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 , .

On the Lamperti transform of the fractional Brownian sheet

2016 ◽  
Vol 19 (6) ◽  
Author(s):  
Marwa Khalil ◽  
Ciprian Tudor ◽  
Mounir Zili
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Gaussian Processes ◽  
Stationary Process ◽  
Similar Process ◽  
Brownian Sheet ◽  
Fractional Brownian Sheet ◽  
Self Similar ◽  
Strictly Stationary Process

AbstractIn 1962 Lamperti introduced a transformation that associates to every non-trivial self-similar process a strictly stationary process. This transform has been widely studied for Gaussian processes and in particular for fractional Brownian motion. Our aim is to analyze various properties of the Lamperti transform of the fractional Brownian sheet. We give the stochastic differential equation satisfied by this transform and we represent it as a series of independent Ornstein-Uhlenbeck sheets.

FRACTIONAL BROWNIAN MOTION WITH STOCHASTIC VARIANCE: MODELING ABSOLUTE RETURNS IN STOCK MARKETS

2008 ◽  
Vol 19 (08) ◽  
pp. 1221-1242 ◽  
Author(s):  
H. E. ROMAN ◽  
M. PORTO
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stock Markets ◽  
Distribution Functions ◽  
Conditional Heteroskedasticity ◽  
Arch Models ◽  
Variance Modeling ◽  
Probability Distribution Functions ◽  
Second Moments ◽  
Long Time

We discuss a model for simulating a long-time memory in time series characterized in addition by a stochastic variance. The model is based on a combination of fractional Brownian motion (FBM) concepts, for dealing with the long-time memory, with an autoregressive scheme with conditional heteroskedasticity (ARCH), responsible for the stochastic variance of the series, and is denoted as FBMARCH. Unlike well-known fractionally integrated autoregressive models, FBMARCH admits finite second moments. The resulting probability distribution functions have power-law tails with exponents similar to ARCH models. This idea is applied to the description of long-time autocorrelations of absolute returns ubiquitously observed in stock markets.

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.

scholarly journals Some fractional and multifractional Gaussian processes: A brief introduction

2015 ◽  
Vol 36 ◽  
pp. 1560001
Author(s):  
S. C. Lim ◽  
C. H. Eab
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Gaussian Processes ◽  
Natural Phenomena ◽  
Liouville Type ◽  
Multifractional Brownian Motion

This paper gives a brief introduction to some important fractional and multifractional Gaussian processes commonly used in modelling natural phenomena and man-made systems. The processes include fractional Brownian motion (both standard and the Riemann-Liouville type), multifractional Brownian motion, fractional and multifractional Ornstein-Uhlenbeck processes, fractional and mutifractional Reisz-Bessel motion. Possible applications of these processes are briefly mentioned.

scholarly journals Time-averaging and emerging nonergodicity upon resetting of fractional Brownian motion and heterogeneous diffusion processes

2021 ◽  
Author(s):  
Wei Wang ◽  
Andrey G. Cherstvy ◽  
Holger Kantz ◽  
Ralf Metzler ◽  
Igor M. Sokolov
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Diffusion Processes ◽  
Stochastic Simulations ◽  
Time Averaging ◽  
Weak Ergodicity ◽  
Experimental Systems ◽  
Long Time ◽  
Breaking Parameter ◽  
Ergodicity Breaking

How different are the results of constant-rate resetting of anomalous-diffusion processes in terms of their ensemble-averaged versus time-averaged mean-squared displacements (MSDs versus TAMSDs) and how does the process of stochastic resetting impact nonergodicity? These are the main questions addressed in this study. Specifically, we examine, both analytically and by stochastic simulations, the implications of resetting on the MSD-and TAMSD-based spreading dynamics of fractional Brownian motion (FBM) with a long-time memory, of heterogeneous diffusion processes (HDPs) with a power-law-like space-dependent diffusivity D(x) = D0 |x| γ, and of their “combined” process of HDP-FBM. We find, i.a., that the resetting dynamics of originally ergodic FBM for superdiffusive choices of the Hurst exponent develops distinct disparities in the scaling behavior and magnitudes of the MSDs and mean TAMSDs, indicating so-called weak ergodicity breaking (WEB). For subdiffusive HDPs we also quantify the nonequivalence of the MSD and TAMSD, and additionally observe a new trimodal form of the probability density function (PDF) of particle’ displacements. For all three reset processes (FBM, HDPs, and HDP-FBM) we compute analytically and verify by stochastic computer simulations the short-time (normal and anomalous) MSD and TAMSD asymptotes (making conclusions about WEB) as well as the long-time MSD and TAMSD plateaus, reminiscent of those for “confined” processes. We show that certain characteristics of the reset processes studied are functionally similar, despite the very different stochastic nature of their nonreset variants. Importantly, we discover nonmonotonicity of the ergodicity breaking parameter EB as a function of the resetting rate r. For all the reset processes studied, we unveil a pronounced resetting-induced nonergodicity with a maximum of EB at intermediate r and EB ∼ (1/r)-decay at large r values. Together with the emerging MSD-versus-TAMSD disparity, this pronounced r-dependence of the EB parameter can be an experimentally testable prediction. We conclude via discussing some implications of our results to experimental systems featuring resetting dynamics.

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