scholarly journals Hurst exponents, Markov processes, and fractional Brownian motion

2007 ◽  
Vol 379 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Joseph L. McCauley ◽  
Gemunu H. Gunaratne ◽  
Kevin E. Bassler
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Markov Processes ◽  
Hurst Exponents

scholarly journals On the Hurst exponents, Markov processes, and fractional Brownian motion

2021 ◽  
Author(s):  
Ginno Millán
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Markov Processes ◽  
The Other ◽  
Time Interval ◽  
Infinite Time Interval ◽  
Point Density ◽  
The Difference ◽  
The One ◽  
Hurst Exponents

There is much confusion in the literature over Hurst exponent (H). The purpose of this paper is to illustrate the difference between fractional Brownian motion (fBm) on the one hand and Gaussian Markov processes where H is different to 1/2 on the other. The difference lies in the increments, which are stationary and correlated in one case and nonstationary and uncorrelated in the other. The two- and one-point densities of fBm are constructed explicitly. The two-point density does not scale. The one-point density for a semi-infinite time interval is identical to that for a scaling Gaussian Markov process with H different to 1/2 over a finite time interval. We conclude that both Hurst exponents and one-point densities are inadequate for deducing the underlying dynamics from empirical data. We apply these conclusions in the end to make a focused statement about nonlinear diffusion.

Conditional Distributions of Processes Related to Fractional Brownian Motion

2013 ◽  
Vol 50 (01) ◽  
pp. 166-183 ◽  
Author(s):  
Holger Fink ◽  
Claudia Klüppelberg ◽  
Martina Zähle
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Markov Processes ◽  
Bond Market ◽  
Market Model ◽  
Standard Brownian Motion ◽  
Conditional Distributions ◽  
Prediction Formula ◽  
Defaultable Bond ◽  
The Core

Conditional distributions for affine Markov processes are at the core of present (defaultable) bond pricing. There is, however, evidence that Markov processes may not be realistic models for short rates. Fractional Brownian motion (FBM) can be introduced by an integral representation with respect to standard Brownian motion. Using a simple prediction formula for the conditional expectation of an FBM and its Gaussianity, we derive the conditional distributions of FBM and related processes. We derive conditional distributions for fractional analogies of prominent affine processes, including important examples like fractional Ornstein–Uhlenbeck or fractional Cox–Ingersoll–Ross processes. As an application, we propose a fractional Vasicek bond market model and compare prices of zero-coupon bonds to those achieved in the classical Vasicek model.

scholarly journals Conditional Distributions of Processes Related to Fractional Brownian Motion

2013 ◽  
Vol 50 (1) ◽  
pp. 166-183 ◽  
Author(s):  
Holger Fink ◽  
Claudia Klüppelberg ◽  
Martina Zähle
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Markov Processes ◽  
Bond Market ◽  
Market Model ◽  
Standard Brownian Motion ◽  
Conditional Distributions ◽  
Prediction Formula ◽  
Defaultable Bond ◽  
The Core

Conditional distributions for affine Markov processes are at the core of present (defaultable) bond pricing. There is, however, evidence that Markov processes may not be realistic models for short rates. Fractional Brownian motion (FBM) can be introduced by an integral representation with respect to standard Brownian motion. Using a simple prediction formula for the conditional expectation of an FBM and its Gaussianity, we derive the conditional distributions of FBM and related processes. We derive conditional distributions for fractional analogies of prominent affine processes, including important examples like fractional Ornstein–Uhlenbeck or fractional Cox–Ingersoll–Ross processes. As an application, we propose a fractional Vasicek bond market model and compare prices of zero-coupon bonds to those achieved in the classical Vasicek model.

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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