The Measure Preserving Martingale Sinkhorn Algorithm
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 Brownian motion. We link it with the semimartingale optimal transport and deduce an alternative way to arrive at the dual formulation recently obtained in Backhoff-Beraguas et al. (2023). We then consider computational methods to compute the Bass martingale. We propose an approach analogous to the famous Sinkhorn algorithm used to solve the entropic optimal transport problem. Our algorithm, dubbed the measure preserving martingale Sinkhorn (MPMS), is equivalent to the fixed-point method of Conze and Henry-Labordere (2021) and performs very well on a range of examples, including a market data case study.
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