Maximum of a fractional Brownian motion: Probabilities of small values
type: Item
instance of: scholarly article
title: Maximum of a fractional Brownian motion: Probabilities of small values
zbMATH Open document ID: 0942.60065
author: G. M. Molchan
published in: Communications in Mathematical Physics
publication date: 1999-12-16T00:00:00Z
review text: Let \(b_\gamma(t)\), \(b_\gamma(0)= 0\), be a fractional Brownian motion, i.e., a Gaussian process with the structure function \(E(|b_\gamma(t)- b_\gamma(s)|)^2=|t- s|^\gamma\), \(0<\gamma<2\). The author establishes the following results: 1. The maximum \(M_t\) of a fractional Brownian motion on \([0,t]\) obeys the asymptotics \(\ln P\{M_t< 1\}= D\ln t^{-1}(1+ O((\ln t)^{-1/2}))\), \(t\to\infty\), where \(D= 1-\gamma/2\). 2. If \(G\) is a bounded convex region on \(\mathbb{R}^d\) which contains a vicinity of the point \(t= 0\), and \(M(G)= \sup\{b_\gamma(t), t\in G\}\), then \(\ln P\{M(TG)< 1\}= -d\ln T(1+ O(\ln T)^{-1/2})\), \(T\to\infty\), where \(TG\) is a region similar to \(G\) and has similarity coefficient \(T\).
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 1379907
zbMATH Keywords: fractional Brownian motion
zbMATH Keywords: small values
reviewed by: Ion C. Vladimirescu
MaRDI profile type: Publication
full work available at URL: s002200050669
OpenAlex ID: W1983336697
DOI: 10.1007/S002200050669
description: scientific article; zbMATH DE number 1379907