Diffusion Parameter Estimation for the Homogenized Equation

TL;DR

This paper introduces a new estimator for the diffusion coefficient in homogenized equations, leveraging local extrema to avoid fixed-interval subsampling and achieving asymptotic unbiasedness with error of order epsilon squared.

Contribution

It proposes a novel estimator based on local extrema, eliminating the need for prior scale separation knowledge and improving estimation accuracy.

Findings

Estimator is asymptotically unbiased.

L2-error of the estimator is of order epsilon squared.

Method avoids fixed-interval subsampling.

Abstract

We construct a novel estimator for the diffusion coefficient of the limiting homogenized equation, when observing the slow dynamics of a multiscale model, in the case when the slow dynamics are of bounded variation. Previous research suggests subsampling the data on fixed intervals and computing the corresponding quadratic variation. However, to achieve optimality, this approach requires knowledge of scale separation variable $\epsilon$. Instead, we suggest computing the quadratic variation corresponding to the local extrema of the slow process. Our approach results to a natural subsampling and avoids the issue of choosing a subsampling rate. We prove that the estimator is asymptotically unbiased and we numerically demonstrate that its $L_2$-error is of order ${\mathcal O}(\epsilon^2)$.