite{Lo23} before our work on this topic started. We noticed that a change of numeraire transformation in the spirit of \cite{CaLaMa14} allows for an alternative viewpoint in the weak optimal transport framework. We make our work public following the publication of Backhoff-Loeper-Obloj's work \cite{BaLoOb24} on arxiv.org. The article \cite{BaLoOb24} derives gSBM using PDE techniques as well as through an independent probabilistic approach which is close to the one we give in the present article.