A general theory of efficient estimation for ergodic diffusion processes sampled at high frequency with an infinite time horizon is presented. High frequency sampling is common in many applications, with finance as a prominent example. The theory is formulated in term of approximate martingale estimating functions and covers a large class of estimators including most of the previously proposed estimators for diffusion processes. Easily checked conditions ensuring that an estimating function is an approximate martingale are derived, and general conditions ensuring consistency and asymptotic normality of estimators are given. Most importantly, simple conditions are given that ensure rate optimality and efficiency. Rate optimal estimators of parameters in the diffusion coefficient converge faster than estimators of drift coefficient parameters because they take advantage of the information in the quadratic variation. The conditions facilitate the choice among the multitude of estimators that have been proposed for diffusion models. Optimal martingale estimating functions in the sense of Godambe and Heyde and their high frequency approximations are, under weak conditions, shown to satisfy the conditions for rate optimality and efficiency. This provides a natural feasible method of constructing explicit rate optimal and efficient estimating functions by solving a linear equation.

翻译：本文提出了在高频采样且时间跨度无限的情况下，遍历扩散过程有效估计的一般理论。高频采样广泛应用于众多领域，其中金融领域尤为突出。该理论基于近似鞅估计函数构建，涵盖了包括以往针对扩散过程提出的大多数估计量在内的广泛估计类别。本文推导了易于验证的条件，确保估计函数为近似鞅，并给出了保证估计量一致性和渐近正态性的一般条件。最重要的是，给出了确保速率最优性和有效性的简洁条件。在扩散系数参数中，速率最优估计量的收敛速度快于漂移系数参数估计量，这是利用了二次变差中的信息。这些条件有助于从众多针对扩散模型提出的估计量中进行选择。在弱条件下，Godambe与Heyde意义下的最优鞅估计函数及其高频近似被证明满足速率最优性和有效性的条件。这提供了一种自然可行的构造显式速率最优且有效估计函数的方法，通过求解线性方程即可实现。