The joint estimation of both drift and diffusion coefficient parameters is treatedunder the situation where the data are discretely observed from an ergodic diffusion processand where the statistical model may or may not include the true diffusion process.We consider the minimum contrast estimator,which is equivalent to the maximum likelihood type estimator, obtained from the contrast function based on a locally Gaussian approximation of the transition density.The asymptotic normality of the minimum contrast estimator is proved.In particular, the rate of convergence for the minimum contrast estimator of diffusion coefficient parameter in a misspecified modelis different from the one in the correctly specified parametric model.

Consider an inhomogeneous Poisson process X on [0, T] whose unknown intensity function ‘switches' from a lower function g∗ to an upper function h∗ at some unknown point θ∗. What is known are continuous bounding functions g and h such that g∗(t) ≤ g(t) ≤ h(t) ≤ h∗(t) for 0 ≤ t ≤ T. It is shown that on the basis of n observations of the process X the maximum likelihood estimate of θ∗ is consistent for n →∞, and also that converges in law and in pth moment to limits described in terms of the unknown functions g∗ and h∗.