For a diffusion process \(X_t=a(X_t,\vartheta)dw_t+b(X_t,\vartheta)dt\) observed at points \(t_i=i\Delta\), \(i=0,\dots,n-1\), an estimator \(\hat\vartheta_n\) of \(\vartheta\) is considered which is a solution of the equation \(F(\hat\vartheta_n)=\sum_{i=0}^{n-1} f(\hat\vartheta_n,X_{t_i})=0\). Conditions of asymptotic normality of \(\hat\vartheta_n\) and its asymptotic variance are obtained in terms of the invariant measure of \(X_t\) and the joint distribution of \((X_0,X_\Delta)\). For one-dimensional \(\vartheta\) the author describes the functions \(f\) which provide the minimal asymptotic variance. E.g., for the Ornstein-Uhlenbeck process the best estimating function is \(F^*(\vartheta)=\sum_{i=0}^{n-1}(2\vartheta X_{t_i}^2-1)\). As an explicit estimator of \(\vartheta\) the author considers \[ \tilde\vartheta_n=-\sum_{i=0}^{n-1} a^2(X_{t_i})h''(X_{t_i}) (2\sum_{i=0}^{n-1} b(X_{t_i})h'(X_{t_i}))^{-1}, \] where \(h\) is any function which satisfies some mild conditions. The estimators \(\hat\vartheta_n\) and \(\tilde\vartheta_n\) are compared with the best possible ML estimator (which is difficult to compute).
The behavior of the proposed estimators is good for the generalized Cox-Ingersoll-Ross process \(dX_t=(\alpha+\vartheta X_t)dt+\sigma X_t^\gamma dw_t\), but for the process \(dX_t=-\vartheta X_tdt+\sqrt{\vartheta+X_t^2}dw_t\) it is bad.