Maximum likelihood estimation for small noise multiscale diffusions

TL;DR

This paper investigates maximum likelihood estimation for multiscale stochastic differential equations with small noise, analyzing different regimes and their asymptotic properties, supported by simulation results.

Contribution

It introduces regime-specific maximum likelihood estimators for small noise multiscale diffusions and studies their consistency and asymptotic normality.

Findings

Establishes consistency of estimators across regimes

Proves asymptotic normality of estimators

Provides simulation validation for Langevin equations

Abstract

We study the problem of parameter estimation for stochastic differential equations with small noise and fast oscillating parameters. Depending on how fast the intensity of the noise goes to zero relative to the homogenization parameter, we consider three different regimes. For each regime, we construct the maximum likelihood estimator and we study its consistency and asymptotic normality properties. A simulation study for the first order Langevin equation with a two scale potential is also provided.