Bridging classical and martingale Schrödinger bridges

Authors: Julio Backhoff, Mathias Beiglböck, Giorgia Bifronte, Armand Ley

Abstract

We investigate the martingale Schrödinger 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 Schrödinger bridge coincides with the Föllmer martingale, that is, with the Doob martingale associated to a suitable Föllmer process. More generally, we relate the martingale Schrödinger 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 Schrödinger bridge.

Complexity vs Empirical Score

Math Complexity: 9.0/10
Empirical Rigor: 1.5/10
Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents highly sophisticated pure mathematical work extending martingale Schrödinger bridges and Föllmer processes to arbitrary dimensions. While mathematically rigorous and highly novel, it is a purely theoretical paper with no empirical data analysis or backtesting included in the excerpt.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend Martingale Schrödinger Bridge MSB to arbitrary dimensions & clarify link to Classical SB] --> B[Input Data: Probability measures in convex order & Brownian motion base measure]
    B --> C[Methodology: Formulate Weak Optimal Transport & Dual Variational Problems]
    C --> D[Continuous-Time Analysis: Minimize weighted quadratic energy deviation from Brownian motion]
    D --> E[Outcome 1: Identify MSB as Doob martingale of a Föllmer process in irreducible case]
    D --> F[Outcome 2: Establish equivalence between Classical SB and MSB via base measure optimization]

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