An explicit formula for the Skorokhod map on $[0,a]$

TL;DR

This paper derives an explicit formula for the Skorokhod map on a finite interval, expressing it as a composition of known maps, and analyzes its properties and comparison results.

Contribution

It provides a new explicit formula for the Skorokhod map on [0,a], extending the understanding of reflecting boundary solutions in stochastic processes.

Findings

Explicit formula for the Skorokhod map on [0,a]

Properties of the map \\Lambda_a are developed

Comparison properties of the Skorokhod map \\Gamma_{0,a} are established

Abstract

The Skorokhod map is a convenient tool for constructing solutions to stochastic differential equations with reflecting boundary conditions. In this work, an explicit formula for the Skorokhod map $\Gamma_{0,a}$ on $[0,a]$ for any $a>0$ is derived. Specifically, it is shown that on the space $\mathcal{D}[0,\infty)$ of right-continuous functions with left limits taking values in $\mathbb{R}$, $\Gamma_{0,a}=\Lambda_a\circ \Gamma_0$, where $\Lambda_a:\mathcal{D}[0,\infty)\to\mathcal{D}[0,\infty)$ is defined by \[\Lambda_a(\phi)(t)=\phi(t)-\sup_{s\in[0,t]}\biggl[\bigl(\ phi(s)-a\bigr)^+\wedge\inf_{u\in[s,t]}\phi(u)\biggr]\] and $\Gamma_0:\mathcal{D}[0,\infty)\to\mathcal{D}[0,\infty)$ is the Skorokhod map on $[0,\infty)$, which is given explicitly by \[\Gamma_0(\psi)(t)=\psi(t)+\sup_{s\in[0,t]}[-\psi(s)]^+.\] In addition, properties of $\Lambda_a$ are developed and comparison properties of $\Gamma_{0,a}$ are established.