Maximum Likelihood Drift Estimation for Multiscale Diffusions

TL;DR

This paper investigates the effectiveness of maximum likelihood estimation for parameters in multiscale stochastic differential equations, revealing conditions under which it succeeds or fails, especially in homogenization scenarios, with applications to molecular dynamics.

Contribution

It provides a rigorous analysis of maximum likelihood estimation in multiscale diffusions, distinguishing between averaging and homogenization cases, and offers explicit error formulas.

Findings

MLE is asymptotically unbiased for averaging problems

MLE fails for homogenization unless data is subsampled appropriately

Explicit asymptotic error formula for the log likelihood

Abstract

We study the problem of parameter estimation using maximum likelihood for fast/slow systems of stochastic differential equations. Our aim is to shed light on the problem of model/data mismatch at small scales. We consider two classes of fast/slow problems for which a closed coarse-grained equation for the slow variables can be rigorously derived, which we refer to as averaging and homogenization problems. We ask whether, given data from the slow variable in the fast/slow system, we can correctly estimate parameters in the drift of the coarse-grained equation for the slow variable, using maximum likelihood. We show that, whereas the maximum likelihood estimator is asymptotically unbiased for the averaging problem, for the homogenization problem maximum likelihood fails unless we subsample the data at an appropriate rate. An explicit formula for the asymptotic error in the log likelihood function is presented. Our theory is applied to two simple examples from molecular dynamics.