On level crossings for a general class of piecewise-deterministic Markov processes
Abstract
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 (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 (b). While our proof of the limit theorem requires additional assumptions, Rice's formula holds whenever the (stationary) overall intensity of jumps is finite.