Optimal estimation of a signal perturbed by a fractional Brownian noise

TL;DR

This paper develops optimal estimation methods for a vector parameter in a fractional Brownian motion model, providing maximum likelihood and Bayesian estimates for different prior distributions.

Contribution

It introduces explicit estimation formulas for the drift parameters in fractional Brownian motion, including maximum likelihood and Bayesian approaches for normal and uniform priors.

Findings

Derived explicit maximum likelihood estimators.

Provided Bayesian estimators for normal and uniform priors.

Enhanced understanding of parameter estimation in fractional Brownian motion.

Abstract

We consider the problem of optimal estimation of the value of a vector parameter $\thetavector=(\theta_0,\ldots,\theta_n)^{\top}$ of the drift term in a fractional Brownian motion represented by the finite sum $\sum_{i=0}^{n}\theta_{i}\varphi_{i}(t)$ over known functions $\varphi_i(t)$, $\alli$. For the value of parameter $\thetavector$, we obtain a maximum likelihood estimate as well as Bayesian estimates for normal and uniform a priori distributions.