The geometry of multi-marginal Skorokhod Embedding

TL;DR

This paper extends the geometric theory of the Skorokhod Embedding Problem to the multi-marginal case, revealing a unified structure that encompasses classical solutions and impacts related areas like martingale transport and peacock problems.

Contribution

It develops a comprehensive geometric framework for multi-marginal Skorokhod embeddings, generalizing classical solutions and connecting them through a joint structure.

Findings

Unified geometric structure for multi-marginal embeddings

Classical solutions have natural multi-marginal counterparts

Implications for martingale transport and peacock problems

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.