ENGINEERING APPLICATIONS OF FRACTIONAL BROWNIAN MOTION: SELF-AFFINE AND SELF-SIMILAR RANDOM PROCESSES

Fractals ◽  
1999 ◽  
Vol 07 (02) ◽  
pp. 151-157 ◽  
Author(s):  
P. S. ADDISON ◽  
A. S. NDUMU
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Random Processes ◽  
The Other ◽  
Engineering Applications ◽  
Self Similar ◽  
Diffusive Processes ◽  
Spatial Trajectories

The purpose of this paper is to explain the connection between fractional Brownian motion (fBm) and non-Fickian diffusive processes, and at the same time, highlight three engineering applications: two requiring self-affine fBm trace functions and the other requiring self-similar fBm spatial trajectories.

scholarly journals A DYNAMICAL APPROACH TO FRACTIONAL BROWNIAN MOTION

Fractals ◽  
1994 ◽  
Vol 02 (01) ◽  
pp. 81-94 ◽  
Author(s):  
RICCARDO MANNELLA ◽  
PAOLO GRIGOLINI ◽  
BRUCE J. WEST
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Correlation Function ◽  
Fractional Brownian Motion ◽  
Gaussian Processes ◽  
The Other ◽  
Hurst Coefficient ◽  
And Diffusion ◽  
Clear Relation

Herein we develop a dynamical foundation for fractional Brownian motion. A clear relation is established between the asymptotic behavior of the correlation function and diffusion in a dynamical system. Then, assuming that scaling is applicable, we establish a connection between diffusion (either standard or anomalous) and the dynamical indicator known as the Hurst coefficient. We argue on the basis of numerical simulations that although we have been able to prove scaling only for "Gaussian" processes, our conclusions may well apply to a wider class of systems. On the other hand, systems exist for which scaling might not hold, so we speculate on the possible consequences of the various relations derived in the paper on such systems.

On Delayed Logistic Equation Driven by Fractional Brownian Motion

2012 ◽  
Vol 7 (3) ◽  
Author(s):  
Nguyen Tien Dung
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Approximate Method ◽  
Explicit Expression ◽  
Stochastic Integral ◽  
Logistic Equation ◽  
The Other ◽  
Stochastic Integration

In this paper we use the fractional stochastic integral given by Carmona et al. (2003, “Stochastic Integration With Respect to Fractional Brownian Motion,” Ann. I.H.P. Probab. Stat., 39(1), pp. 27–68) to study a delayed logistic equation driven by fractional Brownian motion which is a generalization of the classical delayed logistic equation. We introduce an approximate method to find the explicit expression for the solution. Our proposed method can also be applied to the other models and to illustrate this, two models in physiology are discussed.

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.

scholarly journals Least squares type estimations for discretely observed nonergodic Gaussian Ornstein-Uhlenbeck processes of the second kind

2021 ◽  
Vol 7 (1) ◽  
pp. 1095-1114
Author(s):  
Huantian Xie ◽  
◽  
Nenghui Kuang ◽  
Keyword(s):  
Brownian Motion ◽  
Least Squares ◽  
Fractional Brownian Motion ◽  
Unknown Parameter ◽  
Discrete Observations ◽  
Bifractional Brownian Motion ◽  
Similar Index ◽  
Self Similar ◽  
Strongly Consistent ◽  
Theoretical Results

<abstract><p>We consider the nonergodic Gaussian Ornstein-Uhlenbeck processes of the second kind defined by $ dX_t = \theta X_tdt+dY_t^{(1)}, t\geq 0, X_0 = 0 $ with an unknown parameter $ \theta &gt; 0, $ where $ dY_t^{(1)} = e^{-t}dG_{a_{t}} $ and $ \{G_t, t\geq 0\} $ is a mean zero Gaussian process with the self-similar index $ \gamma\in (\frac{1}{2}, 1) $ and $ a_t = \gamma e^{\frac{t}{\gamma}} $. Based on the discrete observations $ \{X_{t_i}:t_i = i\Delta_n, i = 0, 1, \cdots, n\} $, two least squares type estimators $ \hat{\theta}_n $ and $ \tilde{\theta}_n $ of $ \theta $ are constructed and proved to be strongly consistent and rate consistent. We apply our results to the cases such as fractional Brownian motion, sub-fractional Brownian motion, bifractional Brownian motion and sub-bifractional Brownian motion. Moreover, the numerical simulations confirm the theoretical results.</p></abstract>

Generate Breath Flow Having Fractal Signal Feature Using Weierstrass Function Combination

2011 ◽  
Vol 393-395 ◽  
pp. 796-799
Author(s):  
Meng Chao Li ◽  
Zhong Hai He
Keyword(s):  
Mechanical Ventilation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Medical Patient ◽  
Motion Simulation ◽  
Weierstrass Function ◽  
Patient Simulator ◽  
Function Combination ◽  
Fractal Signal ◽  
Self Similar

Fractal signal feature in breath flow is verified by many articles. So the generate fractal feature have two meanings, one to decrease damage to lung in mechanical ventilation because of natural similar, two to increase similarity in breath simulation used in medical patient simulator. The main feature of fractal signal is self-similar. Some algorithms have been proposed using fractional Brownian motion simulation. In this paper we use Weierstrass function combination to generate fractal signal. The method includes all fractal features and easy to realize in algorithm compared with fractional Brownian motion.

scholarly journals Simulation of a strictly φ-sub-Gaussian generalized fractional Brownian motion

2021 ◽  
pp. 11-19
Author(s):  
O. I. Vasylyk ◽  
I. I. Lovytska
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Series Representation ◽  
Random Processes ◽  
Specific Class ◽  
Programming Environment ◽  
Motion Simulation ◽  
Actuarial Mathematics ◽  
R Programming ◽  
Simulation Parameters

In the paper, we consider the problem of simulation of a strictly φ-sub-Gaussian generalized fractional Brownian motion. Simulation of random processes and fields is used in many areas of natural and social sciences. A special place is occupied by methods of simulation of the Wiener process and fractional Brownian motion, as these processes are widely used in financial and actuarial mathematics, queueing theory etc. We study some specific class of processes of generalized fractional Brownian motion and derive conditions, under which the model based on a series representation approximates a strictly φ-sub-Gaussian generalized fractional Brownian motion with given reliability and accuracy in the space C([0; 1]) in the case, when φ(x) = (|x|^p)/p, |x| ≥ 1, p > 1. In order to obtain these results, we use some results from the theory of φ-sub-Gaussian random processes. Necessary simulation parameters are calculated and models of sample pathes of corresponding processes are constructed for various values of the Hurst parameter H and for given reliability and accuracy using the R programming environment.

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