Estimation for the parameter of a class of diffusion processes

Abstract

This paper is concerned with the parameter estimation problem for a stationary ergodic diffusion process with drift coefficient a(X_t, θ) and diffusion coefficient b(X_t) under the case of continuous-time observations.  Firstly, we find a closed interval on which the likelihood function is continuous and does not attain the maximum at two endpoints of this interval.  Secondly, we prove that the maximum likelihood estimator exists in the interval when the sample size is large enough.  Finally, the strong consistency of the estimator and the asymptotic normality of the error of estimation are proved.  All of the results are obtained by applying the maximal inequality for martingales, Borel-Cantelli lemma and uniform ergodic theorem.