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 $Γ_{0,a}$ 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 $\mathbb{R}$, $Γ_{0,a}=Λ_a\circ Γ_0$, where $Λ_a:\mathcal{D}[0,\infty)\to\mathcal{D}[0,\infty)$ is defined by $Λ_a(ϕ)(t)=ϕ(t)-\sup_{s\in[0,t]}\biggl[\bigl(\ phi(s)-a\bigr)^+\wedge\inf_{u\in[s,t]}ϕ(u)\biggr]$ and $Γ_0:\mathcal{D}[0,\infty)\to\mathcal{D}[0,\infty)$ is the Skorokhod map on $[0,\infty)$, which is given explicitly by $Γ_0(ψ)(t)=ψ(t)+\sup_{s\in[0,t]}[-ψ(s)]^+.$ In addition, properties of $Λ_a$ are developed and comparison properties of $Γ_{0,a}$ are established.

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