Table Of ContentLecture Notes in Statistics 216
Corinne Berzin
Alain Latour
José R. León
Inference on the Hurst
Parameter and the
Variance of Diff usions
Driven by Fractional
Brownian Motion
Lecture Notes in Statistics 216
EditedbyP.Bickel,P.J.Diggle,S.E.Fienberg,U.Gather,
I.Olkin,S.Zeger
Forfurthervolumes:
http://www.springer.com/series/694
Corinne Berzin • Alain Latour (cid:129) José R. León
Inference on the Hurst
Parameter and the Variance
of Diffusions Driven
by Fractional Brownian
Motion
123
CorinneBerzin JoséR.León
AlainLatour EscueladeMatemática
UniversitéGrenoble-Alpes FacultaddeCiencias
LaboratoireJean-Kuntzmann UniversidadCentraldeVenezuela
F-38000Grenoble,France Caracas,Venezuela
ISSN0930-0325 ISSN2197-7186(electronic)
ISBN978-3-319-07874-8 ISBN978-3-319-07875-5(eBook)
DOI10.1007/978-3-319-07875-5
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014946965
©SpringerInternationalPublishingSwitzerland2014
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To thememoryof MarioWschebor,our
friendandoutstandingmathematician
(1939–2011)
Foreword
This book is devoted to some stochastic models that present scale invariance.
It is structured around three issues: probabilistic properties, statistical estimation
and simulation of processes and estimators. The interested reader can be either
a specialist of probability, who will find here a friendly presentation of statistics
tools,orastatistician,whowillhavetheoccasiontotacklethemostrecenttheories
in probabilityin order to developcentrallimit theoremsin this context.Both will
certainly be interested in the last part on simulation, which, to my knowledge, is
highlyoriginal.Algorithmsaredescribedingreatdetail,withconcernofprocedures
thatis notusuallyseen inmathematicaltreaties. Thetheoreticalpartisalso partly
original and finds its origin in previouswork of the first and third authors, which
theyimproveandextendhere.
Models under study are fractional Brownian motions (fBm) and processes
that derive from them through stochastic differential equations. Their use for
modeling financial markets is by now well established in the presence of long-
range dependence: fBm with parameter H larger than 1 may then be preferred
2
to Brownian motion. Other applicationsare not as standard as this first one, even
if some of them, such as the description of network traffic, have played a central
role in the development of the theory, as well as in its extension to multifractal
analysis. The diversity of applications will certainly develop with time. I had the
pleasuretoworkseparatelywiththefirstandthirdauthorsonquestionsthatarose
fromthedescriptionofbonemicro-architecture.Thetissueofabonemaybeseenas
aporousmediumwithsomescaleinvariance,andthesemodelsbyfBmhavebeen
testedbydifferentauthors.FractionalBrownianmotionscanalsobeusedtomodel
environmentalphenomena,suchas,forinstance,diffusionofpollutioninalakeor
a river. It is certainly difficult to predict which application will prove to be really
efficientforpracticalissues,butonehasastrongmotivationtodeveloptheoretical
studiesandstatisticaltoolsinordertoidentifyparameters,whichisdonehere.
Evenifthisbookdoesnotdirectlydealwithapplications,itcouldnothavebeen
writteniftheauthorsdidnothavesomeoftheminmind.Sincethechoiceofterms
can be an indicator of their preferences, it looks significant that they call “local
variance” what is usually called “volatility”, by reference to the financial market!
vii
viii Foreword
Theyborrowthis denominationfromMario Wschebor,whopassed away recently
and marked deeply the first and third authors through longstanding collaboration.
The influence of Wschebor can in particular be felt through the presence of
harmonic analysis all along the book: the authors use what they call mollified
versions of processes and would be called filtered versions in signal processing.
SuchapproximationsofprocessesarecentralintheworkofWschebor.
Mostfieldsofmathematicsare,morethaneverthesedays,theobjectofcollective
work. A huge number of mathematicians have contributed to the study of fractal
Brownianmotions,stochasticintegralsandpropertiesofestimatorsofparameters.
It is one of the merits of the present book to rely on this very rich literature but
guide the readersin such a way that issues and proofsare easily accessible. They
let them profit from their intuition and vision but refer also carefully to previous
work. It is impossible to enumerate here all contributors, but two major scientific
figures,whoalsodiedrecently,deservetobementionedfirstandforemost:Benoît
Mandelbrot and Paul Malliavin. Even if they were opponents in many aspects of
scientific life, and specifically concerningthe use of Brownian motion in finance,
theircontributionsaddupforthe greatesthappinessof specialistsof suchrandom
processes.
Everyone now agrees on the fact that Mandelbrot taught us how to see fractal
patterns everywherein naturalobjects. FBm was first introducedby Kolmogorov,
butMandelbrotisreallythefirsttohaveseenhowitconnectedwithfractalanalysis
andcouldbeusedasamodelfordifferentkindsofphenomena.Recallthatroughly
speakingthefractionalBrownianb hasthepropertythat
H
jbH.tC(cid:2)t/(cid:2)bH.t/j'.(cid:2)t/H:
TheparameterH,whichliesbetween0and1,iscalledtheHurstexponentinhonor
ofthephysicistHaroldE.Hurst.Thelatter,who,asahydrologist,studiedtheflow
ofNileduringthefirsthalfofthelastcentury,remarkedthattherescaleddifference
betweenthemaximumandtheminimumvaluesofthisflowduringalengthoftime
T behaveslike TH, with H approximately0:7. Mandelbrothasdescribedindetail
howhegotinterestedinthediscoveryofHurst,andwasledtothedefinitionofthe
fBm,inhisbook“Fractales,hasardetfinance”.Evenifhisinterestforfinancewas
alreadypresent,environmentalissueshavebeenclearlyevokedfromthebeginning
tojustifytheuseoffBmasamodel.
Malliavin, whocontributeddeeplytoanalysisandprobability,introducedpow-
erful theoretical tools for the study of functionals of Gaussian processes as part
of what is called, “Malliavin calculus”. They may be very useful to establish
central limit theorems and are indirectly used in this book through the “Fourth
Moment Theorem”, which is due to Nualart and Peccati (2005) and has led to
a considerable literature. Roughly speaking, for functionals that belong to some
Wiener chaos (in particular for quadratic variations, which are currently used in
signalprocessing),theconsiderationoffourthmomentsissufficienttoprovecentral
limittheorems.Thisissystematicallyusedinthisbook,whileitnotsowellknown
Foreword ix
from statisticians. It leads the authors to rapid and nice proofs, which deserve to
serveasmodelsforthefuture.
Lastbutnotleast,thereaderwillalsofindelegantandnewproofsforalmostsure
convergence. This is only one example of the many contributions of the authors
(others concern the back and forth between discrete and continuous models, for
instance)thathe/shewilldiscoverallalongthisbook.Nottomentionagainthelast
part, whose approachis likely to change practicesin computationalstatistics. But
nowitistimetostartreading.Asaconclusionletmesaywhatapleasureitisfor
metorecommendthis.
Orléans,France AlineBonami
November23,2013
Reference
Nualart, D., & Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic
integrals.TheAnnalsofProbability,33(1),177–193.
