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September 2020 Martingale Benamou–Brenier: A probabilistic perspective
Julio Backhoff-Veraguas, Mathias Beiglböck, Martin Huesmann, Sigrid Källblad
Ann. Probab. 48(5): 2258-2289 (September 2020). DOI: 10.1214/20-AOP1422


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.

We suggest a Benamou–Brenier type formulation of the martingale transport problem for given $d$-dimensional distributions $\mu $, $\nu $ in convex order. The unique solution $M^{*}=(M_{t}^{*})_{t\in [0,1]}$ of this problem turns out to be a Markov-martingale which has several notable properties: In a specific sense it mimics the movement of a Brownian particle as closely as possible subject to the conditions $M^{*}_{0}\sim \mu $, $M^{*}_{1}\sim \nu $. Similar to McCann’s displacement-interpolation, $M^{*}$ provides a time-consistent interpolation between $\mu $ and $\nu $. For particular choices of the initial and terminal law, $M^{*}$ recovers archetypical martingales such as Brownian motion, geometric Brownian motion, and the Bass martingale. Furthermore, it yields a natural approximation to the local vol model and a new approach to Kellerer’s theorem.

This article is parallel to the work of Huesmann–Trevisan, who consider a related class of problems from a PDE-oriented perspective.


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Julio Backhoff-Veraguas. Mathias Beiglböck. Martin Huesmann. Sigrid Källblad. "Martingale Benamou–Brenier: A probabilistic perspective." Ann. Probab. 48 (5) 2258 - 2289, September 2020.


Received: 1 January 2019; Revised: 1 September 2019; Published: September 2020
First available in Project Euclid: 23 September 2020

MathSciNet: MR4152642
Digital Object Identifier: 10.1214/20-AOP1422

Primary: 60G42 , 60G44
Secondary: 91G20

Keywords: Benamou–Brenier , Brenier’s theorem , causal transport , cyclical monotonicity , Knothe Rosenblatt coupling , Martingales , Optimal transport , Schrödinger problem , weak transport problems

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.48 • No. 5 • September 2020
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