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M. Beiglböck, N. Juillet:
"On a problem of optimal transport under marginal martingale constraints";
Annals of Probability, Volume 44 (2016), Number 1; 42 - 106.

The basic problem of optimal transportation consists in minimizing the expected costs E[c(X1,X2)] by varying the joint distribution (X1,X2) where the marginal distributions of the random variables X1 and X2 are fixed.

Inspired by recent applications in mathematical finance and connections with the peacock problem, we study this problem under the additional condition that (Xi)i=1,2
is a martingale, that is, E[X2|X1]=X1.
We establish a variational principle for this problem which enables us to determine optimal martingale transport plans for specific cost functions. In particular, we identify a martingale coupling that resembles the classic monotone quantile coupling in several respects. In analogy with the celebrated theorem of Brenier, the following behavior can be observed: If the initial distribution is continuous, then this "monotone martingale" is supported by the graphs of two functions T1,T2:R→R.

http://dx.doi.org/10.1214/14-AOP966

http://projecteuclid.org/euclid.aop/1454423035#abstract