Parameter Estimation for the Langevin Equation with Stationary-Increment Gaussian Noise

TL;DR

This paper investigates the estimation of the mean reversion parameter in Langevin equations driven by stationary-increment Gaussian noise, establishing consistency, asymptotic normality, and Berry--Esseen bounds for the estimator.

Contribution

It introduces new theoretical results on the properties of an alternative estimator for the mean reversion parameter under broad Gaussian noise conditions, including fractional and bifractional cases.

Findings

Proves strong consistency of the estimator.

Establishes asymptotic normality with Berry--Esseen bounds.

Applies results to fractional and bifractional Ornstein--Uhlenbeck processes.

Abstract

We study the Langevin equation with stationary-increment Gaussian noise. We show the strong consistency and the asymptotic normality with Berry--Esseen bound of the so-called alternative estimator of the mean reversion parameter. The conditions and results are stated in terms of the variance function of the noise. We consider both the case of continuous and discrete observations. As examples we consider fractional and bifractional Ornstein--Uhlenbeck processes. Finally, we discuss the maximum likelihood and the least squares estimators.