Work

Existence of Bass martingales and the martingale Benamou–Brenier problem in Rd

Year: 2026

Type: article

Abstract: In classical optimal transport, the contributions of Benamou–Brenier and McCann regarding the time-dependent version of the problem are cornerstones of the field and form the basis for a variety of applications in other mathematical areas. In this article we characterize solutions to the martingale Benamou–Brenier problem as Bass martingales, that is, transformations of Brownian motion through the gradient of a convex function. Our result is based on a new (static) Brenier-type theorem for a particular weak martingale optimal transport problem. As in the classical case, the structure of the primal optimizer is derived from its dual counterpart, whose derivation forms the technical core of this article. A key challenge is that dual attainment is a subtle issue in martingale optimal transport, where dual optimizers may fail to exist, even in highly regular settings.

Source: The Annals of Probability

DOI: 10.1214/25-aop1800

Authors: Julio Backhoff-Veraguas, Mathias Beiglböck, Walter Schachermayer, Bertram Tschiderer

Institution: University of Vienna

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Funder: Austrian Science Fund

Award: Transport approach to mimicking processes