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.

翻译：泊松加性过程是一种二元条件加性过程，使得在给定第二个过程时，第一个过程为泊松过程。我们证明了泊松加性过程可预测递增均值强度的存在性与唯一性。此外，我们建立了泊松加性过程的似然比公式。该公式直接表明不存在与标准泊松过程绝对连续的可预期泊松加性过程，这证实了P. Brémaud在其1972年博士论文中提出的猜想。当应用于霍克斯过程时，该结论表明自激励函数为常数。类似结果也在维纳加性过程与马尔可夫加性过程中得到。