Martingale estimating functions for Feller diffusion processes generated by degenerate elliptic operators

We consider the class of diffusion processes governed by degenerate elliptic operators of the type A_θ u = x^2 u'' + θxu', with θ unknown real parameter, acting on the space C[0,+∞] of all real valued continuous functions on [0,+∞) which admit finite limits at 0 and at ∞. After showing that the operator Aθ with a suitable domain generates an analytic Feller semigroup on C[0,+∞], we face the statistical problem of constructing optimal (in the asymptotic sense) estimators of the parameter θ by applying and comparing two recent methods based on determining martingale estimating functions when a sample of discrete observations of the diffusion is available. We prove that the estimators coming from the estimating functions considered are consistent and asymptotically normal. Finally, these asymptotic properties are discussed by means of a simulation study of the qualitative behaviour of the estimators themselves.