Estimating Functions for Discretely Observed Diffusions: A Review

Several estimating functions for discretely observed diffusion processes are reviewed. First we discuss simple explicit estimating functions based on Gaussian approximations to the transition density. The corresponding estimators often have considerable bias, a problem that can be avoided by using martingale estimating functions. These, on the other hand, are rarely explicit and therefore often require a considerable computational effort. We review results on how to choose an optimal martingale estimating function and on asymptotic properties of the estimators. Martingale estimating functions based on polynomials of the increments of the observed process or on eigenfunctions for the generator of the diffusion model are considered in more detail. The theory is illustrated by examples. In particular, the Cox-Ingersoll-Ross model is considered.