On Gaussian Processes Equivalent in Law to Fractional Brownian Motion

2004 ◽  
Vol 17 (2) ◽  
pp. 309-325 ◽  
Author(s):  
T. Sottinen
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Gaussian Processes

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 Limit theorems for a quadratic variation of Gaussian processes

2011 ◽  
Vol 16 (4) ◽  
pp. 435-452 ◽  
Author(s):  
Raimondas Malukas
Keyword(s):  
Brownian Motion ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Fractional Brownian Motion ◽  
Central Limit ◽  
Gaussian Processes ◽  
Limit Theorems ◽  
Quadratic Variation

In the paper a weighted quadratic variation based on a sequence of partitions for a class of Gaussian processes is considered. Conditions on the sequence of partitions and the process are established for the quadratic variation to converge almost surely and for a central limit theorem to be true. Also applications to bifractional and sub-fractional Brownian motion and the estimation of their parameters are provided.

scholarly journals On the Integral of the Fractional Brownian Motion and Some Pseudo-Fractional Gaussian Processes

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 991 ◽  
Author(s):  
Mario Abundo ◽  
Enrica Pirozzi
Keyword(s):  
Brownian Motion ◽  
Markov Process ◽  
Gaussian Process ◽  
Fractional Brownian Motion ◽  
Gaussian Processes ◽  
Statistical Parameters ◽  
Uhlenbeck Process ◽  
Ornstein Uhlenbeck Process ◽  
Over Time

We investigate the main statistical parameters of the integral over time of the fractional Brownian motion and of a kind of pseudo-fractional Gaussian process, obtained as a classical Gauss–Markov process from Doob representation by replacing Brownian motion with fractional Brownian motion. Possible applications in the context of neuronal models are highlighted. A fractional Ornstein–Uhlenbeck process is considered and relations with the integral of the pseudo-fractional Gaussian process are provided.

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 , .

scholarly journals Spectral asymptotics for a class of integro-differential equations arising in the theory of fractional Gaussian processes

2020 ◽  
pp. 2050049
Author(s):  
Alexander I. Nazarov
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Fractional Brownian Motion ◽  
Gaussian Processes ◽  
Spectral Asymptotics ◽  
Eigenvalue Asymptotics ◽  
Small Ball ◽  
Small Ball Probabilities ◽  
Spectral Problems

We study spectral problems for integro-differential equations arising in the theory of Gaussian processes similar to the fractional Brownian motion. We generalize the method of Chigansky–Kleptsyna and obtain the two-term eigenvalue asymptotics for such equations. Application to the small ball probabilities in [Formula: see text]-norm is given.

scholarly journals Collision of eigenvalues for matrix-valued processes

2019 ◽  
Vol 09 (04) ◽  
pp. 2030001
Author(s):  
Arturo Jaramillo ◽  
David Nualart
Keyword(s):  
Brownian Motion ◽  
Hausdorff Dimension ◽  
Gaussian Process ◽  
Fractional Brownian Motion ◽  
Gaussian Processes ◽  
Hermitian Matrix ◽  
Symmetric Matrix ◽  
Hurst Parameter ◽  
Real Symmetric Matrix ◽  
Hitting Probabilities

We examine the probability that at least two eigenvalues of a Hermitian matrix-valued Gaussian process, collide. In particular, we determine sharp conditions under which such probability is zero. As an application, we show that the eigenvalues of a real symmetric matrix-valued fractional Brownian motion of Hurst parameter [Formula: see text], collide when [Formula: see text] and do not collide when [Formula: see text], while those of a complex Hermitian fractional Brownian motion collide when [Formula: see text] and do not collide when [Formula: see text]. Our approach is based on the relation between hitting probabilities for Gaussian processes with the capacity and Hausdorff dimension of measurable sets.

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