Open Access
September 2007 An explicit formula for the Skorokhod map on [0, a]
Lukasz Kruk, John Lehoczky, Kavita Ramanan, Steven Shreve
Ann. Probab. 35(5): 1740-1768 (September 2007). DOI: 10.1214/009117906000000890


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 Γ0, 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, Γ0, aa○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 Λa are developed and comparison properties of Γ0, a are established.


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Lukasz Kruk. John Lehoczky. Kavita Ramanan. Steven Shreve. "An explicit formula for the Skorokhod map on [0, a]." Ann. Probab. 35 (5) 1740 - 1768, September 2007.


Published: September 2007
First available in Project Euclid: 5 September 2007

zbMATH: 1139.60017
MathSciNet: MR2349573
Digital Object Identifier: 10.1214/009117906000000890

Primary: 60G05 , 60G17
Secondary: 60J60 , 90B05 , 90B22

Keywords: Comparison principle , double-sided reflection map , Reflecting Brownian motion , reflection map , Skorokhod map

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 5 • September 2007
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