The uniform modulus of continuity of iterated Brownian motion
type: Item
instance of: scholarly article
title: The uniform modulus of continuity of iterated Brownian motion
zbMATH Open document ID: 0880.60081
DOI: 10.1007/BF02214652
author: D. Khoshnevisan
author: Thomas M. Lewis
published in: Journal of Theoretical Probability
publication date: 1998-01-06T00:00:00Z
review text: Let \(X\) be a Brownian motion on \(R^1\) with \(B_0=0\) and \(Y\) be another Brownian motion on \(R^{+}\), independent of \(X\). The iterated Brownian motion (IBM) is defined by \(Z_t=X_{Y_t}.\) IBM was introduced by \textit{P. Deheuvels} and \textit{D. M. Mason} [in: Probability in Banach Spaces, 8. Prog. Probab. 30, 255-266 (1992; Zbl 0844.60012)]. The uniform modulus of Z is studied in the present paper. For \(\delta\in (0,1),\) let \(\omega (\delta )\) \(=\sup_{0\leq s,t\leq 1}\) \(\sup_{0\leq {}s-t{}\leq\delta}\) \({}Z_t\) \(-Z_s{}\) and \(\psi ( \delta )=\delta^{1/4}(\ln(1/\delta))^{3/4}\). It is proven that \(P(\lim_{\delta\to 0}(\omega (\delta)/\psi(\delta))=1)=1\).
reviewed by: Gong Guanglu
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 1000171
zbMATH Keywords: Iterated Brownian motion
zbMATH Keywords: uniform modulus of continuity
zbMATH Keywords: Ray-Knight theorem
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02214652
OpenAlex ID: W2077488641
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description: scientific article; zbMATH DE number 1000171