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Useful Martingales for Stochastic Storage Processes with Lévy-Type Input

Part of: Stochastic analysis Special processes

Published online by Cambridge University Press:  30 January 2018

Offer Kella  and
Onno Boxma
Offer Kella*
Affiliation:
The Hebrew University of Jerusalem
Onno Boxma*
Affiliation:
EURANDOM and Eindhoven University of Technology
*
∗ Postal address: Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel. Email address: offer.kella@huji.ac.il
∗∗ Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: boxma@win.tue.nl
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Abstract

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In this paper we generalize the martingale of Kella and Whitt to the setting of Lévy-type processes and show that the (local) martingales obtained are in fact square-integrable martingales which upon dividing by the time index converge to zero almost surely and in L2. The reflected Lévy-type process is considered as an example.

Keywords

Lévy-type processLévy storage systemKella-Whitt martingale

MSC classification

Primary: 60K25: Queueing theory 60K37: Processes in random environments 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 2 , June 2013 , pp. 439 - 449
Copyright
© Applied Probability Trust 

Footnotes

Supported in part by grant 434/09 from the Israel Science Foundation, the Vigevani Chair in Statistics, and visitor grant no. 040.11.257 from The Netherlands Organisation for Scientific Research.

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