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 ${\mathrm{\Gamma }}_{0,a}$ on $[0,a]$ for any $a>0$ is derived. Specifically, it is shown that on the space $\mathcal{D}[0,\mathrm{\infty })$ of right-continuous functions with left limits taking values in $\mathbb{R}$, ${\mathrm{\Gamma }}_{0,a}={\mathrm{\Lambda }}_a\circ {\mathrm{\Gamma }}_0$, where ${\mathrm{\Lambda }}_a:\mathcal{D}[0,\mathrm{\infty })\to \mathcal{D}[0,\mathrm{\infty })$ is defined by $${\mathrm{\Lambda }}_a(\varphi )(t)=\varphi (t)-\underset{s\in [0,t]}{\sup }\left[\right(phi(s)-a{)}^{+}\wedge \underset{u\in [s,t]}{\inf }\varphi (u)]$$ and ${\mathrm{\Gamma }}_0:\mathcal{D}[0,\mathrm{\infty })\to \mathcal{D}[0,\mathrm{\infty })$ is the Skorokhod map on $[0,\mathrm{\infty })$, which is given explicitly by $${\mathrm{\Gamma }}_0(\psi )(t)=\psi (t)+\underset{s\in [0,t]}{\sup }[-\psi (s){]}^{+}\mathrm{.}$$ In addition, properties of ${\mathrm{\Lambda }}_a$ are developed and comparison properties of ${\mathrm{\Gamma }}_{0,a}$ are established.