A note on the planar Skorokhod embedding problem

2025/12/20 by Maher Boudabra, Boudabra, Maher
Mathematics · #Mathematical Analysis and Transform Methods #Random Matrices and Applications #Spectral Theory in Mathematical Physics

paper · doi:10.48550/arxiv.2512.18353

Abstract

The planar Skorokhod embedding problem was first proposed and solved by R. Gross in 2019 [#gross2019]. Gross worked with probability distributions having finite second moment. In [#boudabra2019remarks, #Boudabra2020], the solutions extended to all distributions with a finite p^th moment for p>1. The case p=1 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.

Citations

• Remarks on Gross' technique for obtaining a conformal Skorohod embedding of planar Brownian motion
• A conformal Skorokhod embedding
• The exit time of planar Brownian motion and the Phragmen-Lindelof principle

Related

• On the boundedness of Gross' solution to the planar Skorokhod embedding problem
• A Numerical scheme to approximate the solution of the planar Skorokhod embedding problem
• Optimal design for the planar Skorokhod embedding problem
• Conformal Skorokhod embeddings and related extremal problems
• A conformal Skorokhod embedding