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A Numerical scheme to approximate the solution of the planar Skorokhod embedding problem

Mrabet Becher, Maher Boudabra, Fathi Haggui

TL;DR

This paper addresses approximating the μ-domain in the planar Skorokhod embedding problem (PSEP) by studying stability of the Gross domain under weak convergence $μ_n ightarrow μ$ with bounded support. A key methodological core is expressing the construction in terms of the quantile function $q$ and its approximants $q_n$, and proving that $q_n o q$ in $L^1$ implies the corresponding univalent maps $G_n(z) = abla ext{(coefficients)}$ converge to $G(z)$ uniformly on compact subsets of the unit disk, yielding convergence of the μ-domains. The paper introduces alpha_p-convergence as a general framework, offers truncation-based schemes for unbounded support, and derives convergence rates (typically $O(1/n)$) with explicit error terms depending on the cdf and its density. Numerical simulations across several distributions demonstrate robustness and practical viability, and the approach is compatible with established constructions such as Boudabra–Markowsky.

Abstract

We present a numerical framework to approximate the $μ$-domain in the planar Skorokhod embedding problem (PSEP), recently appeared in \cite{gross2019}. Our approach investigates the continuity and convergence properties of the solutions with respect to the underlying distribution $μ$. We establish that, under weak convergence of a sequence of probability measures $(μ_n)$ with bounded support, the corresponding sequence of $μ_n$-domains converges to the domain associated with $μ$, limit of $(μ_n)$. We derive explicit convergence results in the $L^1$ norm, supported by a generalization using the concept of $α_p$-convergence. Furthermore, we provide practical implementation techniques, convergence rate estimates, and numerical simulations using various distributions. The method proves robust and adaptable, offering a concrete computational pathway for approximating $μ$-domains in the PSEP.

A Numerical scheme to approximate the solution of the planar Skorokhod embedding problem

TL;DR

This paper addresses approximating the μ-domain in the planar Skorokhod embedding problem (PSEP) by studying stability of the Gross domain under weak convergence with bounded support. A key methodological core is expressing the construction in terms of the quantile function and its approximants , and proving that in implies the corresponding univalent maps converge to uniformly on compact subsets of the unit disk, yielding convergence of the μ-domains. The paper introduces alpha_p-convergence as a general framework, offers truncation-based schemes for unbounded support, and derives convergence rates (typically ) with explicit error terms depending on the cdf and its density. Numerical simulations across several distributions demonstrate robustness and practical viability, and the approach is compatible with established constructions such as Boudabra–Markowsky.

Abstract

We present a numerical framework to approximate the -domain in the planar Skorokhod embedding problem (PSEP), recently appeared in \cite{gross2019}. Our approach investigates the continuity and convergence properties of the solutions with respect to the underlying distribution . We establish that, under weak convergence of a sequence of probability measures with bounded support, the corresponding sequence of -domains converges to the domain associated with , limit of . We derive explicit convergence results in the norm, supported by a generalization using the concept of -convergence. Furthermore, we provide practical implementation techniques, convergence rate estimates, and numerical simulations using various distributions. The method proves robust and adaptable, offering a concrete computational pathway for approximating -domains in the PSEP.
Paper Structure (4 sections, 13 theorems, 62 equations, 5 figures)

This paper contains 4 sections, 13 theorems, 62 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Implementation
  3. Examples
  4. Comments

Key Result

Theorem 2

The sequence $q_{n}$ converges almost everywhere to $q$ on $(0,1)$ .

Figures (5)

  • Figure 1.1: For the uniform distribution on $(-1,1)$, the left domain is Boudabra-Markowsky's solution while Gross's solution is on the right.
  • Figure 3.1: Approximation of the $\mu$-domain generated from $\mu=\text{Uni}(-1,1)$ for $n=5,15,30,200$ (top to bottom)
  • Figure 3.2: Approximation of the $\mu$-domain generated from $\mu=\text{Beta}(2,5)$ for $n=5,15,30,200$ (top to bottom)
  • Figure 3.3: Approximation of the $\mu$-domain generated from $\mu=\text{Uni}((-2,-1)\cup(1,2))$ for $n=5,15,30,200$ (top to bottom). The $\mu$-domain contains the vertical strip $\{-1<x<1\}$. The green horizontal segments appear due to the discrete step-wise nature of the calculation used in the scheme.
  • Figure 3.4: Approximation of the $\mu$-domain generated from $\mu=\mathcal{N}(0,1)_{\mid(-3,3)}$ (truncated normal distribution) for $n=5,15,30,200$ (top to bottom).

Theorems & Definitions (28)

  • Definition 1
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Example 4
  • Theorem 5
  • proof
  • Definition 6
  • Proposition 7
  • ...and 18 more