scholarly journals Super Fractional Brownian Motion, Fractional Super Brownian Motion and Related Self-Similar (Super) Processes

1994 ◽  
Author(s):  
Robert J. Adler ◽  
Gennady Samorodnitsky
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Super Brownian Motion ◽  
Self Similar

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.

AN INTRODUCTION TO THE THEORY OF SELF-SIMILAR STOCHASTIC PROCESSES

2000 ◽  
Vol 14 (12n13) ◽  
pp. 1399-1420 ◽  
Author(s):  
PAUL EMBRECHTS ◽  
MAKOTO MAEJIMA
Keyword(s):  
Probability Theory ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Fractional Brownian Motion ◽  
High Energy Physics ◽  
High Energy ◽  
Point Of View ◽  
Long Range Dependence ◽  
Self Similar ◽  
Energy Physics

Self-similar processes such as fractional Brownian motion are stochastic processes that are invariant in distribution under suitable scaling of time and space. These processes can typically be used to model random phenomena with long-range dependence. Naturally, these processes are closely related to the notion of renormalization in statistical and high energy physics. They are also increasingly important in many other fields of application, as there are economics and finance. This paper starts with some basic aspects on self-similar processes and discusses several topics from the point of view of probability theory.

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.

LOCALLY SELF-SIMILAR FRACTIONAL OSCILLATOR PROCESSES

2007 ◽  
Vol 07 (02) ◽  
pp. L169-L179 ◽  
Author(s):  
S. C. LIM ◽  
MING LI ◽  
L. P. TEO
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Hausdorff Dimensions ◽  
Fractional Oscillator ◽  
Self Similar

Fractional oscillator processes of Riemann-Liouville and Weyl type are shown to be locally self-similar. The fractal or Hausdorff dimensions of these processes are determined. Relationship between fractional oscillator processes and the corresponding fractional Brownian motion processes is established.

scholarly journals Survival Exponents for Some Gaussian Processes

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
G. Molchan
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Power Law ◽  
Gaussian Processes ◽  
Similar Process ◽  
Fixed Level ◽  
Long Time ◽  
Self Similar

The problem is a power-law asymptotics of the probability that a self-similar process does not exceed a fixed level during long time. The exponent in such asymptotics is estimated for some Gaussian processes, including the fractional Brownian motion (FBM) in , and the integrated FBM in , .

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