scholarly journals Stochastic calculus for fractional Brownian motion with Hurst exponent H>¼: A rough path method by analytic extension

2009 ◽  
Vol 37 (2) ◽  
pp. 565-614 ◽  
Author(s):  
Jérémie Unterberger
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Hurst Exponent ◽  
Stochastic Calculus ◽  
Analytic Extension ◽  
Rough Path

Stochastic Calculus for Fractional Brownian Motion and Applications

2008 ◽  
Author(s):  
Francesca Biagini ◽  
Yaozhong Hu ◽  
Bernt Øksendal ◽  
Tusheng Zhang
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stochastic Calculus

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 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.

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.

scholarly journals On some statistics of fractional Brownian motion

2021 ◽  
pp. 131-138
Author(s):  
Viktor Bondarenko
Keyword(s):  
Stochastic Process ◽  
Time Series ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Hurst Exponent ◽  
Limit Theorems ◽  
Goodness Of Fit ◽  
Mean Square ◽  
One Step ◽  
Optimal Forecasting

Fractional Brownian motion as a method for estimating the parameters of a stochastic process by variance and one-step increment covariance is proposed and substantiated. The root-mean-square consistency of the constructed estimates has been proven. The obtained results complement and generalize the consequences of limit theorems for fractional Brownian motion, that have been proved in the number of articles. The necessity to estimate the variance is caused by the absence of a base unit of time and the estimation of the covariance allows one to determine the Hurst exponent. The established results let the known limit theorems to be used to construct goodness-of-fit criteria for the hypothesis “the observed time series is a transformation of fractional Brownian motion” and to estimate the error of optimal forecasting for time series.

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