Remarks on the Poisson additive process

Abstract: The Poisson additive process is a binary conditionally additive process such that the first is the Poisson process provided the second is given. We prove the existence and uniqueness of predictable increasing mean intensity for the Poisson additive process. Besides, we establish a likelihood ratio formula for the Poisson additive process. It directly implies there doesn't exist an anticipative Poisson additive process which is absolutely continuous with respect to the standard Poisson process, which confirms a conjecture proposed by P. Br\'emaud in his PhD thesis in 1972. When applied to the Hawkes process, it concludes that the self-exciting function is constant. Similar results are also obtained for the Wiener additive process and Markov additive process.