Some fractional and multifractional Gaussian processes: A brief introduction

S. C. Lim,C. H. Eab

doi:10.1142/s2010194515600010

Abstract

This paper gives a brief introduction to some important fractional and multifractional Gaussian processes commonly used in modelling natural phenomena and man-made systems. The processes include fractional Brownian motion (both standard and the Riemann-Liouville type), multifractional Brownian motion, fractional and multifractional Ornstein-Uhlenbeck processes, fractional and mutifractional Reisz-Bessel motion. Possible applications of these processes are briefly mentioned.

Highlights

During the past few decades, fractional calculus[1,2,3,4] has found applications in diverse fields ranging from physical and biological sciences, engineering to internet traffic and economics
Absence of stationary property for its increments implies that XH can not have a harmonizable representation, and it is not possible to associate to XH a generalized spectrum of power-law type as in the case of standard Fractional Brownian motion (FBM)
The stationary Gaussian process defined by the following generalized Cauchy (GC) covariance parametrized by two indices

Summary

Introduction

During the past few decades, fractional calculus[1,2,3,4] has found applications in diverse fields ranging from physical and biological sciences, engineering to internet traffic and economics. In the case of the generalized Cauchy process both have the advantage that the two indices provide separate characterization of the fractal dimension or self-similar property, a local property, and the long-range dependence, a global property This is in contrast to models based on fractional Brownian motion which characterize these two properties with a single parameter. Further generalization of fractional process can be carried out by replacing the constant index by a continuous function of time In this way, one obtains multifractional Brownian motion[47, 48] and multifractional Ornstein-Uhlenbeck process.[49] it is possible to have the multifractional extension of Riesz-Bessel motion[50, 51] and generalized Cauchy process. The non-Gaussian fractional and multifractional Levy motion will not be considered here

Fractional Brownian Motion

Multifractional Brownian Motion

Fractional and Multifractional Ornstein-Uhlenbeck Process

Fractional and Multifractional Riesz-Bessel Motion

Generalized Cauchy Process