Most real-world systems exhibit a multiscale behaviour that needs to be taken into consideration when fitting the effective dynamics to data sampled at a given scale. In the case of stochastic multiscale systems driven by Brownian motion, it has been shown that in order for the Maximum Likelihood Estimators of the parameters of the limiting dynamics to be consistent, data needs to be subsampled at an appropriate rate. Recent advances in extracting effective dynamics for fractional multiscale systems make the same question relevant in the fractional diffusion setting. We study the problem of parameter estimation of the diffusion coefficient in this context. In particular, we consider the multiscale fractional Ornstein-Uhlenbeck system (fractional kinetic Brownian motion) and we provide convergence results for the Maximum Likelihood Estimator of the diffusion coefficient of the limiting dynamics, using multiscale data. To do so, we derive asymptotic bounds for the spectral norm of the inverse covariance matrix of fractional Gaussian noise.

翻译：大多数真实世界系统表现出多尺度行为，当将有效动力学拟合到给定尺度采样的数据时，必须考虑这一特性。对于由布朗运动驱动的随机多尺度系统，已有研究表明，为使极限动力学参数的最大似然估计量具有一致性，需要以适当的速率对数据进行二次采样。在分数阶多尺度系统有效动力学提取方面的最新进展，使得同一问题在分数阶扩散背景下同样具有重要意义。我们在此背景下研究扩散系数的参数估计问题。具体而言，我们考虑多尺度分数阶Ornstein-Uhlenbeck系统（分数阶动力学布朗运动），并利用多尺度数据给出了极限动力学扩散系数最大似然估计量的收敛性结果。为此，我们推导了分数阶高斯噪声逆协方差矩阵谱范数的渐近界。