On the Continuity of the Root Barrier

TL;DR

This paper proves that the barrier function in Root's solution to the Skorokhod embedding problem is continuous and finite under certain conditions on the target measure, ensuring better understanding of the solution's properties.

Contribution

It establishes the continuity and finiteness of the barrier function in Root's solution when the target measure has no atoms and a locally bounded away from zero absolutely continuous part.

Findings

Barrier function is continuous where the measure has no atoms.

Barrier function is finite under specified measure conditions.

Provides insights into the structure of Root's solution.

Abstract

We show that the barrier function in Root's solution to the Skorokhod embedding problem is continuous and finite at every point where the target measure has no atom and its absolutely continuous part is locally bounded away from zero.