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

Abstract

We consider a piecewise-deterministic Markov process (X1) 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 N+b of upcrossings of some level b by (X1). We prove a version of Rice's formula relating the stationary density of (X1) to level crossing intensities and show that, for a wide class of processes (X1), as b → ∞, the scaled point process(N+b (v+(b)-1t)), where v+(b) denotes the intensity of upcrossings of b, converges weakly to a geometrically compound Poisson process. © Applied Probability Trust 2008.