Approximation at first and second order of {\(m\)}-order integrals of the fractional {B}rownian motion and of certain semimartingales
type: Item
instance of: scholarly article
title: Approximation at first and second order of {\(m\)}-order integrals of the fractional {B}rownian motion and of certain semimartingales
zbMATH Open document ID: 1063.60079
author: Mihai Gradinaru
author: Ivan Nourdin
published in: Electronic Journal of Probability
publication date: 2005-03-08T00:00:00Z
full work available at URL: https://eudml.org/doc/124686
full work available at URL: paper18.abs.html
review text: The authors investigate the accurate convergence of some approximations of \(m\)-order integrals which appear when one performs stochastic calculus with respect to processes which are not semimartingales, for instance the fractional Brownian motion. Let \(X\) be the fractional Brownian motion of any Hurst index \(H\) in \((0,1)\) (resp. a semimartingale). Let \(Y\) be a continuous process and let \(m\) be a positive integer. The authors study the existence of the limit of the approximation of \(m\)-order integral of \(Y\) with respect to \(X\) and prove that the limits are almost sure, uniformly on each compact interval, and are in terms of the \(m\)th moment of the Gaussian standard random variable. In particular, if \(m\) is an odd integer, the limit equals to zero. They also show that the limit is a Brownian motion when \(X\) is the fractional Brownian motion of index \(H\) in \((0,1]\), and it is in terms of a two-dimensional standard Brownian motion when \(X\) is a semimartingale.
reviewed by: Kun Soo Chang
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 2142122
zbMATH Keywords: \(m\)-order integrals
zbMATH Keywords: fractional Brownian motion
zbMATH Keywords: limit theorems
zbMATH Keywords: stochastic integrals
MaRDI profile type: Publication
DOI: 10.1214/EJP.V8-166
description: scientific article; zbMATH DE number 2142122