What question did this study set out to answer?

The research aims to solve the planar Skorokhod embedding problem specifically for cases where p=1, which had previously been left unresolved.

February 14, 2026Open Access

A Note on the Planar Skorokhod Embedding Problem

Key Points

The research aims to solve the planar Skorokhod embedding problem specifically for cases where p=1, which had previously been left unresolved.
Analyzed properties of probability distributions with finite moments
Utilized the Hilbert transform of the quantile function
Developed a solution framework for planar Brownian motion
Established solvability of the planar Skorokhod embedding for p=1
Identified conditions related to the integrability of the Hilbert transform
Closed the gap in existing literature regarding embeddings for p=1

Abstract

Abstract The planar Skorokhod embedding problem was first proposed and solved by Gross ‘A conformal Skorokhod embedding’, Electron. Commun. Probab. 24 (2019), 11 pages; doi: 10. 1214/19-ECP272. Gross worked with probability distributions having finite second moment. Boudabra and Markowsky ‘Remarks on Gross’ technique for obtaining a conformal Skorokhod embedding of planar Brownian motion’, Electron. Commun. Probab. 25 (2020), 13 pages; doi: 10. 1214/20-ECP300 extended the solution to all distributions with a finite p th moment for p>1. The case p=1 has remained uncovered since then. In this note, we show that the planar Skorokhod embedding problem is solvable for p=1 when the Hilbert transform of its quantile function is integrable, effectively closing this line of investigation.

Bookmark

View Full Paper

Cite This Study

Maher Boudabra (Thu,) studied this question.

synapsesocial.com/papers/699011602ccff479cfe57f16 https://doi.org/https://doi.org/10.1017/s0004972726100987

Bookmark

View Full Paper