  We contribute to the recent studies of the so-called Bass martingale.
Backhoff-Veraguas et al. (2020) showed it is the solution to the martingale
Benamou-Brenier (mBB) problem, i.e., among all martingales with prescribed
initial and terminal distributions it is the one closest to the Brownian
motion. We link it with semimartingale optimal transport and deduce an
alternative way to derive the dual formulation recently obtained in
Backhoff-Veraguas et al. (2023). We then consider computational methods to
compute the Bass martingale. The dual formulation of the transport problem
leads to an iterative scheme that mirrors to the celebrated Sinkhorn algorithm
for entropic optimal transport. We call it the measure preserving martingale
Sinkhorn (MPMS) algorithm. We prove that in any dimension, each step of the
algorithm improves the value of the dual problem, which implies its
convergence. Our MPMS algorithm is equivalent to the fixed-point method of
Conze and Henry-Labordere (2021), studied in Acciaio et al. (2023), and
performs very well on a range of examples, including real market data.
