On quasi likelihood for semimartingales

Abstract

A new general approach to constructing a quasi score function for a class of stochastic processes is proposed. A crucial point in the construction is the separate treatment of the continuous martingale part and the purely discontinuous martingale part. The proposed estimating function fits into the general quasi likelihood framework given by Godambe and Heyde (1987) and is shown to be optimal within the class of essentially all martingale estimating functions according to these authors fixed sample criterion as well as their asymptotic criterion. Relations to other work on quasi likelihood for stochastic processes are discussed. The theory is illustrated by examples.