The geometry of multi-marginal Skorokhod Embedding

Area of Science:

• Stochastic Processes
• Optimal Transport Theory
• Mathematical Finance

Background:

• The Skorokhod Embedding Problem is a classical problem in stochastic processes with broad applications.
• Multi-marginal extensions of the Skorokhod problem have been challenging to solve with existing techniques.
• Recent advancements have provided partial solutions, but a complete theory for the multi-marginal case remained elusive.

Purpose of the Study:

• To extend the theory of optimal transport to the multi-marginal Skorokhod problem.
• To develop a powerful new framework for analyzing multi-marginal embeddings.
• To establish connections between classical optimal embeddings and their multi-marginal counterparts.

Main Methods:

• Extension of the theory developed by Beiglböck et al. to the multi-marginal setup.
• Utilizing a viewpoint comparable to the multi-marginal optimal transport problem.
• Analysis of the joint geometric structure linking different constructions.

Main Results:

• Demonstration that all classical optimal embeddings have natural multi-marginal counterparts.
• Identification of a joint geometric structure that links these constructions.
• Recovery of classical solutions as particular cases within the new framework.

Conclusions:

• The developed multi-marginal framework provides a powerful approach to the Skorokhod problem.
• The results offer new insights into martingale transport and the peacock problem.
• This work establishes a significant theoretical advancement in stochastic processes and optimal transport.