On the LAMN property for continuous observations of some diffusion processes with jumps

In this paper, we consider a diffusion process with jumps whose drift and jump coefficient depend on an unknown parameter. We then give a self-contained proof of the local asymptotic mixed normality (LAMN) property when the process is observed continuously in a time interval $[0; T]$ as $T\to\infty$, and derive, as a consequence, the local asymptotic normality (LAN) property in the ergodic case. For this, we give a proof of a Girsanov's theorem and a Central Limit theorem for a pure jump martingale. Our results could be viewed as a consequence of the LAMN property for semimartingales proved by Luschgy [15], using the Girsanov's theorem for semimartingales obtained in Jacod and Shiryaev [9], and the Central Limit theorem for semimartingales established by S{\o}rensen [21] and Feigin [3]. The aim of this paper is to present a proof of these results without using this abstract semimartingale theory but integral equations with respect to Poisson random measures.