Stochastic models for fractional calculus PDF
Preview Stochastic models for fractional calculus
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
The list of books you might like
