Statistical Inference for the Rough Homogenization Limit of Multiscale Fractional Ornstein-Uhlenbeck Processes

TL;DR

This paper develops a new methodology for consistent parameter estimation in multiscale fractional stochastic systems, extending classical results from Brownian motion to fractional dynamics and addressing the challenges of spectral analysis of fractional Gaussian noise.

Contribution

It introduces a novel approach for maximum likelihood estimation in fractional multiscale systems, overcoming limitations of existing methods designed for Brownian motion.

Findings

Convergence of MLE for fractional diffusion coefficient established

New spectral norm control technique for fractional Gaussian noise

Extension of homogenization inference to fractional stochastic systems

Abstract

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.