On the Approximate Maximum Likelihood Estimation for Diffusion Processes

摘要: The transition density of a diffusion process does not admit an explicit expression in general, which prevents the full maximum likelihood estimation (MLE) based on discretely observed sample paths. Ait-Sahalia [J. Finance 54 (1999) 1361–1395; Econometrica 70 (2002) 223–262] proposed asymptotic expansions to densities processes, lead approximate (AMLE) for parameters. Built Ait-Sahalia’s [Econometrica 223–262; Ann. Statist. 36 (2008) 906–937] proposal and analysis AMLE, we establish consistency convergence rate reveal roles played by number terms used sampling interval between successive observations. We find conditions under AMLE has same distribution as that MLE. A first order approximation Fisher information matrix is proposed.