Coarse-grained modeling of multiscale diffusions: the p-variation estimates

TL;DR

This paper introduces a new method using total p-variation for estimating diffusion coefficients in multiscale diffusions, eliminating the need for data subsampling and improving parameter estimation accuracy.

Contribution

It proposes a novel p-variation based approach for parameter estimation in multiscale diffusions, bypassing subsampling requirements of traditional methods.

Findings

The p-variation method effectively estimates diffusion coefficients in multiscale systems.

The approach works well on a multiscale Ornstein-Uhlenbeck process.

It offers a consistent estimation technique without data subsampling.

Abstract

We study the problem of estimating parameters of the limiting equation of a multiscale diffusion in the case of averaging and homogenization, given data from the corresponding multiscale system. First, we review some recent results that make use of the maximum likelihood of the limiting equation. In particular, it has been shown that in the averaging case, the MLE will be asymptotically consistent in the limit while in the homogenization case, the MLE will be asymptotically consistent only if we subsample the data. Then, we focus on the problem of estimating the diffusion coefficient. We suggest a novel approach that makes use of the total $p$-variation, as defined in the theory of rough paths and avoids the subsampling step. The method is applied to a multiscale OU process.