🤖 AI Summary
This paper addresses the problem of constructing a measure-preserving martingale interpolation between prescribed marginal distributions in arbitrary dimensions, such that the resulting process is as close as possible to Brownian motion. Departing from prior work—largely restricted to one dimension and requiring finite second moments—we propose the first algorithm applicable under the significantly weaker assumption that marginals possess finite moments of order strictly greater than one. Our method establishes a novel connection between Bass martingales and semimartingale optimal transport, leading to the Measure-Preserving Martingale Sinkhorn (MPMS) algorithm. MPMS is derived from a dual formulation and integrates stochastic analysis, optimal transport, and fixed-point theory. We rigorously prove its monotonic convergence and dimensional scalability. Empirical evaluations on both synthetic and real financial datasets demonstrate that MPMS consistently outperforms existing approaches in accuracy and stability.
📝 Abstract
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.