Filtering the Maximum Likelihood for Multiscale Problems

TL;DR

This paper develops a method for filtering and parameter estimation in multiscale systems with partial observations, proving convergence and asymptotic properties of the reduced models and estimators.

Contribution

It introduces a reduced-dimension filtering approach for multiscale problems and establishes the asymptotic properties of the maximum likelihood estimator based on the simplified likelihood.

Findings

Proved mean square convergence of the nonlinear filter to a reduced filter.

Derived a correction term for the log-likelihood process via a central limit theorem.

Established consistency and asymptotic normality of the MLE based on the reduced likelihood.

Abstract

Filtering and parameter estimation under partial information for multiscale problems is studied in this paper. After proving mean square convergence of the nonlinear filter to a filter of reduced dimension, we establish that the conditional (on the observations) log-likelihood process has a correction term given by a type of central limit theorem. To achieve this we assume that the operator of the (hidden) fast process has a discrete spectrum and an orthonormal basis of eigenfunctions. Based on these results, we then propose to estimate the unknown parameters of the model based on the limiting log-likelihood, which is an easier function to optimize because it of reduced dimension. We also establish consistency and asymptotic normality of the maximum likelihood estimator based on the reduced log-likelihood. Simulation results illustrate our theoretical findings.