On minimal predictable intensity of point processes

TL;DR

The paper characterizes the minimal predictable intensity of certain point processes, showing it corresponds to standard Poisson processes under measure transformations.

Contribution

It provides a precise characterization of minimal predictable intensities for adapted point processes, linking them to measure transformations of Poisson processes.

Findings

Minimal predictable intensity exists if and only if the process is a standard Poisson process under an absolutely continuous measure change.

The characterization applies to adapted, right-continuous, non-decreasing, integer-valued processes with unit jumps.

This result bridges measure transformation techniques with the structure of point process intensities.

Abstract

An adapted, right-continuous, non-decreasing, integer-valued process with unit jumps and starting at zero has a minimal predictable intensity if and only if it is a standard Poisson process under an absolutely continuous transformation of measures.