scholarly journals Fractal Stochastic Processes on Thin Cantor-Like Sets

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 613
Author(s):  
Alireza Khalili Golmankhaneh ◽  
Renat Timergalievich Sibatov
Keyword(s):  
Brownian Motion ◽  
Fractional Derivative ◽  
Stochastic Processes ◽  
Fractional Brownian Motion ◽  
Hurst Exponent ◽  
Fourier Transformation ◽  
One Dimensional ◽  
Fractal Derivative ◽  
Fractal Calculus ◽  
Non Local

We review the basics of fractal calculus, define fractal Fourier transformation on thin Cantor-like sets and introduce fractal versions of Brownian motion and fractional Brownian motion. Fractional Brownian motion on thin Cantor-like sets is defined with the use of non-local fractal derivatives. The fractal Hurst exponent is suggested, and its relation with the order of non-local fractal derivatives is established. We relate the Gangal fractal derivative defined on a one-dimensional stochastic fractal to the fractional derivative after an averaging procedure over the ensemble of random realizations. That means the fractal derivative is the progenitor of the fractional derivative, which arises if we deal with a certain stochastic fractal.

scholarly journals On the Solution of Fractional Option Pricing Model by Convolution Theorem

2019 ◽  
pp. 143-157
Author(s):  
A. I. Chukwunezu ◽  
B. O. Osu ◽  
C. Olunkwa ◽  
C. N. Obi
Keyword(s):  
Brownian Motion ◽  
Option Pricing ◽  
Fractional Brownian Motion ◽  
Long Memory ◽  
Hurst Exponent ◽  
Convolution Theorem ◽  
Pricing Model ◽  
One Dimensional ◽  
Black Scholes ◽  
Option Pricing Model

The classical Black-Scholes equation driven by Brownian motion has no memory, therefore it is proper to replace the Brownian motion with fractional Brownian motion (FBM) which has long-memory due to the presence of the Hurst exponent. In this paper, the option pricing equation modeled by fractional Brownian motion is obtained. It is further reduced to a one-dimensional heat equation using Fourier transform and then a solution is obtained by applying the convolution theorem.

scholarly journals Dimensions of Fractional Brownian Images

2021 ◽  
Author(s):  
Stuart A. Burrell
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Stochastic Processes ◽  
Fractional Brownian Motion ◽  
Box Dimension ◽  
General Strategy ◽  
Borel Sets ◽  
Exceptional Sets ◽  
Explicit Condition ◽  
Box Dimensions

AbstractThis paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential-theoretic methods are used to produce dimension bounds for images of sets under Hölder maps and certain stochastic processes. We apply this to compute the almost-sure value of the dimension of Borel sets under index-$$\alpha $$ α fractional Brownian motion in terms of dimension profiles defined using capacities. As a corollary, this establishes continuity of the profiles for Borel sets and allows us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Hölder images such as projections. The methods used propose a general strategy for related problems; dimensional information about a set may be learned from analysing particular fractional Brownian images of that set. To conclude, we obtain bounds on the Hausdorff dimension of exceptional sets, with respect to intermediate dimensions, in the setting of projections.

Fractional Brownian motion with two-variable Hurst exponent

2020 ◽  
pp. 113262
Author(s):  
H. Maleki Almani ◽  
S.M. Hosseini ◽  
M. Tahmasebi
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Hurst Exponent

scholarly journals Fractional Laplace motion

2006 ◽  
Vol 38 (02) ◽  
pp. 451-464 ◽  
Author(s):  
T. J. Kozubowski ◽  
M. M. Meerschaert ◽  
K. Podgórski
Keyword(s):  
Time Series ◽  
Brownian Motion ◽  
Hydraulic Conductivity ◽  
Fractional Brownian Motion ◽  
Long Range ◽  
Financial Time Series ◽  
Long Range Dependence ◽  
Self Similarity ◽  
One Dimensional ◽  
Financial Time

Fractional Laplace motion is obtained by subordinating fractional Brownian motion to a gamma process. Used recently to model hydraulic conductivity fields in geophysics, it might also prove useful in modeling financial time series. Its one-dimensional distributions are scale mixtures of normal laws, where the stochastic variance has the generalized gamma distribution. These one-dimensional distributions are more peaked at the mode than is a Gaussian distribution, and their tails are heavier. In this paper we derive the basic properties of the process, including a new property called stochastic self-similarity. We also study the corresponding fractional Laplace noise, which may exhibit long-range dependence. Finally, we discuss practical methods for simulation.

