The Martingale Sinkhorn Algorithm

ArXiv ID: 2310.13797 “View on arXiv”

Authors: Unknown

Abstract

We develop a numerical method for the martingale analogue of the Benamou-Brenier optimal transport problem, which seeks a martingale interpolating two prescribed marginals which is closest to the Brownian motion. Recent contributions have established existence and uniqueness for the optimal martingale under finite second moment assumptions on the marginals, but numerical methods exist only in the one-dimensional setting. We introduce an iterative scheme, a martingale analogue of the celebrated Sinkhorn algorithm, and prove its convergence in arbitrary dimension under minimal assumptions. In particular, we show that convergence holds when the marginals have finite moments of order $p > 1$, thereby extending the known theory beyond the finite-second-moment regime. The proof relies on a strict descent property for the dual value of the martingale Benamou–Brenier problem. While the descent property admits a direct verification in the case of compactly supported marginals, obtaining uniform control on the iterates without assuming compact support is substantially more delicate and constitutes the main technical challenge.

Keywords: Optimal Transport, Martingale, Benamou-Brenier, Sinkhorn Algorithm, Brownian Motion, Quantitative Analysis

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats
  • Why: The paper introduces a novel, convergent iterative algorithm for a martingale optimal transport problem, requiring advanced stochastic analysis, convex duality, and proofs of convergence in arbitrary dimensions, placing high demand on mathematical complexity. However, the work is purely theoretical, presenting a convergence proof under minimal assumptions without any numerical experiments, backtests, or empirical data analysis, resulting in low empirical rigor.
  flowchart TD
    A["Research Goal:<br>Develop a numerical method<br>for the Martingale Benamou-Brenier problem"] --> B["Key Methodology:<br>Introduce a martingale analogue<br>of the Sinkhorn algorithm"]
    B --> C["Data/Inputs:<br>Prescribed marginals<br>Finite moment assumptions p > 1"]
    C --> D["Computational Process:<br>Iterative dual descent<br>proving strict descent property"]
    D --> E["Key Findings:<br>Convergence in arbitrary dimension<br>Extension beyond finite-second-moment regime"]