A Special Study of the Mixed Weighted Fractional Brownian Motion

Salih Djilali,Hashim M. Alshehri,Tareq Saeed,Anwar Zeb,Anas D. Khalaf,Mahmoud Abouagwa

doi:10.3390/fractalfract5040192

Abstract

In this work, we present the analysis of a mixed weighted fractional Brownian motion, defined by ηt:=Bt+ξt, where B is a Brownian motion and ξ is an independent weighted fractional Brownian motion. We also consider the parameter estimation problem for the drift parameter θ>0 in the mixed weighted fractional Ornstein–Uhlenbeck model of the form X0=0;Xt=θXtdt+dηt. Moreover, a simulation is given of sample paths of the mixed weighted fractional Ornstein–Uhlenbeck process.

Highlights

Academic Editor: Hijaz AhmadReceived: 18 September 2021Accepted: 20 October 2021Published: 31 October 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Over the course of the last two decades, the investigation of a system of particles moving in Rd under the influence of a symmetric α-stable Lèvy noise, 0 < α 6 2, has attracted many scholars’ attention (e.g., [1,2,3,4,5])
Inspired by the aforementioned monographs and facts, in this work, we begin by introducing a new stochastic process named as the mixed weighted fractional Brownian motion and establish the stochastic integral and the canonical representation for such process. We apply this process to one of the most interesting problem in mathematical statistics which is parameter estimation problem, such that we investigate the problem of parameter estimation for Ornstein–Uhlenbek process in which the dynamics follows mixed-weighted fractional Brownian motion (mwfBm)
New results have been established for the proposed estimator of parameter θ that are different from those that have previously been obtained for both fBm and wfBm

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. We apply this process to one of the most interesting problem in mathematical statistics which is parameter estimation problem, such that we investigate the problem of parameter estimation for Ornstein–Uhlenbek process in which the dynamics follows mwfBm. new results have been established for the proposed estimator of parameter θ that are different from those that have previously been obtained for both fBm and wfBm (see Section 3). We used the most recent results for the numerical simulation (such as in [21,22,23,24]) to discuss the simulation of the sample paths of the mixed weighted fractional Ornstein–Uhlenbeck process.

Notions and Auxiliary Results

A Canonical Innovation Representation for mwfBm

Least Square Estimator for the mwfOU Process

Numerical Simulations

Discussion

Conclusions