*J. Japan Statist. Soc.,* Vol. 33 (No. 2), pp. 157-167, 2003

## Estimation for dynamical systems with small noise from discrete observations

Masayuki Uchida

**Abstract. **
We consider an efficient estimation of an unknown parameter appearing in both the drift and the diffusion coefficient for a *d*-dimensional dynamical system with small noise. Asymptotic properties of an *M*-estimator obtained from an approximate quadratic martingale estimating function are stated. The sample path is observed at equidistant times *k/n, k=0,1, …,n*. The type of asymptotics considered is when a small dispersion parameter *e* goes to *0* and *n* goes to ∞ simultaneously.

*Key words and phrases*:
Discrete time observation, dynamical systems with small noise, *M*-estimator, parametric inference, quadratic martingale estimating function

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