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

Abstract

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

Rice formula for processes with jumps and applications

“…Borovkov and Last in [8] are interested in the continuous crossings through a level u by a stationary Piecewise Deterministic Markov Process. A process X of this kind, starts at a random position, then jumps a random quantity at random times but moves deterministically between jumps.…”

“…The case of discontinuous processes has been treated only recently in the literature. Borovkov and Last [8,9] consider the number of continuous crossings of a discontinuous process with random jumps and deterministic evolution between jumps. The continuous and discontinuous parts of the process are not independent, hence, our main result does not apply.…”

From rainfed agriculture to stress-avoidance irrigation: I. A generalized irrigation scheme with stochastic soil moisture

“…We focus here on the frequency of excursions of the soil moisture process below and above a generic threshold n, i.e., on the frequencies of downcrossing, m # n , and upcrossing, m " n . Because irrigation does not alter the crossing during a soil moisture dry down, the frequency of downcrossings of a generic threshold n Ps is as in [40,62] m # n ¼ qðnÞpðnÞ:…”

A Lévy input model with additional state-dependent services

“…When X is a compound Poisson process with negative drift p, for the process V with absolutely continuous stationary distribution one can identify Rice's formula relating the density of V (∞) with the intensity of up-and down crossings of a fixed level; for details see [8,35].…”