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On the Dynamics of Semimartingales with Two Reflecting Barriers

  • Mats Pihlsgård (a1) and Peter W. Glynn (a2)

Abstract

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).

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Copyright

Corresponding author

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

References

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[1] Andersen, L. N. (2011). Subexponential loss rate asymptotics for Lévy processes. Math. Meth. Operat. Res. 73, 91108.
[2] Andersen, L. N. and Asmussen, S. (2011). Local time asymptotics for centered Lévy processes with two-sided reflection. Stoch. Models 27, 202219.
[3] Andersen, L. N. and Mandjes, M. (2009). Structural properties of reflected Lévy processes. Queueing Systems 63, 301322.
[4] Asmussen, S. (1995). Stationary distributions via first passage times. In Advances in Queueing, ed. Dshalalow, J., CRC, Boca Raton, FL, pp. 79102.
[5] Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.
[6] Asmussen, S. and Pihlsgård, M. (2007). Loss rates for Lévy processes with two reflecting barriers. Math. Operat. Res. 32, 308321.
[7] Bekker, R. and Zwart, B. (2005). On an equivalence between loss rates and cycle maxima in queues and dams. Prob. Eng. Inf. Sci. 19, 241255.
[8] Billingsley, P. (1999). Convergence of Probability Measures, 2nd edn. John Wiley, New York.
[9] Cooper, W. L., Schmidt, V. and Serfozo, R. F. (2001). Skorohod-Loynes characterizations of queueing, fluid, and inventory processes. Queueing Systems 37, 233257.
[10] 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, 15661581.
[11] Jelenković, P. R. (1999). Subexponential loss rates in a GI/GI/1 queue with applications. Queueing Systems 33, 91123.
[12] Kella, O. (2006). Reflecting thoughts. Statist. Prob. Lett. 76, 18081811.
[13] Kella, O., Boxma, O. and Mandjes, M. (2006). A Lévy process reflected at a Poisson age process. J. Appl. Prob. 43, 221230.
[14] Kim, H. S. and Shroff, N. B. (2001). On the asymptotic relationship between the overflow probability and the loss ratio. J. Appl. Prob. 33, 836863.
[15] Kruk, L., Lehoczky, J., Ramanan, K. and Shreve, S. (2007). An explicit formula for the Skorokhod map on [0,a]. Ann. Prob. 35, 17401768.
[16] Moran, P. A. P. (1959). The Theory of Storage. Methuen, London.
[17] 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. 99108.
[18] Pihlsgård, M. (2005). Loss rate asymptotics in a GI/G/1 queue with finite buffer. Stoch. Models 21, 913931.
[19] Protter, P. E. (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.
[20] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
[21] Stadje, W. (1993). A new look at the Moran dam. J. Appl. Prob. 30, 489495.
[22] Zwart, A. P. (2000). A fluid queue with a finite buffer and subexponential input. Ann. Prob. 32, 221243.

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On the Dynamics of Semimartingales with Two Reflecting Barriers

  • Mats Pihlsgård (a1) and Peter W. Glynn (a2)

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