Bridging classical and martingale Schr\"odinger bridges

Abstract

We investigate the martingale Schr\"odinger bridge, recently introduced by Nutz and Wiesel as a distinguished martingale transport plan between two probability measures in convex order. We show that this construction extends naturally to arbitrary dimension and admits several equivalent characterizations. In particular, we identify its continuous-time counterpart as the continuous martingale with prescribed marginals that minimizes a weighted quadratic energy measuring the deviation from Brownian motion. In the irreducible case, we prove that this continuous martingale Schr\"odinger bridge coincides with the F\"ollmer martingale, that is, with the Doob martingale associated to a suitable F\"ollmer process. More generally, we relate the martingale Schr\"odinger bridge to a variational problem over base measures and to the dual formulation of the corresponding weak optimal transport problem, thereby clarifying its connection with the classical Schr\"odinger bridge.