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The Annals of Probability

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

Lukasz Kruk, John Lehoczky, Kavita Ramanan, and Steven Shreve
Source: Ann. Probab. Volume 35, Number 5 (2007), 1740-1768.

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 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, a=Λa○Γ0, where $\Lambda_{a}\dvtx \mathcal{D}[0,\infty)\rightarrow\mathcal{D}[0,\infty )$ 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 $\Gamma_{0}\dvtx \mathcal{D}[0,\infty)\rightarrow\mathcal{D}[0,\infty)$ 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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Primary Subjects: 60G05, 60G17
Secondary Subjects: 60J60, 90B05, 90B22
Full-text: Open access
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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aop/1189000926
Digital Object Identifier: doi:10.1214/009117906000000890
Mathematical Reviews number (MathSciNet): MR2349573
Zentralblatt MATH identifier: 1139.60017

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