On the Root solution to the Skorokhod embedding problem given full marginals

TL;DR

This paper characterizes the Root solution to the Skorokhod embedding problem with full marginals using a parabolic PDE, extending previous finite-marginal results through limiting arguments.

Contribution

It provides a novel PDE-based characterization of the Root solution for full marginals, advancing the theoretical understanding of the problem.

Findings

Characterization of the potential function via a parabolic PDE

Extension of finite-marginal results to full marginals

Limiting arguments connecting finite and full marginal solutions

Abstract

This paper examines the Root solution of the Skorohod embedding problem given full marginals on some compact time interval. Our results are obtained by limiting arguments based on finitely-many marginals Root solution of Cox, Obl\'oj and Touzi. Our main result provides a characterization of the corresponding potential function by means of a convenient parabolic PDE.