Hölder conditions for the local times of multiscale fractional Brownian motion
type: Item
instance of: scholarly article
title: Hölder conditions for the local times of multiscale fractional Brownian motion
zbMATH Open document ID: 1106.60039
author: Raby Guerbaz
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2006-12-14T00:00:00Z
review text: For \(K\in \mathbb{N}\) an \(M_K\)-multiscale fractional Brownian motion is defined: it is the Gaussian process with stationary increments defined by \[ X_\rho(t)=\int_{\mathbb{R}} \frac{e^{it\lambda}-1}{\rho(\lambda)}\,dW(\lambda), \] where \(dW(\lambda)\) is the Brownian measure on \(L_2(\mathbb{R})\), and for \(i=0,\dots,K\) there exist \((w_i,\sigma_i,H_i)\in (\mathbb{R}_+,\mathbb{R}_+,]0,1[)\) such that \(\rho(x)=\sigma_i^{-1}| x| ^{H_i+1/2}\) for \(| x| \in D_i=[w_i,w_{i+1}[\) with \(w_0=0<w_1<\dots <w_K<w_{K+1}=\infty\). The law of the iterated logarithm for \(X_\rho\) is proved. Further, \(X_\rho\) admits a.s. jointly continuous local time \(L(t,x)\), and the estimates on the modulus of continuity of \(L(x,t)\) are found. Also, some corollaries from these two results are derived.
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60J55
zbMATH DE Number: 5080200
zbMATH Keywords: multiscale Brownian motion
zbMATH Keywords: law of the iterated logarithm
reviewed by: V. Knopova
MaRDI profile type: Publication
full work available at URL: j.crma.2006.09.026
OpenAlex ID: W1993976507
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DOI: 10.1016/J.CRMA.2006.09.026
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description: scientific article; zbMATH DE number 5080200