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

Pablo Ramses Alonso-Martin, Horatio Boedihardjo, Anastasia Papavasiliou

TL;DR

This work addresses parameter inference for the rough homogenization limit of multiscale systems driven by fractional dynamics, focusing on a kinetic fractional Brownian motion model where the coarse-grained limit is $X^{\epsilon}_t \approx \overline{\sigma} B^H_t$. The authors propose an MLE-type estimator $\hat{\sigma}^2_{\delta,\epsilon}$ based on discretized observations and analyze its behavior under subsampling, establishing that unbiased consistency requires an $\epsilon$-dependent sampling rate $\delta = \epsilon^{\alpha}$ with $0<\alpha<\min\{1, H/(1-H)\}$ (with sharper thresholds depending on $H$). A key technical contribution is new bounds on the spectral norm of the inverse covariance $P^{-1}$ of discretized fractional Gaussian noise, derived via Wiener-integral representations and fractional calculus, which underpin the convergence analysis. The paper also provides a practical method to estimate the Hurst index $H$ from data and discusses robustness of the estimator to mis-specification of $H$, supported by numerical experiments. Overall, the results guide sampling design and inference for multiscale fractional systems undergoing rough homogenization, enabling consistent estimation of the effective diffusion coefficient from multiscale observations.

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.

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

TL;DR

This work addresses parameter inference for the rough homogenization limit of multiscale systems driven by fractional dynamics, focusing on a kinetic fractional Brownian motion model where the coarse-grained limit is . The authors propose an MLE-type estimator based on discretized observations and analyze its behavior under subsampling, establishing that unbiased consistency requires an -dependent sampling rate with (with sharper thresholds depending on ). A key technical contribution is new bounds on the spectral norm of the inverse covariance of discretized fractional Gaussian noise, derived via Wiener-integral representations and fractional calculus, which underpin the convergence analysis. The paper also provides a practical method to estimate the Hurst index from data and discusses robustness of the estimator to mis-specification of , supported by numerical experiments. Overall, the results guide sampling design and inference for multiscale fractional systems undergoing rough homogenization, enabling consistent estimation of the effective diffusion coefficient from multiscale observations.

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.
Paper Structure (20 sections, 6 theorems, 74 equations, 1 figure, 9 tables)

This paper contains 20 sections, 6 theorems, 74 equations, 1 figure, 9 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Estimators for $\overline{\sigma}$
  4. Main results
  5. Asymptotic biasedness without subsampling
  6. Consistency with appropriate subsampling
  7. Auxiliary Tools: asymptotic bounds on the spectral norm of $P^{-1}$.
  8. Proof of Lemma \ref{['orderlemma']}
  9. The Wiener Integral with respect to fractional Brownian Motion
  10. Case $H>1/2$.
  11. Case $H<1/2$.
  12. Proof of Theorem \ref{['noconvergence']}
  13. Proof of Theorem \ref{['maintheorem']}
  14. Some considerations on the practical usage of $\hat{\sigma}_{\epsilon,\delta}$
  15. Supplementary Material: Bounds for $A_i$ and $B_{j_2}$ in Proof 3.3.
  16. ...and 5 more sections

Key Result

Proposition 2.4

Let $X^{\epsilon}_t$ be the solution of eq: kfbm. For any $H\in(0,1)$, $p>1$ and $T>0$ it holds that where $\lesssim$ denotes less or equal up to a positive constant and $\overline{\sigma}$ is the homogenized diffusion coefficient $\overline{\sigma} = \sigma_1\cdot\sigma_2$.

Figures (1)

  • Figure 1: Numerical simulation of the Mean Squared Error in the estimation for different values of $\epsilon$ and $\alpha$ when $\sigma = T = 1$.

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4: ximeli
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.3
  • Definition 4.1
  • Remark 4.2
  • Proposition 4.3
  • ...and 5 more