Martingale Benamou–Brenier: A probabilistic perspective

By Julio Backhoff-Veraguas, Mathias Beiglbock, Martin Huesmann, Sigrid Kallblad. Annals of Probability, 2017

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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.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/6d71f13c0fa22bee705306ec532661f5f5f2700c

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Citations in the corpus

All adapted topologies are equal (2019, background): “The article [7] connects adapted Wasserstein distance (in continuous time) to martingale optimal transport (cf.”
Estimating processes in adapted Wasserstein distance (2020, background): “and Talay [10] and Gigli [17, Chapter 4]. Adapted distances / topologies on laws of processes are of fundamental importance in questions of stability in mathematical nance and stochastic control, see [13, 5, 18, 1, 9, 3, 7]. Notably, all these notions (and in fact several more that we do not discuss”
The Wasserstein space of stochastic processes (2021, background): “the survey of Leonard [55]), martingale optimal transport, the weak transport problem (see [36, 6] among many others), the theory of linear transfers ([25]), and mathematical finance / martingale optimal transport, we refer to [10, 15, 2, 12].”