Stochastic Models for Fractional Calculus

340 Pages·2019·4.053 MB·English

by  Mark M. Meerschaert, Alla Sikorskii

Mark M. Meerschaert, Alla Sikorskii
Stochastic Models for Fractional Calculus
De Gruyter Studies in
Mathematics
Edited by
Carsten Carstensen, Berlin, Germany
Gavril Farkas, Berlin, Germany
Nicola Fusco, Napoli, Italy
Fritz Gesztesy, Waco, Texas, USA
Niels Jacob, Swansea, United Kingdom
Zenghu Li, Beijing, China
Karl-Hermann Neeb, Erlangen, Germany
Volume 43

Mark M. Meerschaert, Alla Sikorskii
Stochastic Models for
Fractional Calculus
2nd Edition
Mathematics Subject Classification 2010
60F05, 26A33, 82C31, 60G05, 60H30
Authors
Prof. Dr. Mark M. Meerschaert
Michigan State University
Department of Statistics and Probability
C430 Wells Hall
619 Red Cedar Road
East Lansing, MI 48824-1027
USA
[email protected]
Prof. Dr. Alla Sikorskii
Michigan State University
Department of Statistics and Probability
C430 Wells Hall
619 Red Cedar Road
East Lansing, MI 48824-1027
USA
[email protected]
ISBN 978-3-11-055907-1
e-ISBN (PDF) 978-3-11-056024-4
e-ISBN (EPUB) 978-3-11-055914-9
ISSN 0179-0986
Library of Congress Control Number: 2019914385
Bibliographic information published by the Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available on the Internet at http://dnb.dnb.de.
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Printing and binding: CPI books GmbH, Leck
www.degruyter.com
|
Thisbookisdedicatedtoourspousesandfamilies.Yourlove,support,andpatience
makeseverythingpossible.

Preface
ThisbookisbasedonaseriesoflecturenotesforagraduatecourseintheDepartment
ofStatisticsandProbabilityatMichiganStateUniversity.Thegoalistopreparegradu-
atestudentsforresearchintheareaoffractionalcalculus,anomalousdiffusion,and
heavytails.Thebookcoversbasiclimittheoremsforrandomvariablesandrandom
vectorswithheavytails.Thisincludesregularvariation,triangulararrays,infinitely
divisiblelaws,randomwalks,andstochasticprocessconvergenceintheSkorokhod
topology. The basicideasof fractionalcalculusand anomalousdiffusionareintro-
ducedinthecontextofprobabilitytheory.Eachsectionofthebookprovidesmaterial
roughly equivalenttoonelecture.Mostsectionsconcludewithsomeadditionalde-
tails,intendedforindividualreading,andtomakethebookrelativelyself-contained.
Heavytailsareappliedinfinance,insurance,physics,geophysics,cellbiology,
ecology, medicine,andcomputerengineering. Arandomvariablehasheavytailsif
P X x Cx−αforsomeα 0.Ifα 2,thenthesecondmomentofXisundefined,
(| |> )≈ > <
sotheusualcentrallimittheoremdoesnotapply.Aheavy-tailedversionofthecen-
trallimittheoremleadstoastabledistribution.Randomwalkswithheavytailscon-
vergetoastableLévymotion,similartoBrownianmotion.ThedensitiesofBrownian
motionsolvethediffusionequation,whichprovidesapowerfullinkbetweendiffer-
entialequationsandstochasticprocesses.DensitiesofastableLévymotionsolvea
fractionaldiffusionequationlike
∂ ∂α
p x,t c p x,t
∂t ( )= ∂xα ( )
usingafractionalderivativeoforderα.Fractionalderivativesarelimitsoffractional
difference quotients, using the same fractional difference operator that appears in
timeseriesmodelsoflongrangedependence.
Vector models with heavy tails are useful in many applications, since physical
spaceisnotonedimensional.Aheavytailedversionofthecentrallimittheoremfor
randomvectorsleadstoanoperatorstablelimitdistribution,whichcanhaveacombi-
nationofnormalandstablecomponents.Vectorrandomwalkswithheavytailscon-
verge to an operator stable Lévy motion. Probabilitydensities of this random walk
limitsolveavectorfractionaldiffusionequationlike
∂ ∂α ∂β
p x,y,t c p x,y,t c p x,y,t
∂t ( )= 1∂xα ( )+ 2∂yβ ( )
withadifferentfractionalderivativeineachcoordinate.
Many interesting research problems in this area remain open. This book will
guidethemotivatedreadertogaintheessentialbackground,insights,andtechniques
neededtoreadandunderstandcurrentresearchpapersandbeginmakingtheirown
contributionstothisrapidlygrowingfield.
Inthissecondeditionofthetext,wehavecorrectedallknownerrorsandadded
threesectionsonapplications:ThefractionalPoissonprocess;distributedorderfrac-
VIII |
tionalderivatives;andthecorrelationstructureoffractionalprocesses.Wewouldlike
tothankallthepeoplewhowrotetousaboutthefirsteditionwithhelpfulideasand
corrections.Itisourhopethatthesecondeditioncontinuestobeausefulcontribution
totheliteratureonthisinterestingareaoftheoreticalandappliedresearch.
EastLansing,April2019 MarkM.Meerschaert,AllaSikorskii