Statistical inference for the rough homogenization limit of multiscale fractional Ornstein-Uhlenbeck processes

We study the problem of parameter estimation for the homogenization limit of multiscale systems involving fractional dynamics. 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. We extend these results to a class of fractional multiscale systems, often described as scaled fractional kinetic Brownian motions. We provide convergence results for the MLE of the diffusion coefficient of the limiting dynamics, computed using multiscale data. This requires the development of a different methodology to that used in the standard Brownian motion case, which is based on controlling the spectral norm of the inverse covariance matrix of a discretized fractional Gaussian noise on an interval.