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

Abstract

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 ℝ, Γ_0, a=Λ_a○Γ₀, where $\Lambda_{a}: \mathcal{D}[0,\infty)\rightarrow\mathcal{D}[0,\infty )$ 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 $\Gamma_{0}: \mathcal{D}[0,\infty)\rightarrow\mathcal{D}[0,\infty)$ is the Skorokhod map on [0, ∞), which is given explicitly by $$\Gamma_{0}(\psi)(t)=\psi(t)+\sup_{s\in[0,t]}[-\psi(s)]^{+}.$$ In addition, properties of Λ_a are developed and comparison properties of Γ_0, a are established.