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The transition density of a diffusion process does not admit an explicitexpression in general, which prevents the full maximum likelihood estimation(MLE) based on discretely observed sample paths. A\"{\i}t-Sahalia [J. Finance54 (1999) 1361--1395; Econometrica 70 (2002) 223--262] proposed asymptoticexpansions to the transition densities of diffusion processes, which lead to anapproximate maximum likelihood estimation (AMLE) for parameters. Built onA\"{\i}t-Sahalia's [Econometrica 70 (2002) 223--262; Ann. Statist. 36 (2008)906--937] proposal and analysis on the AMLE, we establish the consistency andconvergence rate of the AMLE, which reveal the roles played by the number ofterms used in the asymptotic density expansions and the sampling intervalbetween successive observations. We find conditions under which the AMLE hasthe same asymptotic distribution as that of the full MLE. A first orderapproximation to the Fisher information matrix is proposed.

On the approximate maximum likelihood estimation for diffusion processes