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

Abstract

Let X be a stochastic process obeying a stochastic differential equation of the form dX_t = b(X_t, θ)dt + dY_t, 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 (tⁿ_i)ⁿ_i=0. We suppose h_n := max_{1 ≤ i ≤ n}(tⁿ_i − tⁿ_{i − 1}) → 0 and tⁿ_n → ∞ as n → ∞: the data becomes more high-frequency as its size increases. Under some regularity conditions including the ergodicity of X, we obtain √nh_n-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.