Maximum likelihood drift estimation for Gaussian process with stationary increments

TL;DR

This paper develops a maximum likelihood estimation method for the drift parameter in Gaussian processes with stationary increments, providing explicit formulas and proving strong consistency, generalizing previous results for fractional Brownian motion.

Contribution

It introduces a new likelihood-based estimator for the drift in Gaussian processes with stationary increments, extending known results to broader classes of processes.

Findings

Derived the likelihood function in terms of an integral equation

Established the maximum likelihood estimator for the drift parameter

Proved the strong consistency of the estimator

Abstract

The paper deals with the regression model $X_t = \theta t + B_t$, $t\in[0, T ]$, where $B=\{B_t, t\geq 0\}$ is a centered Gaussian process with stationary increments. We study the estimation of the unknown parameter $\theta$ and establish the formula for the likelihood function in terms of a solution to an integral equation. Then we find the maximum likelihood estimator and prove its strong consistency. The results obtained generalize the known results for fractional and mixed fractional Brownian motion.