In classical optimal transport, the contributions of Benamou–Brenier and
Mc-Cann 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.
Based on a weak length relaxation we suggest a Benamou-Brenier type
formulation of martingale optimal transport. We give an explicit
probabilistic representation of the optimizer for a specific cost
function leading to a continuous Markov-martingale M with several
notable properties: In a specific sense it mimics the movement of a
Brownian particle as closely as possible subject to the marginal
conditions a time 0 and 1. 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 new approach to Kellerer’s theorem.
(based on joint work with J. Backhoff, M. Beiglböck, S. Källblad, and D.
- Mathematical and Computational Finance Seminar