Consistency and asymptotic normality of an approximate maximum likelihood estimator for discretely observed diffusion processes

Abstract

Most often the likelihood function based on discrete observations of a diffusion process is unknown, and estimators alternative to the well-behaved maximum likelihood estimator must be found. Traditionally, such estimators are defined with origin in the theory for continuous observation of the diffusion process, and are as a consequence strongly biased unless the discrete observation time-points are close. In contrast to these estimators, an estimator based on an approximation to the (unknown) likelihood function was proposed in Pedersen (1994). We prove consistency and asymptotic normality of this estimator with no assumptions on the distance between the discrete observation time-points.