The geometry of multi-marginal Skorokhod Embedding.

ABSTRACT

The Skorokhod Embedding Problem is one of the classical problems in the theory of stochastic processes, with applications in many different fields [cf.  the surveys (Hobson in Paris-Princeton lectures on mathematical finance 2010, Volume 2003 of Lecture Notes in Mathematics, Springer, Berlin, 2011; Oblój in Probab Surv 1321-390, 2004)]. Many of these applications have natural multi-marginal extensions leading to the (optimal) multi-marginal Skorokhod problem. Some of the first papers to consider this problem are Brown et al. (Probab Theory Relat Fields 119(4)558-578, 2001), Hobson (Séminaire de Probabilités, XXXII, Volume 1686 of Lecture Notes in Mathematics, Springer, Berlin, 1998), Madan and Yor (Bernoulli 8(4)509-536, 2002). However, this turns out to be difficult using existing techniques only recently a complete solution was be obtained in Cox et al. (Probab Theory Relat Fields 173211-259, 2018) establishing an extension of the Root construction, while other instances are only partially answered or remain wide open. In this paper, we extend the theory developed in Beiglböck et al. (Invent Math 208(2)327-400, 2017) to the multi-marginal setup which is comparable to the extension of the optimal transport problem to the multi-marginal optimal transport problem. As for the one-marginal case, this viewpoint turns out to be very powerful. In particular, we are able to show that all classical optimal embeddings have natural multi-marginal counterparts. Notably these different constructions are linked through a joint geometric structure and the classical solutions are recovered as particular cases. Moreover, our results also have consequences for the study of the martingale transport problem as well as the peacock problem.