Certain aspects of maximum likelihood estimation for ergodic diffusions are studied via recently developed empirical process theory for martingales. This approach enables us to remove some undesirable regularity conditions that usually appear in the statistical literature on ergodic diffusions. In particular, dimension dependent conditions for the existence of a continuous likelihood and for consistency of the maximum likelihood estimator turn out to be unnecessary.
Zanten, van, J. H. (2001). A note on consistent estimation of multivariate parameters in ergodic diffusion models. Scandinavian Journal of Statistics, 28(4), 617-623. https://doi.org/10.1111/1467-9469.00258