Journal
SIAM JOURNAL ON OPTIMIZATION
Volume 27, Issue 4, Pages 2528-2562Publisher
SIAM PUBLICATIONS
DOI: 10.1137/16M1080197
Keywords
optimal transport; causality; nested distance; general transport costs; Knothe-Rosenblatt rearrangement; transport inequalities
Categories
Funding
- Austrian Science Fund (FWF) [P26736, Y782-N25]
- European Research Council (ERC) [FA506041, 321111]
- Doktoratskolleg of the Austrian Science Fund (FWF) [W1245]
- Austrian Science Fund (FWF) [W1245] Funding Source: Austrian Science Fund (FWF)
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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 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.
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