Asymptotic properties of an intensity estimator of an inhomogeneous Poisson process in a combined model

A stochastic process with a drift, a diffusion, and a Poisson component is considered, where the last is an inhomogeneous process with unknown intensity $\lambda=\lambda(t)$ belonging to a compact of a Sobolev space. By observations over the process within a time interval $[0,T]$ we construct the maximum likelihood estimator (MLE) of $\lambda$. Conditions providing consistency of the estimator and asymptotic normality of the functionals of it are studied. A comparison is given of the MLEs constructed by the observations over the whole process and over its individual components.