A new estimating function for discretely sampled diffusions

Abstract

This paper shows that discretization after the application of Itô formula in the Girsanov likelihood produces estimators of the drift which have faster rates of convergence than the Euler estimator for stationary ergodic diffusions and is free of approximating the stochastic integral. The discretization schemes are related to the Hausdorff moment problem. Interalia I introduce a new stochastic integral which will be of independent interest. I show strong consistency, asymptotic normality and a Berry-Esseen bound for the corresponding approximate maximum likelihood estimators of the drift parameter from high frequency data observed over a long time period.