Large deviations for the Ornstein–Uhlenbeck process without tears

Our goal is to establish large deviations for the maximum likelihood estimator of the drift parameter of the Ornstein–Uhlenbeck process without tears. We propose a new strategy to establish large deviation results which allows us, via a suitable transformation, to circumvent the classical difficulty of non-steepness. Our approach holds in the stable case where the process is positive recurrent as well as in the unstable and explosive cases where the process is respectively null recurrent and transient. It can also be successfully implemented for more complex diffusion processes.