Change of numeraire for weak martingale transport

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 Lo23 before our work on this topic started. We noticed that a change of numeraire transformation in the spirit of 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 BaLoOb24 on arxiv. org. The article 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.