We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coefficient of the homogenized equation requires pre-processing of the data, often in the form of subsampling; this is because the two-scale equation and the homogenized single-scale equation are incompatible at small scales, generating mutually singular measures on the path space. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel function, and compute maximum likelihood estimators based on the filtered process. We show that the estimators we propose are asymptotically unbiased and demonstrate numerically the advantages of our method with respect to subsampling. Finally, we show how our filtered data methodology can be combined with Bayesian techniques and provide a full uncertainty quantification of the inference procedure.

我们研究了两尺度连续时间序列的漂移估计问题。我们将自己置于过阻尼 Langevin 方程的框架中，其中存在单尺度替代均质方程。在这种情况下，估计均质方程的漂移系数需要对数据进行预处理，通常采用二次采样的形式；这是因为二尺度方程和同质化的单尺度方程在小尺度上不兼容，在路径空间上产生相互奇异的测度。我们避免二次采样，而是使用通过应用适当的核函数找到的过滤数据，并根据过滤过程计算最大似然估计量。我们表明我们提出的估计量是渐近无偏的，并在数值上证明了我们的方法在子采样方面的优势。最后，我们展示了我们的过滤数据方法如何与贝叶斯技术相结合，并提供推理过程的完整不确定性量化。