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 $\scr{D}[0,\infty)$ of right-continuous functions with left limits taking values in ${\Bbb R}$, $\Gamma _{0,a}=\Lambda _{a}\circ \Gamma _{0}$, where $\Lambda _{a}\colon \scr{D}[0,\infty)\rightarrow \scr{D}[0,\infty)$ is defined by $\Lambda _{a}(\phi)(t)=\phi (t)-\underset s\in [0,t]\to{\text{sup}}\left[(\phi (s)-a)^{+}\wedge \underset u\in [s,t]\to{\text{inf}}\phi (u)\right]$ and $\Gamma _{0}\colon \scr{D}[0,\infty)\rightarrow \scr{D}[0,\infty)$ is the Skorokhod map on [0, ∞), which is given explicitly by $\Gamma _{0}(\psi)(t)=\psi (t)+\underset s\in [0,t]\to{\text{sup}}[-\psi (s)]^{+}$. In addition, properties of Λₐ are developed and comparison properties of $\Gamma _{0,a}$ are established.