A NOTE ON THE PLANAR SKOROKHOD EMBEDDING PROBLEM

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 pth 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.