$Some linear fractional stochastic equations$

Some linear fractional stochastic equations

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Some linear fractional stochastic equations Ivan Nourdin? LPMA, Universite Pierre et Marie Curie Paris 6, Boıte courrier 188, 4 Place Jussieu, 75252 Paris Cedex 5, France Ciprian A. Tudor SAMOS/MATISSE, Centre d'Economie de La Sorbonne, Universite de Pantheon-Sorbonne Paris 1 90, rue de Tolbiac, 75634 Paris Cedex 13, France Abstract Using the multiple stochastic integrals we prove an existence and uniqueness result for a linear stochastic equation driven by the fractional Brownian motion with any Hurst parameter. We study both the one parameter and two parameter cases. When the drift is zero, we show that in the one-parameter case the solution in an exponential, thus positive, function while in the two-parameter settings the solution is negative on a non-negligible set. Key words: Fractional Brownian motion, fractional Brownian sheet, multiple stochastic in- tegral, Girsanov transform. 2000 Mathematics Subject Classification: 60H05, 60G15, 60G18. 1 Introduction The significant amount of applications where the fractional Brownian motion (fBm) is used led to the intensive development of the stochastic calculus with respect to this process and its planar version. The study of stochastic differential equations (SDEs) driven by a ?Corresponding author 1

universite de pantheon-sorbonne paris

skorohod stochastic

fractional brownian

wiener-ito multiple

hilbert space

standard wiener

techniques like fractional

brownian sheet

indicator functions

standard brownian

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Publié par

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Nur al-Din Hilbert space

Some linear fractional stochastic equations

∗
Ivan Nourdin
LPMA, Universit´ Pierre et Marie Curie Paris 6,
Boıˆtecourrier188,4PlaceJussieu,75252ParisCedex5,France
nourdin@ccr.jussieu.fr

Ciprian A. Tudor
SAMOS/MATISSE, Centre d’Economie de La Sorbonne,
Universit´ de Panth´on-Sorbonne Paris 1
90, rue de Tolbiac, 75634 Paris C´dex 13, France
tudor@univ-paris1.fr

Abstract
Using the multiple stochastic integrals we prove an existence and uniqueness result
for a linear stochastic equation driven by the fractional Brownian motion with any
Hurst parameter.We study both the one parameter and two parameter cases.When
the drift is zero, we show that in the one-parameter case the solution in an exponential,
thus positive, function while in the two-parameter settings the solution is negative on
a non-negligible set.

Key words:Fractional Brownian motion, fractional Brownian sheet, multiple stochastic
integral, Girsanov transform.

2000 Mathematics Subject Classiﬁcation:60H05, 60G15, 60G18.

Introduction

The signiﬁcant amount of applications where the fractional Brownian motion (fBm) is used
led to the intensive development of the stochastic calculus with respect to this process
and its planar version.The study of stochastic diﬀerential equations (SDEs) driven by a

∗
Corresponding author

Voir