## Journal of Applied Probability

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

### On Rice's formula for stationary multivariate piecewise smooth processes

K. Borovkov and G. Last

#### Abstract

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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 16 June 2012

**Permanent link to this document**

http://projecteuclid.org/euclid.jap/1339878791

**Digital Object Identifier**

doi:10.1239/jap/1339878791

**Mathematical Reviews number (MathSciNet)**

MR2977800

**Zentralblatt MATH identifier**

1279.60112

**Subjects**

Primary: 60J75: Jump processes

Secondary: 60G55: Point processes

**Keywords**

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

#### Citation

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. http://projecteuclid.org/euclid.jap/1339878791.