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On Rice's Formula for Stationary Multivariate Piecewise Smooth Processes

Part of:Markov processesStochastic 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 = {X t : t ≥ 0} be a stationary piecewise continuous R d -valued process that moves between jumps along the integral curves of a given continuous vector field, and let S ⊂ R d 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 X 0. 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

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 2 , June 2012, pp. 351 - 363
Copyright
© Applied Probability Trust 

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