Geometric Martingale Benamou-Brenier transport and geometric Bass martingales

We introduce and study geometric Bass martingales. Bass martingales were introduced in \cite{Ba83} and studied recently in a series of works, including \cite{BaBeHuKa20,BaBeScTs23}, where they appear as solutions to the martingale version of the Benamou-Brenier optimal transport formulation. These arithmetic, as well as our novel geometric, Bass martingales are continuous martingale on $[0,1]$ with prescribed initial and terminal distributions. An arithmetic Bass martingale is the one closest to Brownian motion: its quadratic variation is as close as possible to being linear in the averaged $L^2$ sense. Its geometric counterpart we develop here, is the one closest to a geometric Brownian motion: the quadratic variation of its logarithm is as close as possible to being linear. By analogy between Bachelier and Black-Scholes models in mathematical finance, the newly obtained geometric Bass martingales} have the potential to be of more practical importance in a number of applications. Our main contribution is to exhibit an explicit bijection between geometric Bass martingales and their arithmetic counterparts. This allows us, in particular, to translate fine properties of the latter into the new geometric setting. We obtain an explicit representation for a geometric Bass martingale for given initial and terminal marginals, we characterise it as a solution to an SDE, and we show that geometric Brownian motion is the only process which is both an arithmetic and a geometric Bass martingale. Finally, we deduce a dual formulation for our geometric martingale Benamou-Brenier problem. Our main proof is probabilistic in nature and uses a suitable change of measure, but we also provide PDE arguments relying on the dual formulation of the problem, which offer a rigorous proof under suitable regularity assumptions.