  An approximate maximum likelihood method of estimation of diffusion
parameters $(\vartheta,\sigma)$ based on discrete observations of a diffusion
$X$ along fixed time-interval $[0,T]$ and Euler approximation of integrals is
analyzed. We assume that $X$ satisfies a SDE of form $dX_t =\mu (X_t ,\vartheta
)\, dt+\sqrt{\sigma} b(X_t )\, dW_t$, with non-random initial condition. SDE is
nonlinear in $\vartheta$ generally. Based on assumption that maximum likelihood
estimator $\hat{\vartheta}_T$ of the drift parameter based on continuous
observation of a path over $[0,T]$ exists we prove that measurable estimator
$(\hat{\vartheta}_{n,T},\hat{\sigma}_{n,T})$ of the parameters obtained from
discrete observations of $X$ along $[0,T]$ by maximization of the approximate
log-likelihood function exists, $\hat{\sigma}_{n,T}$ being consistent and
asymptotically normal, and $\hat{\vartheta}_{n,T}-\hat{\vartheta}_T$ tends to
zero with rate $\sqrt{\delta}_{n,T}$ in probability when $\delta_{n,T}
=\max_{0\leq i<n}(t_{i+1}-t_i )$ tends to zero with $T$ fixed. The same holds
in case of an ergodic diffusion when $T$ goes to infinity in a way that
$T\delta_n$ goes to zero with equidistant sampling, and we applied these to
show consistency and asymptotical normality of $\hat{\vartheta}_{n,T}$,
$\hat{\sigma}_{n,T}$ and asymptotic efficiency of $\hat{\vartheta}_{n,T}$ in
this case.
