The Bass functional of martingale transport

TL;DR

This paper investigates the Bass functional in martingale optimal transport, establishing conditions for the existence of minimizers and connecting them to Bass martingales, with implications for variational methods and convexity in Wasserstein space.

Contribution

It introduces a variational approach to determine initial conditions for Bass martingales, complementing existing duality methods, and proves displacement convexity of the Bass functional.

Findings

Existence of minimizers linked to Bass martingales

Infinitesimal version of the main result established

Displacement convexity of the Bass functional proven

Abstract

An interesting question in the field of martingale optimal transport, is to determine the martingale with prescribed initial and terminal marginals which is most correlated to Brownian motion. Under a necessary and sufficient irreducibility condition, the answer to this question is given by a $\textit{Bass martingale}$. At an intuitive level, the latter can be imagined as an order-preserving and martingale-preserving space transformation of an underlying Brownian motion starting with an initial law $\alpha$ which is tuned to ensure the marginal constraints. In this article we study how to determine the aforementioned initial condition $\alpha$. This is done by a careful study of what we dub the $\textit{Bass functional}$. In our main result we show the equivalence between the existence of minimizers of the Bass functional and the existence of a Bass martingale with prescribed marginals. This complements the convex duality approach in a companion paper by the present authors together with M. Beiglb\"ock, with a purely variational perspective. We also establish an infinitesimal version of this result, and furthermore prove the displacement convexity of the Bass functional along certain generalized geodesics in the $2$-Wasserstein space.