Remarks on Gross’ technique for obtaining a conformal Skorohod embedding of planar Brownian motion

Abstract

In [7] it was proved that, given a distribution $\mu $ with zero mean and finite second moment, there exists a simply connected domain $\Omega $ such that if $Z_{t}$ is a standard planar Brownian motion, then $\mathcal{R} e(Z_{\tau _{\Omega }})$ has the distribution $\mu $, where $\tau _{\Omega }$ denotes the exit time of $Z_{t}$ from $\Omega $. In this note, we extend this method to prove that if $\mu $ has a finite $p$-th moment then the first exit time $\tau _{\Omega }$ from $\Omega $ has a finite moment of order $\frac{p} {2}$. We also prove a uniqueness principle for this construction, and use it to give several examples.