Summary: First we consider a process given by a SDE
with parameter , where and is a standard Wiener process. We study the asymptotic behavior of the MLE of based on the observation as . We formulate sufficient conditions under which converges to the distribution of , where denotes the Fisher information for contained in the sample is a standard Wiener process, and or . We also weaken the sufficient conditions due to H. Luschgy [Probab. Theory Relat. Fields 92, No. 2, 151–176 (1992; Zbl 0768.62067), Section 4.2)] under which converges to a Cauchy distribution. Furthermore, we give sufficient conditions so that the MLE of is asymptotically normal with some appropriate random normalizing factor. Next we study a SDE
with a perturbed drift satisfying with some . We give again sufficient conditions under which converges to the distribution of . We emphasize that our results are valid in both cases and , and we develop a unified approach to handle these cases.