Martingale optimal transport - discrete to continous

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. 
Trevisan)