Maximum likelihood estimation for the drift parameter in diffusion processes

In this paper, parameter estimation for a stationary ergodic diffusion process with drift coefficient and diffusion coefficient is investigated. The likelihood function is given based on the Girsanov theorem and the existence of the maximum likelihood estimator is proved by applying the uniform ergodic theorem, Borel-Cantelli lemma and Rolle’s theorem. The consistency in probability of the estimator and asymptotic normality of the error of estimation are proved with the help of the uniform ergodic theorem, maximal inequality for martingale and Borel-Cantelli lemma.