Upper and lower classes for \(\mathbb{L}^ 2\)- and \(\mathbb{L}^ p\)-norms of Brownian motion and norms of \(\alpha\)-stable motion
type: Item
instance of: scholarly article
title: Upper and lower classes for \(\mathbb{L}^ 2\)- and \(\mathbb{L}^ p\)-norms of Brownian motion and norms of \(\alpha\)-stable motion
zbMATH Open document ID: 0839.60072
DOI: 10.1016/0304-4149(94)00009-I
author: J. M. P. Albin
published in: Stochastic Processes and their Applications
publication date: 1996-06-12T00:00:00Z
review text: Let \(X (\theta, t) = \int^t_{\theta t} |B (s) |^pds\), \(p \geq 1\), \(t \geq 0\), \(\theta \in [0,1)\), where \(B\) is standard Brownian motion. Integral tests for upper and lower functions for \(X (\theta,t)\) are given in the case when \(p = 2\). Next, the growth of \(X (\theta,t)\) is studied for any \(p \geq 1\). Finally, the growth of \(|\xi (\cdot) I_{[\theta t,t)} (\cdot) |\) is studied. Here \(|\cdot |\) is a measurable norm on \(\{(f : {\mathbf R}^+ \to {\mathbf R}) : f\) right-continuous with compact support\} satisfying some regularity assumptions, and \(\xi\) is a right-continuous \(\alpha\)-stable motion with \(\alpha \in (0,1) \cup (1,2)\) and skewness \(\beta \in [- 1,1]\), or with \((\alpha, \beta) = (1,0)\).
reviewed by: Marius Iosifescu
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 803379
zbMATH Keywords: Brownian motion
zbMATH Keywords: compact support
zbMATH Keywords: regularity assumptions
MaRDI profile type: Publication
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full work available at URL: 0304-4149(94)00009-i
OpenAlex ID: W2094368029
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description: scientific article; zbMATH DE number 803379