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

Abstract

We consider a piecewise-deterministic Markov process (Xt) 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. The paper deals with the point processof upcrossings of some levelbby (Xt). We prove a version of Rice's formula relating the stationary density of (Xt) to level crossing intensities and show that, for a wide class of processes (Xt), asb→ ∞, the scaled point processwhere ν+(b) denotes the intensity of upcrossings ofb, converges weakly to a geometrically compound Poisson process.