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On Rice's Formula for Stationary Multivariate Piecewise Smooth Processes

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  04 February 2016

K. Borovkov  and
G. Last
K. Borovkov*
Affiliation:
University of Melbourne
G. Last*
Affiliation:
Karlsruher Institut für Technologie
*
∗ Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia. Email address: k.borovkov@ms.unimelb.edu.au
∗∗ Postal address: Institut für Stochastik, Karlsruher Institut für Technologie, D-76128 Karlsruhe, Germany. Email address: guenter.last@kit.edu
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Abstract

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Let X = {Xt: t ≥ 0} be a stationary piecewise continuous Rd-valued process that moves between jumps along the integral curves of a given continuous vector field, and let S ⊂ Rd be a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the intensity of the point process of (localized) continuous crossings of S by X to the distribution of X0. Our result is illustrated by examples relating to queueing networks and stress release network models.

Keywords

Level crossingRice's formulastationarityPalm probabilitypiecewise-deterministic processstochastic network

MSC classification

Primary: 60J75: Jump processes
Secondary: 60G55: Point processes
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 2 , June 2012 , pp. 351 - 363
Copyright
© Applied Probability Trust 

References

Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer, New York.Google Scholar
Azaı¨s, J.-M. and Wschebor, M. (2009). Level Sets and Extrema of Random Processes and Fields. John Wiley, Hoboken, NJ.Google Scholar
Baccelli, F. and Brémaud, P. (1994). Elements of Queueing Theory. Springer, Berlin.Google Scholar
Bar-David, I. and Nemirovsky, A. (1972). Level crossings of nondifferentiable shot processes. IEEE Trans. Inf. Theory 18, 27–34.Google Scholar
Borovkov, K. and Bebbington, M. S. (2003). A stochastic two-node stress transfer model reproducing Omori's law. Pure Appl. Geophys. 160, 1429–1445.Google Scholar
Borovkov, K. and Last, G. (2008). On level crossings for a general class of piecewise-deterministic Markov processes. Adv. Appl. Prob. 40, 815–834.Google Scholar
Borovkov, K. and Vere-Jones, D. (2000). Explicit formulae for stationary distributions of stress release processes. J. Appl. Prob. 37, 315–321.Google Scholar
Hartman, P. (2002). Ordinary Differential Equations, 2nd edn. SIAM, Philadelphia, PA.CrossRefGoogle Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.Google Scholar
Leadbetter, M. R. and Spaniolo, G. V. (2004). Reflections on Rice's formulae for level crossings—history, extensions and use. Austral. N. Z. J. Statist. 46, 173–180.Google Scholar
Rice, S. O. (1944). Mathematical analysis of random noise. Bell System Tech. J. 24, 282–332.CrossRefGoogle Scholar
Rychlik, I. (2000). On some reliability applications of Rice's formula for the intensity of level crossings. Extremes 3, 331–348.Google Scholar
Zähle, U. (1984). A general Rice formula, Palm measures, and horizontal-window conditioning for random fields. Stoch. Process. Appl. 17, 265–283.Google Scholar