A note on the planar Skorokhod embedding problem

TL;DR

This paper extends the solvability of the planar Skorokhod embedding problem to include distributions with a finite first moment under certain conditions, completing previous research that focused on higher moments.

Contribution

It demonstrates that the problem is solvable for p=1 when the Hilbert transform of the quantile function is integrable, filling a gap in existing literature.

Findings

Solvability for p=1 established under new conditions

Completes the classification for finite moment cases

Provides a criterion involving the Hilbert transform

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.