# A conformal Skorokhod embedding

## TL;DR

This paper introduces a new complex-analytic method for the Skorokhod embedding theorem, allowing the construction of barriers that produce Brownian motion hitting points with any prescribed centered distribution with finite variance.

## Contribution

It provides a constructive, explicit approach to the Skorokhod embedding problem using complex analysis, expanding the toolkit for stochastic process embedding techniques.

## Key findings

- Constructed barriers for prescribed distributions
- Provided explicit descriptions of barriers
- Offered a new proof of the Skorokhod embedding theorem

## Abstract

Start a planar Brownian motion and let it run until it hits some given barrier. We show that the barrier may be crafted so that the x coordinate at the hitting time has any prescribed centered distribution with finite variance. This provides a new, complex-analytic proof of the Skorokhod embedding theorem. Our method is constructive and can give an explicit description of the barrier.