Lévy Matters V: Functionals of Lévy Processes PDF
Preview Lévy Matters V: Functionals of Lévy Processes
Lecture Notes in Mathematics 2149 Lévy Matters – A subseries on Lévy Processes Lars Nørvang Andersen · Søren Asmussen Frank Aurzada · Peter W. Glynn Makoto Maejima · Mats Pihlsgård Thomas Simon Lévy Matters V Functionals of Lévy Processes Lecture Notes in Mathematics 2149 Editors-in-Chief: J.-M.Morel,Cachan B.Teissier,Paris AdvisoryBoard: CamilloDeLellis,Zurich MariodiBernardo,Bristol AlessioFigalli,Austin DavarKhoshnevisan,SaltLakeCity IoannisKontoyiannis,Athens GaborLugosi,Barcelona MarkPodolskij,Aarhus SylviaSerfaty,ParisandNY CatharinaStroppel,Bonn AnnaWienhard,Heidelberg Moreinformationaboutthisseriesathttp://www.springer.com/series/304 “Lévy Matters” is a subseries of the Springer Lecture Notes in Mathematics, devoted to the dissemination ofimportantdevelopments intheareaofStochastics thatarerootedinthetheoryof Lévyprocesses.Eachvolumewillcontainstate-of-the-art theoretical resultsaswellasapplications ofthisrapidlyevolvingfield,withspecialemphasisonthecaseofdiscontinuouspaths.Contributions tothisseries byleading experts willpresentorsurveynew andexciting areas ofrecent theoretical developments,orwillfocusonsomeofthemorepromisingapplicationsinrelatedfields.Inthisway eachvolumewillconstituteareferencetextthatwillservePhDstudents,postdoctoralresearchersand seasonedresearchersalike. Editors OleE.Barndorff-Nielsen JeanJacod ThieleCentreforAppliedMathematics InstitutdeMathématiquesdeJussieu inNaturalScience CNRS-UMR7586 DepartmentofMathematicalSciences UniversitéParis6-PierreetMarieCurie AarhusUniversity 75252ParisCedex05,France 8000AarhusC,Denmark [email protected] [email protected] JeanBertoin ClaudiaKlüppelberg InstitutfürMathematik ZentrumMathematik UniversitätZürich TechnischeUniversitätMünchen 8057Zürich,Switzerland 85747GarchingbeiMünchen,Germany [email protected] [email protected] ManagingEditor ErichBaur ENSLyon UnitédeMathématiquesPuresetAppliquées UMRCNRS5669 46,alléed’Italie 69364Lyon,France [email protected] ThevolumesinthissubseriesarepublishedundertheauspicesoftheBernoulliSociety Lars Nørvang Andersen (cid:129) Søren Asmussen (cid:129) Frank Aurzada (cid:129) Peter W. Glynn (cid:129) Makoto Maejima (cid:129) Mats Pihlsgård (cid:129) Thomas Simon Lévy Matters V Functionals of Lévy Processes 123 LarsNørvangAndersen SørenAsmussen DepartmentofMathematics DepartmentofMathematics AarhusUniversity AarhusUniversity Aarhus,Denmark Aarhus,Denmark FrankAurzada PeterW.Glynn FachbereichMathematik DeptofManagementScience TechnischeUniversitaRtDarmstadt &Engineering Darmstadt,Germany StanfordUniversity Stanford,CA,USA MakotoMaejima DepartmentofMathematics MatsPihlsgård KeioUniversity FacultyofMedicine Yokohama,Japan LundUniversity Lund,Sweden ThomasSimon LaboratoirePaulPainlevé UniversitédeLille1 Villeneuved’AscqCedex,France ISSN0075-8434 ISSN1617-9692 (electronic) LectureNotesinMathematics ISBN978-3-319-23137-2 ISBN978-3-319-23138-9 (eBook) DOI10.1007/978-3-319-23138-9 LibraryofCongressControlNumber:2015954465 MathematicsSubjectClassification(2010):60E07,60G18,60G51,60K20,60K35 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) Preface to the Series Lévy Matters Overthepast10–15years,wehaveseenarevivalofgeneralLévyprocessestheory aswellasaburstofnewapplications.Inthepast,BrownianmotionorthePoisson process had been considered as appropriate models for most applications. Nowa- days,theneedformorerealisticmodellingofirregularbehaviourofphenomenain natureandsocietysuchasjumps,burstsandextremeshasledtoarenaissanceofthe theory of general Lévy processes. Theoreticaland applied researchers in fields as diverse as quantum theory, statistical physics, meteorology,seismology, statistics, insurance, finance and telecommunication have realized the enormous flexibility ofLévymodelsinmodellingjumps,tails, dependenceandsamplepathbehaviour. Lévy processes or Lévy-driven processes feature slow or rapid structural breaks, extremalbehaviour,clusteringandclumpingofpoints. Tools and techniques from related but distinct mathematical fields, such as point processes, stochastic integration, probability theory in abstract spaces and differential geometry, have contributed to a better understanding of Lévy jump processes. As in many other fields, the enormous power of modern computers has also changed the view of Lévy processes. Simulation methods for paths of Lévy processes and realizations of their functionals have been developed. Monte Carlo simulationmakesitpossibletodeterminethedistributionoffunctionalsofsample pathsofLévyprocessestoahighlevelofaccuracy. ThisdevelopmentofLévyprocesseswasaccompaniedandtriggeredbyaseries ofConferencesonLévyProcesses:TheoryandApplications.Thefirstandsecond conferences were held in Aarhus (1999, 2002, respectively), the third in Paris (2003),thefourthinManchester(2005)andthefifthinCopenhagen(2007). v vi PrefacetotheSeriesLévyMatters