On the Martingale Schrödinger Bridge between Two Distributions

Paper: arXiv 2401.05209

Authors: Marcel Nutz, Johannes Wiesel

Abstract

We study a martingale Schrödinger bridge problem: given two probability distributions, find their martingale coupling with minimal relative entropy. Our main result provides Schrödinger potentials for this coupling. Namely, under certain conditions, the log-density of the optimal coupling is given by a triplet of real functions representing the marginal and martingale constraints. The potentials are also described as the solution of a dual problem.

Complexity vs Empirical Score

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

Why this score: The paper presents a highly sophisticated theoretical framework extending the Schrödinger bridge problem to include martingale constraints, featuring advanced measure theory and dual formulation proofs. However, as a purely theoretical probability paper, it lacks empirical validation, real-world financial data application, or accompanying backtest code.

Research Flowchart

  flowchart TD
    A[Research Goal: Find Martingale Coupling with Minimal Relative Entropy] --> B[Inputs: Two Probability Distributions & Martingale Constraint]
    B --> C[Methodology: Formulate Primal Schrödinger Bridge Problem]
    C --> D[Computational Process: Solve Dual Optimization Problem]
    D --> E[Outcome: Derive Triplet of Real Schrödinger Potentials]
    E --> F[Key Finding: Log-Density Characterization for Multi-Asset Derivatives]