The geometry of multi-marginal Skorokhod embedding
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Publication:2174667
DOI10.1007/S00440-019-00935-ZzbMath1469.60126arXiv1705.09505OpenAlexW2964649716WikidataQ94474638 ScholiaQ94474638MaRDI QIDQ2174667
Mathias Beiglböck, Martin Huesmann, Alexander Matthew Gordon Cox
Publication date: 21 April 2020
Published in: Unnamed Author (Search for Journal in Brave)
Abstract: The Skorokhod Embedding Problem (SEP) is one of the classical problems in the study of stochastic processes, with applications in many different fields (cf.~ the surveys cite{Ob04,Ho11}). Many of these applications have natural multi-marginal extensions leading to the emph{(optimal) multi-marginal Skorokhod problem} (MSEP). Some of the first papers to consider this problem are cite{Ho98b, BrHoRo01b, MaYo02}. However, this turns out to be difficult using existing techniques: only recently a complete solution was be obtained in cite{CoObTo15} 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 cite{BeCoHu14} 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.
Full work available at URL: https://arxiv.org/abs/1705.09505
Martingales with discrete parameter (60G42) Martingales with continuous parameter (60G44) Derivative securities (option pricing, hedging, etc.) (91G20)
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Related Items (14)
Fine properties of the optimal Skorokhod embedding problem ⋮ On the continuity of the root barrier ⋮ Shadows and barriers ⋮ Supermartingale Brenier's theorem with full-marginals constraint ⋮ Supermartingale shadow couplings: the decreasing case ⋮ A potential-based construction of the increasing supermartingale coupling ⋮ Martingale Benamou-Brenier: a probabilistic perspective ⋮ On the Root Solution to the Skorokhod Embedding Problem Given Full Marginals ⋮ Martingale optimal transport in the discrete case via simple linear programming techniques ⋮ A free boundary characterisation of the root barrier for Markov processes ⋮ Stopping with expectation constraints: 3 points suffice ⋮ Shadow couplings ⋮ Shadow martingales -- a stochastic mass transport approach to the peacock problem ⋮ Adapted Wasserstein distances and stability in mathematical finance
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