Change of numeraire for weak martingale transport

Change of numeraire is a classical tool in mathematical finance. Campi–Laachir–Martini (Campi et al., 2017) established its applicability to martingale optimal transport. We note that the results of Campi et al. (2017) extend to the case of weak martingale transport. We apply this to shadow couplings (in the sense of Beiglböck and Juillet (2021)), continuous time martingale transport problems in the framework of Huesmann–Trevisan (Huesmann and Trevisan, 2019) and in particular to establish the correspondence of stretched Brownian motion with its geometric counterpart. From a mathematical finance perspective, the geometric (stretched) Brownian motion and the corresponding geometric Bass local volatility model are more natural, and via the change of numeraire transform the efficient and well-understood algorithm for the Bass local volatility model can be adapted to this geometric counterpart.