Drift Estimation of Multiscale Diffusions Based on Filtered Data

TL;DR

This paper introduces a new method for estimating the drift in multiscale diffusions using filtered data, avoiding subsampling, and provides theoretical and numerical validation of its effectiveness.

Contribution

It proposes a novel filtered data approach for drift estimation in multiscale diffusions, bypassing the need for subsampling and enabling Bayesian uncertainty quantification.

Findings

Estimators are asymptotically unbiased.

Filtered data method outperforms subsampling.

Method integrates with Bayesian techniques for uncertainty quantification.

Abstract

We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coefficient of the homogenized equation requires pre-processing of the data, often in the form of subsampling; this is because the two-scale equation and the homogenized single-scale equation are incompatible at small scales, generating mutually singular measures on the path space. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel function, and compute maximum likelihood estimators based on the filtered process. We show that the estimators we propose are asymptotically unbiased and demonstrate numerically the advantages of our method with respect to subsampling. Finally, we show how our filtered data methodology can be combined with Bayesian techniques and provide a full uncertainty quantification of the inference procedure.