Journal of Applied Probability

On Rice's formula for stationary multivariate piecewise smooth processes

K. Borovkov and G. Last

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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 SRd 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.

Article information

J. Appl. Probab. Volume 49, Number 2 (2012), 351-363.

First available in Project Euclid: 16 June 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J75: Jump processes
Secondary: 60G55: Point processes

Level crossing Rice's formula stationarity Palm probability piecewise-deterministic process stochastic network


Borovkov, K.; Last, G. On Rice's formula for stationary multivariate piecewise smooth processes. J. Appl. Probab. 49 (2012), no. 2, 351--363. doi:10.1239/jap/1339878791.

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