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: ParisPrinceton lectures on mathematical finance 2010, Volume 2003 of Lecture Notes in Mathematics, Springer, Berlin, 2011; Obłój in: Probab Surv 1:321–390, 2004)]. Many of these applications have natural multimarginal extensions leading to the (optimal) multimarginal 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 173:211–259, 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 multimarginal setup which is comparable to the extension of the optimal transport problem to the multimarginal optimal transport problem. As for the onemarginal case, this viewpoint turns out to be very powerful. In particular, we are able to show that all classical optimal embeddings have natural multimarginal 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.
Original language  English 

Pages (fromto)  10451096 
Number of pages  52 
Journal  Probability Theory and Related Fields 
Volume  176 
Issue number  34 
Early online date  1 Aug 2019 
DOIs  
Publication status  Published  30 Apr 2020 
Keywords
 math.PR
 qfin.PR
 60G42, 60G44, 91G20
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Alex Cox
 Department of Mathematical Sciences  Deputy Head of Department
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Probability Laboratory at Bath
Person: Research & Teaching