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A note on the planar Skorokhod embedding problem

Maher Boudabra

TL;DR

The paper advances the planar Skorokhod embedding problem by establishing solvability for p=1 under a Hilbert-transform integrability condition on the quantile function, filling a gap left by prior results for p>1. It constructs a univalent map G via the Cauchy-Poisson integral to produce a simply connected domain U=G(D) whose Brownian exit distribution matches the target μ, with finite exit-time moments guaranteed by membership in an appropriate Hardy space. The work connects analytic criteria (Hilbert transform of the quantile) to probabilistic embedding feasibility and provides practical criteria (via Zygmund-type estimates) and explicit examples, including laws with only a finite first moment. These insights broaden the known scope of PSEP solvability and suggest directions for necessity questions and extensions to smaller moments (p<1) and proper mappings.

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.

A note on the planar Skorokhod embedding problem

TL;DR

The paper advances the planar Skorokhod embedding problem by establishing solvability for p=1 under a Hilbert-transform integrability condition on the quantile function, filling a gap left by prior results for p>1. It constructs a univalent map G via the Cauchy-Poisson integral to produce a simply connected domain U=G(D) whose Brownian exit distribution matches the target μ, with finite exit-time moments guaranteed by membership in an appropriate Hardy space. The work connects analytic criteria (Hilbert transform of the quantile) to probabilistic embedding feasibility and provides practical criteria (via Zygmund-type estimates) and explicit examples, including laws with only a finite first moment. These insights broaden the known scope of PSEP solvability and suggest directions for necessity questions and extensions to smaller moments (p<1) and proper mappings.

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 moment for . The case remained uncovered since then. In this note we show that the planar Skorokhod embedding problem is solvable for when the Hilbert transform of its quantile function is integrable, effectively closing this line of investigation.

Paper Structure

This paper contains 5 sections, 5 theorems, 22 equations.

Table of Contents

  1. Introduction and main results
  2. Proofs
  3. Proof of theorem \ref{['boundary distribution']}
  4. Proof of theorem \ref{['thm:main']}
  5. Comments

Key Result

Theorem 3

Let $f$ be a univalent function on $\mathbb{D}$ with $f(0)=0$ and set $U=f(\mathbb{D})$. Let $(Z_{t})_{t\geq0}$ be standard planar Brownian motion and $\tau_{U}$ its exit time from $U$. Then $Z_{\tau_{U}}$ and $f(\xi)$ share the same distribution where $\xi$ is any random variable uniformly distribu

Theorems & Definitions (8)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Remark 8