Mixed fractional Brownian motion
type: Item
instance of: scholarly article
title: Mixed fractional Brownian motion
zbMATH Open document ID: 1005.60053
DOI: 10.2307/3318626
author: Patrick Cheridito
published in: Bernoulli
publication date: 2003-02-19T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bj/1078951129
review text: Let \(B\) be the standard Brownian motion and \(B^H\) fractional Brownian motion with Hurst index \(H\in (0,1]\). If the Brownian motion \(B\) and the fractional Brownian motion \(B^H\) are independent and \(\alpha\in\mathbb{R} \setminus \{0\}\), define the mixed fractional Brownian motion \(M^{H,\alpha}\) by \(M^{H,\alpha} \doteq B+\alpha B^H\). If \(X\) is a stochastic process, denote by \(\mathbb{F}^X\) the filtration generated by \(X\). If \(f=\sum^{n-1}_{j=0} f_jI_{(t_j, t_{j+1}]}\) is a simple \(\mathbb{F}^X\)-predictable process on \((0,1]\): \(|f|\leq 1\), \(f_j\in F_{t_j}^X\) and \(0\leq t_0<t_1 <\cdots< t_n\leq 1\). If for every simple predictable process the random variables \(\sum^{n-1}_{j=0} f_j(X_{t_{j+1}}-X_{t_j})\) are bounded, the process \(X\) is a weak semimartingale. Two continuous processes \(X,Y\) are equivalent, if the measures \(\mathbb{P}_X\) and \(\mathbb{P}_Y\) are equivalent on \(C[0,1]\). The main results of this paper are: If \(H\in(0,1) \setminus \{\frac 12\}\), then the fractional Brownian motion \(B^H\) is not a weak semimartingale. If \(H\in(0,\frac 12) \cup(\frac 12,\frac 34]\), then the mixed fractional Brownian motion \(M^{H,\alpha}\) is not a weak semimartingale. If \(H\in (\frac 34,1]\), then the mixed fractional Brownian motion \(M^{H,\alpha}\) is equivalent to standard Brownian motion \(B\). The paper ends with some applications to pricing models in finance.
reviewed by: Esko Valkeila
Mathematics Subject Classification ID: 60G30
Mathematics Subject Classification ID: 60J65
Mathematics Subject Classification ID: 60G15
zbMATH DE Number: 1786702
zbMATH Keywords: fractional Brownian motion
zbMATH Keywords: equivalent measures
zbMATH Keywords: semimartingales
MaRDI profile type: Publication
OpenAlex ID: W2016193062
description: scientific article; zbMATH DE number 1786702