Eigenfunction martingale estimating functions and filtered data for drift estimation of discretely observed multiscale diffusions

TL;DR

This paper introduces new estimators for the drift of multiscale diffusions from discrete data, utilizing eigenfunctions and filtering techniques to improve bias and sampling rate issues.

Contribution

It develops two novel estimators based on eigenfunctions and filtering, with the second estimator being asymptotically unbiased regardless of sampling rate.

Findings

The second estimator is asymptotically unbiased for any sampling rate.

Numerical experiments demonstrate the estimators' reliability and efficiency.

Eigenfunction-based methods improve drift estimation in multiscale diffusions.

Abstract

We propose a novel method for drift estimation of multiscale diffusion processes when a sequence of discrete observations is given. For the Langevin dynamics in a two-scale potential, our approach relies on the eigenvalues and the eigenfunctions of the homogenized dynamics. Our first estimator is derived from a martingale estimating function of the generator of the homogenized diffusion process. However, the unbiasedness of the estimator depends on the rate with which the observations are sampled. We therefore introduce a second estimator which relies also on filtering the data and we prove that it is asymptotically unbiased independently of the sampling rate. A series of numerical experiments illustrate the reliability and efficiency of our different estimators.