Fractional Brownian motion with variable Hurst parameter: definition and properties
type: Item
instance of: scholarly article
title: Fractional Brownian motion with variable Hurst parameter: definition and properties
zbMATH Open document ID: 1333.60077
author: Jelena Ryvkina
published in: Journal of Theoretical Probability
publication date: 2015-12-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1306.2870
review text: The author introduces a class of Gaussian processes generalizing the fractional Brownian motion with Hurst index \(H\in (1/2,1).\) Any measurable function attaining values in the interval \((1/2,1)\) can be chosen as a variable Hurst parameter for this new class. These processes allow modeling of phenomena where long-range dependence is present and self-similarity changes as the phenomenon evolves. Sample path properties of these processes are investigated. Fokker-Plank-type equations for such processes are derived. It is found that the regularity properties of the chosen Hurst function directly correspond to the regularity of the sample paths of the process. Reviewer's comments: Such processes are also called locally self-similar. \textit{P. Gonçalves} and \textit{P. Flandrin} [in: Progress in wavelet analysis and applications. Proceedings of the 3rd international conference on wavelets and applications, Toulouse, France, 1992. Gif-sur-Yvette: Editions Frontières, 271--276 (1993; Zbl 0878.94024)] proposed such processes. \textit{Y.-Z. Wang} et al. [J. Stat. Plann. Inference 99, No. 1, 91--110 (2001; Zbl 0989.62045)] studied estimations of the self-similarity index for locally self-similar processes (cf. [\textit{B. L. S. Prakasa Rao}, Statistical inference for fractional diffusion processes. Wiley Series in Probability and Statistics. Chichester: John Wiley \& Sons (2010; Zbl 1211.62143)]).
reviewed by: B. L. S. Prakasa Rao
Mathematics Subject Classification ID: 60G22
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G17
Mathematics Subject Classification ID: 60G18
zbMATH DE Number: 6516669
zbMATH Keywords: fractional Brownian motion
zbMATH Keywords: variable Hurst parameter
zbMATH Keywords: Gaussian processes
zbMATH Keywords: self-similarity
zbMATH Keywords: sample path regularity
MaRDI profile type: Publication
OpenAlex ID: W2081671937
arXiv ID: 1306.2870
cites work: Q5558293
cites work: Stochastic calculus with respect to Gaussian processes
cites work: Lévy Processes and Stochastic Calculus
cites work: The random walk's guide to anomalous diffusion: A fractional dynamics approach
cites work: Elliptic Gaussian random processes
cites work: Stochastic Calculus for Fractional Brownian Motion and Applications
cites work: Local time and Tanaka formula for a Volterra-type multifractional Gaussian process
cites work: Some zero-one laws for Gaussian processes
cites work: Estimation and pricing under long-memory stochastic volatility
cites work: Long memory in continuous-time stochastic volatility models
cites work: Q3223360
cites work: Stochastic integration with respect to Volterra processes
cites work: Q4945846
cites work: Q4788946
cites work: On time-changed Gaussian processes and their associated Fokker-Planck-Kolmogorov equations
cites work: Gaussian Hilbert Spaces
cites work: Probability in Banach spaces. Isoperimetry and processes
cites work: Fractional Brownian Motions, Fractional Noises and Applications
cites work: Limit theorems for continuous-time random walks with infinite mean waiting times
cites work: Piecewise fractional Brownian motion
cites work: Q3744481
cites work: Path properties of multifractal Brownian motion
cites work: Chaos, fractional kinetics, and anomalous transport
DOI: 10.1007/S10959-013-0502-3
description: scientific article