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 ${\Gamma }_{0,a}$ 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 \[Λ_a(ϕ)(t)=ϕ(t)-\sup_{s\in[0,t]}\biggl[\bigl(\ phi(s)-a\bigr)^+\wedge\inf_{u\in[s,t]}ϕ(u)\biggr]\] and ${\Gamma }_0:\mathcal{D}[0,\infty )\to \mathcal{D}[0,\infty )$ 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 ${\Lambda }_a$ are developed and comparison properties of ${\Gamma }_{0,a}$ are established.

Published in at http://dx.doi.org/10.1214/009117906000000890 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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