2023/08/09 by Dirk Becherer, Becherer, Dirk, Thomas Bernhardt +3
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2308.05080
openalex publication_date 2023/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.