On level crossings for a general class of piecewise-deterministic Markov processes

Details On level crossings for a general class of piecewise-deterministic Markov processes On level crossings for a general class of piecewise-deterministic Markov processes

2007-05-14

arxiv.org/abs/0705.1863v1

J.Appl.Probab. 40 (2008) 815-834

Mathematics

Probability

25 pages

Scientific paper

We consider a piecewise-deterministic Markov process governed by a jump intensity function, a rate function that determines the behaviour between jumps, and a stochastic kernel describing the conditional distribution of jump sizes. We study the point process of upcrossings of a level $b$ by the Markov process. Our main result shows that, under a suitable scaling $\nu(b)$, the point process converges, as $b$ tends to infinity, weakly to a geometrically compound Poisson process. We also prove a version of Rice's formula relating the stationary density of the process to level crossing intensities. This formula provides an interpretation of the scaling factor $\nu(b)$. While our proof of the limit theorem requires additional assumptions, Rice's formula holds whenever the (stationary) overall intensity of jumps is finite.

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