Parameter estimation for Gaussian processes with application to the model with two independent fractional Brownian motions

TL;DR

This paper reviews recent maximum likelihood estimation methods for Gaussian process regression models and applies these techniques to a model involving two independent fractional Brownian motions, highlighting estimation challenges and solutions.

Contribution

It provides a comprehensive review of estimation techniques for Gaussian processes and extends these methods to a novel model with two independent fractional Brownian motions.

Findings

Estimation techniques for Gaussian process models are summarized.

Application of methods to models with fractional Brownian motions.

Analysis of the model with two independent fractional Brownian motions.

Abstract

The purpose of the article is twofold. Firstly, we review some recent results on the maximum likelihood estimation in the regression model of the form $X_t = \theta G(t) + B_t$, where $B$ is a Gaussian process, $G(t)$ is a known function, and $\theta$ is an unknown drift parameter. The estimation techniques for the cases of discrete-time and continuous-time observations are presented. As examples, models with fractional Brownian motion, mixed fractional Brownian motion, and sub-fractional Brownian motion are considered. Secondly, we study in detail the model with two independent fractional Brownian motions and apply the general results mentioned above to this model.