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

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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 ⊂ Rp is an unknown parameter. We consider estimation of θ when X is discretely observed at possibly non-equidistant time-points (tni)ni=0. We suppose hn := max1 ≤ i ≤ n(tni − tni − 1) → 0 and tnn → ∞ 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.

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