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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.

References

Applebaum, D. (2004). Lévy Processes and Stochastic Calculus. Cambridge University Press.Google Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Asmussen, S. and Kella, O. (2000). A multi-dimensional martingale for Markov additive processes and its applications. Adv. Appl. Prob. 32, 376–393.Google Scholar
Asmussen, S. and Kella, O. (2001). On optional stopping of some exponential martingales for Lévy processes with or without reflection. Stoch. Process. Appl. 91, 47–55.Google Scholar
Boxma, O. J., Ivanovs, J., Kosiński, K. M. and Mandjes, M. R. H. (2011). Lévy-driven polling systems and continuous-state branching processes. Stoch. Systems 1, 411–436.Google Scholar
Kella, O. (1998). An exhaustive Lévy storage process with intermittent output. Commun. Statist. Stoch. Models 14, 979–992.Google Scholar
Kella, O. (2006). Reflecting thoughts. Statist. Prob. Lett. 76, 1808–1811.Google Scholar
Kella, O. and Whitt, W. (1991). Queues with server vacations and Lévy processes with secondary Jump input. Ann. Appl. Prob. 1, 104–117.Google Scholar
Kella, O. and Whitt, W. (1992). Useful martingales for stochastic storage processes with Lévy input. J. Appl. Prob. 29, 396–403.Google Scholar
Kella, O. and Whitt, W. (1996). Stability and structural properties of stochastic storage networks. J. Appl. Prob. 33, 1169–1180.Google Scholar
Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
Melamed, B. and Whitt, W. (1990). On arrivals that see time averages: a martingale approach. J. Appl. Prob. 27, 376–384.Google Scholar
Nguyen-Ngoc, L. and Yor, M. (2005). Some martingales associated to reflected Lévy processes. In Séminaire de Probabilités XXXVIII (Lecture Notes Math. 1857), Springer, Berlin, pp. 42–69.Google Scholar
Protter, P. E. (2005). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.Google Scholar
Rüdiger, B. (2004). Stochastic integration with respect to compensated Poisson random measures on separable Banach spaces. Stoch. Stoch. Reports 76, 213–242.Google Scholar