Rice Formula for processes with jumps and applications

TL;DR

This paper extends the Rice Formula to processes combining smooth and jump components, providing explicit crossing counts and tail distribution insights, with applications to non-stationary piecewise deterministic Markov processes.

Contribution

It introduces a generalized Rice Formula for processes with jumps and smooth parts, including explicit crossing calculations and tail distribution analysis.

Findings

Derived formulas for mean number of crossings (continuous and discontinuous)

Explicit examples computing crossing counts for specific processes

Asymptotic analysis of the maximum distribution tail

Abstract

We extend Rice Formula to a process which is the sum of two independent processes: a smooth process and a pure jump process with finitely many jumps. Formulas for the mean number of both continuous and discontinuous crossings through a fixed level on a compact time interval are obtained. We present examples in which we compute explicitly the mean number of crossings and compare which kind of crossings dominate for high levels. In one of the examples the leading term of the tail of the distribution function of the maximum of the process over a compact time interval as the level goes to infinity is obtained. We end giving a generalization, to the non-stationary case, of Borovkov-Last Rice Formula for Piecewise Deterministic Markov Processes.