For point processes comprising i.i.d. copies of a multiplicative intensity process, it is shown that even though log-likelihood functions are unbounded, consistent maximum likelihood estimators of the unknown function in the stochastic intensity can be constructed using the method of sieves. Conditions are given for existence and strong and weak consistency, in the $L^1$-norm, of suitably defined maximum likelihood estimators. A theorem on local asymptotic normality of log-likelihood functions is established, and applied to show that sieve estimators satisfy the same central limit theorem as do associated martingale estimators. Examples are presented. Martingale limit theorems are a principal tool throughout.
"Maximum Likelihood Estimation in the Multiplicative Intensity Model via Sieves." Ann. Statist. 15 (2) 473 - 490, June, 1987. https://doi.org/10.1214/aos/1176350356