A multivariate Weierstrass–Mandelbrot function

1985 ◽  
Vol 400 (1819) ◽  
pp. 331-350 ◽  
Keyword(s):  
Brownian Motion ◽  
Stochastic Processes ◽  
Fractional Brownian Motion ◽  
Continuum Limit ◽  
Small Length ◽  
Stochastic Description ◽  
Fractal Surfaces ◽  
Higher Dimensional ◽  
The Many ◽  
The Continuum

The univariate Weierstrass–Mandelbrot function is generalized to many variables to model higher dimensional stochastic processes such as undersea topography. Because this topography is difficult to measure at small length scales over the many large regions that affect long-ranged acoustic propagation in the ocean, one needs a stochastic description that can be extrapolated to both large and small features. Fractal surfaces are a convenient framework for such a description. Computer-generated plots for the two-variable case are presented. It is shown that in the continuum limit the multivariate function is equivalent to the multivariate fractional Brownian motion.

scholarly journals GENERALIZED (m, k)-Zipf LAW FOR FRACTIONAL BROWNIAN MOTION-LIKE TIME SERIES WITH OR WITHOUT EFFECT OF AN ADDITIONAL LINEAR TREND

2003 ◽  
Vol 14 (03) ◽  
pp. 351-365 ◽  
Author(s):  
PH. BRONLET ◽  
M. AUSLOOS
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Size Effects ◽  
Hurst Exponent ◽  
Linear Trend ◽  
Finite Size ◽  
Time Dependent ◽  
Finite Size Effects ◽  
The Relationship ◽  
Zipf Law

We have translated fractional Brownian motion (FBM) signals into a text based on two "letters", as if the signal fluctuations correspond to a constant stepsize random walk. We have applied the Zipf method to extract the ζ′ exponent relating the word frequency and its rank on a log–log plot. We have studied the variation of the Zipf exponent(s) giving the relationship between the frequency of occurrence of words of length m < 8 made of such two letters: ζ′ is varying as a power law in terms of m. We have also searched how the ζ′ exponent of the Zipf law is influenced by a linear trend and the resulting effect of its slope. We can distinguish finite size effects, and results depending whether the starting FBM is persistent or not, i.e., depending on the FBM Hurst exponent H. It seems then numerically proven that the Zipf exponent of a persistent signal is more influenced by the trend than that of an antipersistent signal. It appears that the conjectured law ζ′ = |2H - 1| only holds near H = 0.5. We have also introduced considerations based on the notion of a time dependent Zipf law along the signal.

Fractional derivatives of multidimensional Colombeau generalized stochastic processes

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Danijela Rajter-Ćirić ◽  
Mirjana Stojanović
Keyword(s):  
Stochastic Process ◽  
Fractional Derivative ◽  
Stochastic Processes ◽  
Compact Support ◽  
Caputo Fractional Derivative ◽  
Fractional Derivatives ◽  
One Dimensional ◽  
Compactly Supported ◽  
Derivatives Of ◽  
Do So

AbstractWe consider fractional derivatives of a Colombeau generalized stochastic process G defined on ℝn. We first introduce the Caputo fractional derivative of a one-dimensional Colombeau generalized stochastic process and then generalize the procedure to the Caputo partial fractional derivatives of a multidimensional Colombeau generalized stochastic process. To do so, the Colombeau generalized stochastic process G has to have a compact support. We prove that an arbitrary Caputo partial fractional derivative of a compactly supported Colombeau generalized stochastic process is a Colombeau generalized stochastic process itself, but not necessarily with a compact support.

scholarly journals Spectral characteristics of high latitude raw 40 MHz cosmic noise signals

2015 ◽  
Vol 2 (4) ◽  
pp. 969-987
Author(s):  
C. M. Hall
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Hurst Exponent ◽  
Spectral Characteristics ◽  
Noise Signal ◽  
Density Profiles ◽  
Generalized Hurst Exponent ◽  
Electron Density Profiles ◽  
Cosmic Noise ◽  
Non Gaussian

Abstract. Cosmic noise at 40 MHz is measured at Ny-Ålesund (79° N, 12° E) using a relative ionospheric opacity meter ("riometer"). A riometer is normally used to determine the degree to which cosmic noise is absorbed by the intervening ionosphere, giving an indication of ionization of the atmosphere at altitudes lower than generally monitored by other instruments. The usual course is to determine a "quiet-day" variation, this representing the galactic noise signal itself in the absence of absorption; the current signal is then subtracted from this to arrive at absorption expressed in dB. By a variety of means and assumptions, it is thereafter possible to estimate electron density profiles in the very lowest reaches of the ionosphere. Here however, the entire signal, i.e. including the cosmic noise itself will be examined and spectral characteristics identified. It will be seen that distinct spectral subranges are evident which can, in turn be identified with non-Gaussian processes characterized by generalized Hurst exponents, α. Considering all periods greater than 1 h, α &amp;approx; 1.24 – an indication of fractional Brownian motion, whereas for periods greater than 1 day α &amp;approx; 0.9 – approximately pink noise and just in the domain of fractional Gaussian noise. The results are compared with other physical processes suggesting that absorption of cosmic noise is characterized by a generalized Hurst exponent &amp;approx; 1.24 and thus non-persistent fractional Brownian motion, whereas generation of cosmic noise is characterized by a generalized Hurst exponent &amp;approx; 1.

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