Bridging classical and martingale Schr\"odinger bridges

TL;DR

This paper extends the concept of martingale Schr"odinger bridges to higher dimensions, characterizes their continuous-time versions, and links them to classical Schr"odinger bridges and weak optimal transport problems.

Contribution

It introduces a natural extension of martingale Schr"odinger bridges to arbitrary dimensions and establishes their connections with classical Schr"odinger bridges and variational problems.

Findings

The continuous-time martingale Schr"odinger bridge minimizes a weighted quadratic energy.

In the irreducible case, it coincides with the F"ollmer martingale.

The bridge relates to a variational problem and dual formulations in weak optimal transport.

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.