The Gradient Flow of the Bass Functional in Martingale Optimal Transport

TL;DR

This paper introduces a gradient flow approach to minimize the Bass functional, which determines the Bass martingale in optimal transport, proving convergence and exponential speed in one dimension.

Contribution

It develops a gradient flow method for the Bass functional and proves convergence to the minimizer, advancing the theoretical understanding of Bass martingales.

Findings

Gradient flow converges in norm to the Bass functional minimizer.

In one dimension, convergence is exponentially fast.

The method provides a practical way to construct Bass martingales.

Abstract

Given $\mu$ and $\nu$, probability measures on $\mathbb R^d$ in convex order, a Bass martingale is arguably the most natural martingale starting with law $\mu$ and finishing with law $\nu$. Indeed, this martingale is obtained by stretching a reference Brownian motion so as to meet the data $\mu,\nu$. Unless $\mu$ is a Dirac, the existence of a Bass martingale is a delicate subject, since for instance the reference Brownian motion must be allowed to have a non-trivial initial distribution $\alpha$, not known in advance. Thus the key to obtaining the Bass martingale, theoretically as well as practically, lies in finding $\alpha$. In \cite{BaSchTsch23} it has been shown that $\alpha$ is determined as the minimizer of the so-called Bass functional. In the present paper we propose to minimize this functional by following its gradient flow, or more precisely, the gradient flow of its $L^2$-lift. In our main result we show that this gradient flow converges in norm to a minimizer of the Bass functional, and when $d=1$ we further establish that convergence is exponentially fast.