SIMPLE ESTIMATORS FOR PARAMETRIC MARKOVIAN TREND OF ERGODIC PROCESSES BASED ON SAMPLED DATA

摘要

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 is contained in R~p is an unknown parameter. We consider estimation of θ when X is discretely observed at possibly non-equidistant time-points (t_i~n)_i~n = o. We suppose h_n : = max_(1 ≤ i ≤ n)(t_i~n - t_(i-1)~n) → 0 and t_n~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)~(1/2)-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.