Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-18T09:28:37.818Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Dynamics of Semimartingales with Two Reflecting Barriers

Part of: Stochastic analysis Stochastic processes

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
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Andersen, L. N. (2011). Subexponential loss rate asymptotics for Lévy processes. Math. Meth. Operat. Res. 73, 91–108.CrossRefGoogle Scholar
Andersen, L. N. and Asmussen, S. (2011). Local time asymptotics for centered Lévy processes with two-sided reflection. Stoch. Models 27, 202–219.Google Scholar
Andersen, L. N. and Mandjes, M. (2009). Structural properties of reflected Lévy processes. Queueing Systems 63, 301–322.Google Scholar
Asmussen, S. (1995). Stationary distributions via first passage times. In Advances in Queueing, ed. Dshalalow, J., CRC, Boca Raton, FL, pp. 79–102.Google Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Asmussen, S. and Pihlsgård, M. (2007). Loss rates for Lévy processes with two reflecting barriers. Math. Operat. Res. 32, 308–321.CrossRefGoogle Scholar
Bekker, R. and Zwart, B. (2005). On an equivalence between loss rates and cycle maxima in queues and dams. Prob. Eng. Inf. Sci. 19, 241–255.Google Scholar
Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.Google Scholar
Cooper, W. L., Schmidt, V. and Serfozo, R. F. (2001). Skorohod-Loynes characterizations of queueing, fluid, and inventory processes. Queueing Systems 37, 233–257.Google Scholar
D'Auria, B. and Kella, O. (2012). Markov modulation of a two-sided reflected Brownian motion with application to fluid queues. Stoch. Process. Appl. 122, 1566–1581.Google Scholar
Jelenković, P. R. (1999). Subexponential loss rates in a GI/GI/1 queue with applications. Queueing Systems 33, 91–123.CrossRefGoogle Scholar
Kella, O. (2006). Reflecting thoughts. Statist. Prob. Lett. 76, 1808–1811.Google Scholar
Kella, O., Boxma, O. and Mandjes, M. (2006). A Lévy process reflected at a Poisson age process. J. Appl. Prob. 43, 221–230.Google Scholar
Kim, H. S. and Shroff, N. B. (2001). On the asymptotic relationship between the overflow probability and the loss ratio. J. Appl. Prob. 33, 836–863.CrossRefGoogle Scholar
Kruk, L., Lehoczky, J., Ramanan, K. and Shreve, S. (2007). An explicit formula for the Skorokhod map on [0,a]. Ann. Prob. 35, 1740–1768.Google Scholar
Moran, P. A. P. (1959). The Theory of Storage. Methuen, London.Google Scholar
Palmowski, Z. and Światek, P. (2011). Loss rate for a general Lévy process with downward periodic barrier. In New Frontiers in Applied Probability (J. Appl. Prob. Spec. Vol. 48A), eds Glynn, P., Mikosch, T. and Rolski, T., Applied Probability Trust, Sheffield, pp. 99–108.Google Scholar
Pihlsgård, M. (2005). Loss rate asymptotics in a GI/G/1 queue with finite buffer. Stoch. Models 21, 913–931.Google Scholar
Protter, P. E. (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Stadje, W. (1993). A new look at the Moran dam. J. Appl. Prob. 30, 489–495.CrossRefGoogle Scholar
Zwart, A. P. (2000). A fluid queue with a finite buffer and subexponential input. Ann. Prob. 32, 221–243.Google Scholar