Martingale Benamou-Brenier: A probabilistic perspective

In classical optimal transport, the contributions of Benamou–Brenier and McCann regarding the time-dependent version of the problem are corner-stones 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 μ, ν in convex order. The unique solution M∗ = (M∗
t )t∈[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 ∼ μ, M∗1 ∼ ν. Similar to McCann’s displacement-interpolation, M∗ provides a time-consistent interpolation between μ and ν. 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.