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Local martingales with two reflecting barriers

Part of: Stochastic analysis Stochastic processes

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 

References

[1] Andersen, L. N. (2011). Subexponential loss rate asymptotics for Lévy processes. Math. Meth. Operat. Res. 1, 91–108.Google Scholar
[2] Andersen, L. N. and Asmussen, S. (2011). Local time asymptotics for centered Lévy processes with two-sided reflection. Stoch. Models 27, 202–219.CrossRefGoogle Scholar
[3] Andersen, L. N. and Mandjes, M. (2009). Structural properties of reflected Lévy processes. Queueing Systems 63, 301–322.CrossRefGoogle Scholar
[4] Anscombe, F. J. (1952). Large-sample theory of sequential estimation. Proc. Camb. Phil. Soc. 48, 600–607.Google Scholar
[5] Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
[6] Asmussen, S. and PihlsgåRd, M. (2007). Loss rates for Lévy processes with two reflecting barriers. Math. Operat. Res. 32, 308–321.CrossRefGoogle Scholar
[7] Cooper, W. L., Schmidt, V. and Serfozo, R. F. (2001). Skorohod-Loynes characterizations of queueing, fluid, and inventory processes. Queueing Systems 37, 233–257.CrossRefGoogle Scholar
[8] 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
[9] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York.CrossRefGoogle Scholar
[10] Friedman, A. (1982). Foundations of Modern Analysis. Dover, New York.Google Scholar
[11] Kella, O. (2006). Reflecting thoughts. Statist. Prob. Lett. 76, 1808–1811.CrossRefGoogle Scholar
[12] Kella, O., Boxma, O. and Mandjes, M. (2006). A Lévy process reflected at a Poisson age process. J. Appl. Prob. 43, 221–230.CrossRefGoogle Scholar
[13] Kim, H. S. and Shroff, N. B. (2001). On the asymptotic relationship between the overflow probability and the loss ratio. Adv. Appl. Prob. 33, 836–863.Google Scholar
[14] 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
[15] Palmowski, Z. and Świa̧tek, 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), Applied Probability Trust, Sheffield, pp. 99–108.Google Scholar
[16] PihlsgåRd, M. (2005). Loss rate asymptotics in a GI/G/1 queue with finite buffer. Stoch. Models 21, 913–931.CrossRefGoogle Scholar
[17] PihlsgåRd, M. and Glynn, P. W. (2013). On the dynamics of semimartingales with two reflecting barriers. J. Appl. Prob. 50, 671–685.Google Scholar
[18] Protter, P. E. (2004). Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin.Google Scholar
[19] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.CrossRefGoogle Scholar
[20] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
[21] Stadje, W. (1993). A new look at the Moran dam. J. Appl. Prob. 30, 489–495.Google Scholar
[22] Zhang, X. and Glynn, P. W. (2011). On the dynamics of a finite buffer queue conditioned on the amount of loss. Queueing Systems 67, 91–110.Google Scholar