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.
Categories: math.PR