The Annals of Probability

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

Lukasz Kruk, John Lehoczky, Kavita Ramanan, and Steven Shreve

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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, aa○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.

Article information

Source
Ann. Probab., Volume 35, Number 5 (2007), 1740-1768.

Dates
First available in Project Euclid: 5 September 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1189000926

Digital Object Identifier
doi:10.1214/009117906000000890

Mathematical Reviews number (MathSciNet)
MR2349573

Zentralblatt MATH identifier
1139.60017

Subjects
Primary: 60G05: Foundations of stochastic processes 60G17: Sample path properties
Secondary: 60J60: Diffusion processes [See also 58J65] 90B05: Inventory, storage, reservoirs 90B22: Queues and service [See also 60K25, 68M20]

Keywords
Skorokhod map reflection map double-sided reflection map comparison principle reflecting Brownian motion

Citation

Kruk, Lukasz; Lehoczky, John; Ramanan, Kavita; Shreve, Steven. An explicit formula for the Skorokhod map on [0, a ]. Ann. Probab. 35 (2007), no. 5, 1740--1768. doi:10.1214/009117906000000890. https://projecteuclid.org/euclid.aop/1189000926


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