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On the Dynamics of Semimartingales with Two Reflecting Barriers

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  30 January 2018

Mats Pihlsgård  and
Peter W. Glynn
Mats Pihlsgård*
Affiliation:
Lund University
Peter W. Glynn*
Affiliation:
Stanford University
*
∗ Postal address: Clinical Research Centre, Lund University, Building 28, Floor 13, Jan Waldenströms gata 35, 20502 Malmö, Sweden. Email address: mats.pihlsgard@med.lu.se
∗∗ Postal address: Management Science and Engineering, Stanford University, Stanford, CA 94305-4121, USA. Email address: glynn@stanford.edu
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Abstract

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We consider a semimartingale X which is reflected at an upper barrier T and a lower barrier S, where S and T are also semimartingales such that T is bounded away from S. First, we present an explicit construction of the reflected process. Then we derive a relationship in terms of stochastic integrals linking the reflected process and the local times at the respective barriers to X, S, and T. This result reveals the fundamental structural properties of the reflection mechanism. We also present a few results showing how the general relationship simplifies under additional assumptions on X, S, and T, e.g. if we take X, S, and T to be independent martingales (which satisfy some extra technical conditions).

Keywords

Skorokhod problemreflectionsemimartingaleLévy processmartingalestochastic integration

MSC classification

Primary: 60G44: Martingales with continuous parameter 60G51: Processes with independent increments; L'evy processes 60H05: Stochastic integrals
Secondary: 60G17: Sample path properties
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 671 - 685
Copyright
© Applied Probability Trust 

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