The Martingale Sinkhorn Algorithm

TL;DR

This paper introduces a multidimensional iterative Sinkhorn-like algorithm for the martingale optimal transport problem, extending existing methods beyond finite second moments and providing convergence guarantees.

Contribution

It develops a novel martingale Sinkhorn algorithm applicable in arbitrary dimensions with minimal assumptions, advancing numerical solutions for martingale optimal transport.

Findings

The algorithm converges to a Bass potential in any dimension.

It works under finite p-th moment conditions with p > 1.

The method extends numerical solutions beyond the finite second moment case.

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 of 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 that it yields a Bass potential in arbitrary dimension under minimal assumptions. In particular, we show that this 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.