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 $delta_{n,T} =max_{0leq i<n}(t_{i+1}-t_i )$ tends to zero with $T$ fixed. The same holds in case of an ergodic diffusion when $T$ goes to infinity in a way that $Tdelta_n$ 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.
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