*J. Japan Statist. Soc.,* Vol. 35 (No. 2), pp. 147-170, 2005

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

Hiroki Masuda

**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
(*t*_{i}^{n})_{i}^{n}=_{0}. We suppose *h*_{n} := max_{1≤i≤ n}(*t*_{i}^{n} - *t*_{i}^{n}_{-1}) → 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 -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.

*Key words and phrases*:
Discrete sampling, parametric estimation, stochastic differential equation, trajectory-fitting.

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