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

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 $mathbb{R}$, $Gamma_{0,a}=Lambda_acirc Gamma_0$, where $Lambda_a:mathcal{D}[0,infty)tomathcal{D}[0,infty)$ is defined by [Lambda_a(phi)(t)=phi(t)-sup_{sin[0,t]}biggl[bigl( phi(s)-abigr)^+wedgeinf_{uin[s,t]}phi(u)biggr]] and $Gamma_0:mathcal{D}[0,infty)tomathcal{D}[0,infty)$ is the Skorokhod map on $[0,infty)$, which is given explicitly by [Gamma_0(psi)(t)=psi(t)+sup_{sin[0,t]}[-psi(s)]^+.] In addition, properties of $Lambda_a$ are developed and comparison properties of $Gamma_{0,a}$ are established.