Construction of maximum likelihood estimator in the mixed fractional--fractional Brownian motion model with double long-range dependence

TL;DR

This paper develops a maximum likelihood estimator for the drift parameter in a model driven by two independent fractional Brownian motions with different Hurst indices, addressing the challenge of double long-range dependence.

Contribution

It reduces the estimation problem to solving a Fredholm integral equation and proves the existence, uniqueness, and asymptotic consistency of the estimator.

Findings

Established the compactness of the integral operator.

Proved the existence and uniqueness of the estimator.

Demonstrated asymptotic consistency of the estimator.

Abstract

We construct an estimator of the unknown drift parameter $\theta\in {\mathbb{R}}$ in the linear model \[X_t=\theta t+\sigma_1B^{H_1}(t)+\sigma_2B^{H_2}(t),\;t\in[0,T],\] where $B^{H_1}$ and $B^{H_2}$ are two independent fractional Brownian motions with Hurst indices $H_1$ and $H_2$ satisfying the condition $\frac{1}{2}\leq H_1<H_2<1.$ Actually, we reduce the problem to the solution of the integral Fredholm equation of the 2nd kind with a specific weakly singular kernel depending on two power exponents. It is proved that the kernel can be presented as the product of a bounded continuous multiplier and weak singular one, and this representation allows us to prove the compactness of the corresponding integral operator. This, in turn, allows us to establish an existence--uniqueness result for the sequence of the equations on the increasing intervals, to construct accordingly a sequence of statistical estimators, and to establish asymptotic consistency.