Standard maximum likelihood drift parameter estimator in the homogeneous diffusion model is always strongly consistent

TL;DR

This paper proves that the standard maximum likelihood estimator for the drift parameter in a homogeneous diffusion model is strongly consistent under mild conditions, including for discretized versions.

Contribution

It establishes the strong consistency of the standard maximum likelihood estimator and its discretized form in homogeneous diffusion models.

Findings

MLE is strongly consistent under mild conditions

Discretized estimators also exhibit strong consistency

Provides theoretical guarantees for parameter estimation in diffusion models

Abstract

We consider the homogeneous stochastic differential equation with unknown parameter to be estimated. We prove that the standard maximum likelihood estimate is strongly consistent under very mild conditions. There are also established the conditions for strong consistency of the discretized estimator.