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Local martingales with two reflecting barriers

Part of: Stochastic processes Stochastic analysis

Published online by Cambridge University Press:  30 March 2016

Mats Pihlsgård
Mats Pihlsgård*
Affiliation:
Lund University
*
∗Postal address: Clinical Research Centre, Bd. 28, Fl. 13, Jan Valdenströms gata 35, 20502 Malmö, Sweden. Email address: [email protected]
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Abstract

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We give an account of the characteristics that result from reflecting a drifting local martingale (i.e. the sum of a local martingale and a multiple of its quadratic variation process) in 0 and b > 0. We present conditions which guarantee the existence of finite moments of what is required to keep the reflected process within its boundaries. Also, we derive an associated law of large numbers and a central limit theorem which apply when the input is continuous. Similar results for integrals of the paths of the reflected process are also presented. These results are in close agreement to what has previously been shown for Brownian motion.

Keywords

Skorokhod problemreflectionstochastic integrationBrownian motionlocal martingalesemimartingale

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 Papers
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 1062 - 1075
Copyright
Copyright © Applied Probability Trust 2015 

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