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

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 process of upcrossings of some level b by (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 ), as b → ∞, the scaled point process where ν+(b) denotes the intensity of upcrossings of b, converges weakly to a geometrically compound Poisson process.