Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Martingale Sinkhorn Algorithm

Manuel Hasenbichler, Benjamin Joseph, Gregoire Loeper, Jan Obloj, Gudmund Pammer

TL;DR

The paper tackles computing the Bass martingale, the optimal martingale interpolation between prescribed marginals in the Benamou–Brenier framework, in arbitrary dimensions. It introduces a multidimensional Martingale Sinkhorn Algorithm, an iterative Sinkhorn-like scheme that updates Brenier–McCann potentials and Bass measures, and proves convergence under minimal moment assumptions (p>1) with irreducibility. A key innovation is a relaxed dual objective ensuring strict descent and a tightness argument that handles non-compact marginals, yielding existence and uniqueness of the Bass solution in higher dimensions. This method provides a practical, principled numerically stable tool for martingale optimal transport calibration and related stochastic control problems with a PDE duality perspective.

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.

The Martingale Sinkhorn Algorithm

TL;DR

The paper tackles computing the Bass martingale, the optimal martingale interpolation between prescribed marginals in the Benamou–Brenier framework, in arbitrary dimensions. It introduces a multidimensional Martingale Sinkhorn Algorithm, an iterative Sinkhorn-like scheme that updates Brenier–McCann potentials and Bass measures, and proves convergence under minimal moment assumptions (p>1) with irreducibility. A key innovation is a relaxed dual objective ensuring strict descent and a tightness argument that handles non-compact marginals, yielding existence and uniqueness of the Bass solution in higher dimensions. This method provides a practical, principled numerically stable tool for martingale optimal transport calibration and related stochastic control problems with a PDE duality perspective.

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 , 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.
Paper Structure (9 sections, 16 theorems, 127 equations, 1 figure, 1 algorithm)

This paper contains 9 sections, 16 theorems, 127 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. The classical OT problem and its Benamou-Brenier formulation
  3. The martingale Benamou-Brenier problem
  4. Main Results
  5. PDE perspective on duality and the Bass martingale
  6. Analogy to the Schrödinger system and the Sinkhorn algorithm
  7. Related Literature
  8. Convergence of the Martingale Sinkhorn Algorithm
  9. Auxiliary Results

Key Result

Theorem 1.1

Let $\mu_0,\mu_1 \in \mathcal{P}_{p}(\mathbb{R}^d)$ for some $p > 1$ be in convex order and irreducible and $v_0 \in L^1(\mu_1)$ be lower semi-continuous convex such that $v_0^{\star} * \gamma$ is properA convex function $\psi\colon E \to \mathbb R\cup\{\pm\infty\}$ is called proper if $\psi(x) > -\

Figures (1)

  • Figure 1: Schematic illustration of the relations characterising the Bass potential $v$ and Bass measure $\alpha$ that solve the Martingale Benamou--Brenier problem (see also \ref{['sec:Sinkhorn']}). Clockwise iteration of this diagram yields \ref{['alg:MSinkhorn']}, which decreases the dual objective function $\mathcal{E}$ after each full cycle.

Theorems & Definitions (33)

  • Theorem 1.1: Convergence of the Martingale Sinkhorn Algorithm
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4: Strict Descent under Compact Support
  • proof
  • Lemma 2.5: Tightness
  • ...and 23 more