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J. Backhoff, M. Beiglböck, Y. Lin, A. Zalashko:
"Causal transport in discrete time and applications";
SIAM Journal on Control and Optimization, Volume 27 (2017), Issue 4; 2528 - 2562.

Loosely speaking, causal transport plans are a relaxation of adapted processes in the same sense as Kantorovich transport plans extend Monge-type transport maps. The corresponding causal version of the transport problem has recently been introduced by Lassalle. Working in a discrete time setup, we establish a dynamic programming principle that links the causal transport problem to the transport problem for general costs recently considered by Gozlan et al. Based on this recursive principle, we give conditions under which the celebrated Knothe-Rosenblatt rearrangement can be viewed as a causal analogue to the Brenier's map. Moreover, these considerations provide transport-information inequalities for the nested distance between stochastic processes pioneered by Pflug and Pichler, and so serve to gauge the discrepancy between stochastic programs driven by different noise distributions.

Optimal transport causality nested distance general transp ort costs Knothe-Rosenblatt rearrangement transport inequalities

http://dx.doi.org/10.1137/16M1080197

http://epubs.siam.org/doi/abs/10.1137/16M1080197