On Watanabe's characterisation and change of intensity \`{a} la Girsanov for Cox processes

TL;DR

This paper explores the theoretical foundations of Cox processes, establishing the equivalence of various definitions, and investigates conditions under which measure changes can modify their intensities using Girsanov's theorem.

Contribution

It provides a necessary and sufficient condition for Watanabe's characterization and details when measure changes can alter Cox process intensities via stochastic exponentials.

Findings

Watanabe's characterization is necessary and sufficient.

Measure change via Girsanov is possible if the target intensity is a product of the original and a non-negative process.

Conditions for the stochastic exponential to be a martingale are established.

Abstract

We discuss the equivalence of definitions for conditional Poisson processes, Cox processes, and stochastic intensities of point processes on the real line. We show that Watanabe's characterisation of conditional Poisson processes in terms of local martingales is necessary and sufficient. Additionally, we consider conditions enabling the measure change method a la Girsanov to alter the intensity of Cox processes to a desired new target intensity, e.g. for the probability reference approach in filtering. Such a measure change exists if a corresponding stochastic exponential is a proper martingale. We show that this holds if the new locally integrable target intensity is the product of the original intensity and another non-negative process.