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

TL;DR

This paper analyzes the level crossing behavior of a broad class of piecewise-deterministic Markov processes, establishing a convergence to a compound Poisson process and deriving a version of Rice's formula relating stationary density to crossing intensities.

Contribution

It introduces a limit theorem for level crossings of PDMPs under scaling and extends Rice's formula to these processes, providing new theoretical insights.

Findings

Point process of level crossings converges to a geometrically compound Poisson process

Rice's formula relates stationary density to crossing intensities

Scaling factor $ u(b)$ is interpreted through the formula

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 $\nu(b)$, 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 $\nu(b)$. While our proof of the limit theorem requires additional assumptions, Rice's formula holds whenever the (stationary) overall intensity of jumps is finite.