Estimating a class of diffusions from discrete observations via approximate maximum likelihood method

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.