Estimating parameters of a diffusion process given continuous-time observations of the process via maximum likelihood approaches or, online, via stochastic gradient descent or Kalman filter formulations constitutes a well-established research area. It has also been established previously that these techniques are, in general, not robust to perturbations in the data in the form of temporal correlations. While the subject is relatively well understood and appropriate modifications have been suggested in the context of multi-scale diffusion processes and their reduced model equations, we consider here an alternative setting where a second-order diffusion process in positions and velocities is only observed via its positions. In this note, we propose a simple modification to standard stochastic gradient descent and Kalman filter formulations, which eliminates the arising systematic estimation biases. The modification can be extended to standard maximum likelihood approaches and avoids computation of previously proposed correction terms.

翻译：基于最大似然方法，或通过随机梯度下降或卡尔曼滤波公式在线估计扩散过程参数，给定过程的连续时间观测，构成了一个成熟的研究领域。先前也已证实，这些技术通常对数据中时间相关性形式的扰动不具有鲁棒性。尽管该主题已得到相对充分的理解，并且在多尺度扩散过程及其简化模型方程的背景下提出了适当的修正，但本文考虑另一种设定：一个在位置和速度上的二阶扩散过程仅通过其位置被观测。在本研究中，我们提出对标准随机梯度下降和卡尔曼滤波公式的简单修改，以消除由此产生的系统性估计偏差。该修改可扩展至标准最大似然方法，并避免了先前提出的修正项的计算。