scholarly journals LIMIT THEOREMS FOR A CLASS OF ADDITIVE FUNCTIONALS OF SYMMETRIC STABLE PROCESS AND FRACTIONAL BROWNIAN MOTION IN BESOV-ORLICZ SPACES

2012 ◽  
Vol 8 (4) ◽  
pp. 506-516
Author(s):  
Ouahra
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Limit Theorems ◽  
Stable Process ◽  
Orlicz Spaces ◽  
Symmetric Stable Process ◽  
Additive Functionals

On limit theorems of some extensions of fractional Brownian motion and their additive functionals

2017 ◽  
Vol 17 (03) ◽  
pp. 1750022
Author(s):  
M. Ait Ouahra ◽  
S. Moussaten ◽  
A. Sghir
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Local Time ◽  
Limit Theorems ◽  
Fractional Derivatives ◽  
Slowly Varying Function ◽  
Bifractional Brownian Motion ◽  
Additive Functionals ◽  
Subfractional Brownian Motion

This paper is divided into two parts. The first deals with some limit theorems to certain extensions of fractional Brownian motion like: bifractional Brownian motion, subfractional Brownian motion and weighted fractional Brownian motion. In the second part we give the similar results of their continuous additive functionals; more precisely, local time and its fractional derivatives involving slowly varying function.

scholarly journals Second-order limit laws for occupation times of fractional Brownian motion

2017 ◽  
Vol 54 (2) ◽  
pp. 444-461 ◽  
Author(s):  
Fangjun Xu
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Method Of Moments ◽  
Second Order ◽  
Hurst Index ◽  
Occupation Times ◽  
Limit Laws ◽  
Additive Functionals ◽  
Finite Dimensional ◽  
Functional Version

Abstract We prove a second-order limit law for additive functionals of a d-dimensional fractional Brownian motion with Hurst index H = 1 / d, using the method of moments and extending the Kallianpur–Robbins law, and then give a functional version of this result. That is, we generalize it to the convergence of the finite-dimensional distributions for corresponding stochastic processes.

scholarly journals On the Fractional Wave Equation

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 874
Author(s):  
Francesco Iafrate ◽  
Enzo Orsingher
Keyword(s):  
Brownian Motion ◽  
Wave Equation ◽  
Wave Equations ◽  
Stable Process ◽  
Space Time ◽  
Probabilistic Interpretation ◽  
Symmetric Stable Process ◽  
Fractional Wave Equation ◽  
Fractional Wave Equations ◽  
Time Fractional Wave Equation

In this paper we study the time-fractional wave equation of order 1 < ν < 2 and give a probabilistic interpretation of its solution. In the case 0 < ν < 1 , d = 1 , the solution can be interpreted as a time-changed Brownian motion, while for 1 < ν < 2 it coincides with the density of a symmetric stable process of order 2 / ν . We give here an interpretation of the fractional wave equation for d > 1 in terms of laws of stable d−dimensional processes. We give a hint at the case of a fractional wave equation for ν > 2 and also at space-time fractional wave equations.

SOME LIMIT THEOREMS ON THE INCREMENTS OF A MULTI-PARAMETER FRACTIONAL BROWNIAN MOTION

2001 ◽  
Vol 19 (4) ◽  
pp. 499-517 ◽  
Author(s):  
Zheng-Yan Lin ◽  
Yong-Kab Choi ◽  
Kyo-Shin Hwang
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Limit Theorems

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.

scholarly journals The synchronization of coupled stochastic systems driven by symmetric α-stable process and Brownian motion

Filomat ◽  
2018 ◽  
Vol 32 (6) ◽  
pp. 2219-2245
Author(s):  
Shahad Al-Azzawi ◽  
Jicheng Liu ◽  
Xianming Liu
Keyword(s):  
Dynamical System ◽  
Mathematical Model ◽  
Brownian Motion ◽  
Gaussian Noise ◽  
Stochastic Systems ◽  
Stable Process ◽  
Symmetric Stable Process ◽  
And Brownian Motion ◽  
Coupled Dynamical System ◽  
Non Gaussian

The synchronization of stochastic differential equations (SDEs) driven by symmetric ?-stable process and Brownian Motion is investigated in pathwise sense. This coupled dynamical system is a new mathematical model, where one of the systems is driven by Gaussian noise, another one is driven by non- Gaussian noise. In this paper, we prove that the synchronization still persists for this coupled dynamical system. Examples and simulations are given.

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.

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