Drift parameter estimation in the double mixed fractional Brownian model via solutions of Fredholm equations with singular kernels

TL;DR

This paper develops a numerical method to compute the maximum likelihood estimator for a drift parameter in a model driven by two independent fractional Brownian motions with different Hurst indices, by solving a Fredholm integral equation with a weakly singular kernel.

Contribution

It introduces a novel numerical approach to approximate the MLE in a complex fractional Brownian motion model using Fredholm equations with singular kernels.

Findings

The method effectively computes the MLE in simulated experiments.

Numerical results demonstrate the accuracy and efficiency of the proposed approach.

Abstract

We consider drift parameter estimation in a model driven by the sum of two independent fractional Brownian motions with different Hurst indices. Although the maximum likelihood estimator (MLE) for this model is known theoretically, its computation requires solving an operator equation involving fractional covariance operators. We develop an effective numerical method for approximating the solution of this equation by reformulating it as a Fredholm integral equation of the second kind with a weakly singular kernel. The resulting algorithm enables practical computation of the MLE. Numerical experiments illustrate the performance of the method.