The application of fractional derivatives in stochastic models driven by fractional Brownian motion

2010 ◽  
Vol 389 (21) ◽  
pp. 4809-4818 ◽  
Author(s):  
Lv Longjin ◽  
Fu-Yao Ren ◽  
Wei-Yuan Qiu
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stochastic Models ◽  
Fractional Derivatives

scholarly journals On the relation between the fractional Brownian motion and the fractional derivatives

2008 ◽  
Vol 372 (7) ◽  
pp. 958-968 ◽  
Author(s):  
Manuel Duarte Ortigueira ◽  
Arnaldo Guimarães Batista
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Fractional Derivatives

A New Look at the Fractional Brownian Motion Definition

2007 ◽  
Author(s):  
Manuel Duarte Ortigueira ◽  
Arnaldo Guimara˜es Batista
Keyword(s):  
Brownian Motion ◽  
Fractional Derivative ◽  
White Noise ◽  
Fractional Brownian Motion ◽  
Fractional Derivatives ◽  
Autocorrelation Functions ◽  
Fractional Noise ◽  
Simulation Procedure ◽  
Definition Of ◽  
Fractional Noises

A reinterpretation of the classic definition of fractional Brownian motion leads to a new definition involving a fractional noise obtained as a fractional derivative of white noise. To obtain this fractional noise, two sets of fractional derivatives are considered: a) the forward and backward and b) the central derivatives. For these derivatives the autocorrelation functions of the corresponding fractional noises have the same representations. The obtained results are used to define and propose a new simulation procedure.

On limit theorems of some extensions of fractional Brownian motion and their additive functionals

2017 ◽  
Vol 17 (03) ◽  
pp. 1750022
Author(s):  
M. Ait Ouahra ◽  
S. Moussaten ◽  
A. Sghir
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Local Time ◽  
Limit Theorems ◽  
Fractional Derivatives ◽  
Slowly Varying Function ◽  
Bifractional Brownian Motion ◽  
Additive Functionals ◽  
Subfractional Brownian Motion

This paper is divided into two parts. The first deals with some limit theorems to certain extensions of fractional Brownian motion like: bifractional Brownian motion, subfractional Brownian motion and weighted fractional Brownian motion. In the second part we give the similar results of their continuous additive functionals; more precisely, local time and its fractional derivatives involving slowly varying function.

STOCHASTIC POROUS MEDIA EQUATION DRIVEN BY FRACTIONAL BROWNIAN MOTION

2013 ◽  
Vol 13 (04) ◽  
pp. 1350010 ◽  
Author(s):  
JAN BÁRTEK ◽  
MARÍA J. GARRIDO-ATIENZA ◽  
BOHDAN MASLOWSKI
Keyword(s):  
Porous Media ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Fractional Derivatives ◽  
Multiplicative Noise ◽  
Stochastic Equation ◽  
Random Dynamical System ◽  
Porous Media Equation ◽  
Media Equation ◽  
Long Time

The present work deals with stochastic porous media equation with multiplicative noise, driven by fractional Brownian motion B(H) with Hurst index H > 1/2. The stochastic integral with integrator B(H) is defined pathwise following the theory developed by Zähle [24], based on the so-called fractional derivatives. It is shown that there is a one-to-one correspondence between solutions to the stochastic equation and solutions to its deterministic counterpart. By means of this correspondence and exploiting properties of the deterministic porous media equation, the existence, uniqueness, regularity and long-time properties of the solution is established. We also prove that the solution forms a random dynamical system in an appropriate function space.

scholarly journals A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

2014 ◽  
Vol 51 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Dawei Hong ◽  
Shushuang Man ◽  
Jean-Camille Birget ◽  
Desmond S. Lun
Keyword(s):  
Brownian Motion ◽  
Convergence Rate ◽  
Fractional Brownian Motion ◽  
Parallel Algorithm ◽  
Uniform Approximation ◽  
Haar Wavelets ◽  
Hurst Index ◽  
Sample Paths ◽  
Vanishing Moment ◽  
Uniform Expansion

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

Global attracting sets of neutral stochastic functional integro-differential equations driven by a fractional Brownian motion

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
A. Bakka ◽  
S. Hajji ◽  
D. Kiouach
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Sufficient Conditions ◽  
Integrodifferential Equations ◽  
Hurst Parameter ◽  
Fixed Point Principle ◽  
Stochastic Functional

Abstract By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {H\in(\frac{1}{2},1)} in a Hilbert space.

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