We consider a one-dimensional diffusion process X, with ergodic property, with drift b(x, è) and diffusion coef®cient a(x, ó) depending on unknown parameters è and ó. We are interested in the joint estimation of (è, ó). For that purpose, we dispose of a discretized trajectory, observed at n equidistant times t n i ih n , 1 < i < n. We assume that h n 3 0 and nh n 3 I. Under the condition nh p n 3 0 for an arbitrary integer p, we exhibit a contrast dependent on p which provides us with an asymptotically normal and ef®cient estimator of (è, ó).

A new type of martingale estimating function is proposed for inference about classes of diffusion processes based on discrete-time observations. These estimating functions can be tailored to a particular class of diffusion processes by utilizing a martingale property of the eigenfunctions of the generators of the diffusions. Optimal estimating functions in the sense of Godambe and Heyde are found. Inference based on these is invariant under transformations of data. A result on consistency and asymptotic normality of the estimators is given for ergodic diffusions. The theory is illustrated by several examples and by a simulation study.

We consider a one-dimensional diffusion process X , with ergodic property, with drift b(x, è) and diffusion coef®cient a(x, è) depending on an unknown parameter è that may be multidimensional. We are interested in the estimation of è and dispose, for that purpose, of a discretized trajectory, observed at n equidistant times t i iÄ, i 0, F F F, n. We study a particular class of estimating functions of the form f(è, X t i21 ) which, under the assumption that the integral of f with respect to the invariant measure is null, provide us with a consistent and asymptotically normal estimator. We determine the choice of f that yields the estimator with minimum asymptotic variance within the class and indicate how to construct explicit estimating functions based on the generator of the diffusion. Finally the theoretical study is completed with simulations.

We study the problem of parameter estimation for time-series possessing two, widely separated, characteristic time scales. The aim is to understand situations where it is desirable to fit a homogenized singlescale model to such multiscale data. We demonstrate, numerically and analytically, that if the data is sampled too finely then the parameter fit will fail, in that the correct parameters in the homogenized model are not identified. We also show, numerically and analytically, that if the data is subsampled at an appropriate rate then it is possible to estimate the coefficients of the homogenized model correctly.

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