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Let X be a stochastic process obeying a stochastic differential equation of the form dXt = b(Xt ,θ )dt+ dYt, where Y is an adapted driving process possibly depending on X’s past history, and θ ∈ Θ ⊂ R p is an unknown parameter. We consider estimation of θ when X is discretely observed at possibly non-equidistant time-points (t n ) n=0. We suppose hn := max1≤i≤n(t n − t n−1) → 0 and t n →∞ as n →∞ : the data becomes more high-frequency as its size increases. Under some regularity conditions including the ergodicity of X, we obtain √ nhn-consistency of trajectory-fitting estimate as well as least-squares estimate, without identifying Y . Also shown is that some additional conditions, which requires Y ’s structure to some extent, lead to asymptotic normality. In particular, a Wiener-Poisson-driven setup is discussed as an important special case.

Simple Estimators for Parametric Markovian Trend of Ergodic Processes Based on Sampled Data