The Gradient Flow of the Bass Functional in Martingale Optimal Transport

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.