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Peer-reviewedCC BYThe Annals of Probability

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

  • Julio Backhoff‐Veraguas1
  • Mathias Beiglböck1
  • Walter Schachermayer B1
  • Bertram Tschiderer1
  1. 1Faculty of Mathematics, University of Vienna
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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.

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Cite

BibTeXRIS
BibTeX
@article{backhoffveraguas2026existence,
  title = {{Existence of Bass martingales and the martingale Benamou–Brenier problem in Rd}},
  author = {Julio Backhoff‐Veraguas and Mathias Beiglböck and Walter Schachermayer B and Bertram Tschiderer},
  journal = {The Annals of Probability},
  year = {2026},
  month = aug,
  doi = {10.1214/25-aop1800},
  url = {https://doi.org/10.1214/25-aop1800},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Existence of Bass martingales and the martingale Benamou–Brenier problem in Rd
AU  - Julio Backhoff‐Veraguas
AU  - Mathias Beiglböck
AU  - Walter Schachermayer B
AU  - Bertram Tschiderer
PY  - 2026
DA  - 2026/08/28
JO  - The Annals of Probability
DO  - 10.1214/25-aop1800
UR  - https://doi.org/10.1214/25-aop1800
AB  - 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.
SN  - 0091-1798
SN  - 2168-894X
ER  -