Maximum likelihood drift estimation for the mixing of two fractional Brownian motions

TL;DR

This paper develops a maximum likelihood estimator for the drift parameter in a model combining two independent fractional Brownian motions with different Hurst indices, providing a new approach for parameter estimation in such stochastic processes.

Contribution

The paper introduces a novel MLE formula for a mixed fractional Brownian motion model with different Hurst parameters, based on solving an integral equation with a weak polar kernel.

Findings

Derived explicit MLE formula for the drift parameter

Established properties of the estimator in the fractional Brownian motion context

Provided a method applicable to models with two independent fractional Brownian motions

Abstract

We construct the maximum likelihood estimator (MLE) of the unknown drift parameter $\theta\in \mathbb{R}$ in the linear model $X_t=\theta t+\sigma B^{H_1}(t)+B^{H_2}(t),\;t\in[0,T],$ where $B^{H_1}$ and $B^{H_2}$ are two independent fractional Brownian motions with Hurst indices $\frac12<H_1<H_2<1.$ The formula for MLE is based on the solution of the integral equation with weak polar kernel.