Toshowthebroadspectrumoftheseconferences,thefollowingtopicsaretaken fromtheannouncementoftheCopenhagenconference: (cid:129) StructuralresultsforLévyprocesses:distributionandpathproperties (cid:129) Lévytrees,superprocessesandbranchingtheory (cid:129) Fractalprocessesandfractalphenomena (cid:129) Stableandinfinitelydivisibleprocessesanddistributions (cid:129) Applicationsinfinance,physics,biosciencesandtelecommunications (cid:129) Lévyprocessesonabstractstructures (cid:129) Statistical,numericalandsimulationaspectsofLévyprocesses (cid:129) Lévyandstablerandomfields AttheConferenceonLévyProcesses:TheoryandApplicationsinCopenhagen, theideawasborntostartaseriesofLectureNotesonLévyprocessestobearwitness oftheexcitingrecentadvancesintheareaofLévyprocessesandtheirapplications. Itsgoalisthedisseminationofimportantdevelopmentsintheoryandapplications. Each volume will describe state-of-the-art results of this rapidly evolving subject with special emphasis on the non-Brownian world. Leading experts will present new exciting fields, or surveys of recent developments, or focus on some of the most promising applications. Despite its special character, each article is written in an expositorystyle, normallywith an extensivebibliographyat the end. In this way,each articlemakesan invaluablecomprehensivereferencetext.The intended audiencearePhDandpostdoctoralstudents,orresearchers,whowanttolearnabout recent advances in the theory of Lévy processes and to get an overview of new applicationsindifferentfields. Now,withthefieldinfullflourishandwithfutureinterestdefinitelyincreasing, itseemedreasonabletostartaseriesofLectureNotesinthisarea,whoseindividual volumes will appear over time under the common name “Lévy Matters”, in tune with the developments in the field. “Lévy Matters” appears as a subseries of the Springer Lecture Notes in Mathematics, thus ensuring wide dissemination of the scientific material. The mainly expository articles should reflect the broadness of theareaofLévyprocesses. We take thepossibilityto acknowledgetheverypositivecollaborationwith the relevantSpringerstaffandtheeditorsoftheLNseriesandthe(anonymous)referees ofthearticles. We hope that the readers of “Lévy Matters” enjoy learning about the high potentialof Lévyprocessesin theoryand applications.Researcherswith ideas for contributionstofurthervolumesintheLévyMattersseriesareinvitedtocontactany oftheeditorswithproposalsorsuggestions. Aarhus,Denmark OleE.Barndorff-Nielsen Paris,France JeanBertoin Paris,France JeanJacod Munich,Germany ClaudiaKüppelberg June2010 A Short Biography of Paul Lévy Avolumeoftheseries“LévyMatters”wouldnotbecompletewithoutashortsketch aboutthelifeandmathematicalachievementsofthemathematicianwhosenamehas beenborrowedandusedhere.ThisismoreaformoftributetoPaulLévy,whonot onlyinventedwhatwe callnowLévyprocesses,butalso isina sense thefounder ofthe waywe are nowlookingatstochastic processes,with emphasisonthe path properties. Paul Lévy was born in 1886, and lived until 1971. He studied at the Ecole PolytechniqueinParis,andwassoonappointedasprofessorofmathematicsinthe sameinstitution,apositionthatheheldfrom1920to1959.Hestartedhiscareeras ananalyst,with20publishedpapersbetween1905(hewasthen19yearsold)and 1914,andhebecameinterestedin probabilitybychance,so tospeak,whenasked to give a series of lectures on this topic in 1919 in that same school: this was the startingpointofanastoundingseriesofcontributionsinthisfield,inparallelwitha continuingactivityinfunctionalanalysis. Very briefly, one can mention that he is the mathematician who introduced characteristicfunctionsinfullgenerality,provinginparticularthecharacterisation theoremandthefirst“Lévy’stheorem”aboutconvergence.Thisnaturallyledhimto studymoredeeplytheconvergenceinlawwithitsmetric,andalsotoconsidersums of independentvariables, a hot topic at the time: Paul Lévy proved a form of the 0-1law,aswellasmanyotherresults,forseriesofindependentvariables.Healso introduced stable and quasi-stable distributions, and unravelled their weak and/or strongdomainsofattractions,simultaneouslywithFeller. Then we arrive at the book “Théorie de l’addition des variables aléatoires”, published in 1937, and in which he summaries his findings about what he called “additive processes” (the homogeneous additive processes are now called Lévy processes,buthedidnotrestricthisattentiontothehomogeneouscase).Thisbook containsahostofnewideasandnewconcepts:thedecompositionintothesumof jumpsatfixedtimesandtherestoftheprocess;thePoissonianstructureofthejumps for an additiveprocess withoutfixed times of discontinuities;the “compensation” ofthosejumpssothatoneisabletosumupallofthem;thefactthattheremaining continuous part is Gaussian. As a consequence, he implicitly gave the formula vii viii AShortBiographyofPaulLévy providing the form of all additive processes without fixed discontinuities, now called the Lévy-Itô Formula, and he proved the Lévy-Khintchine formula for the characteristicfunctionsofallinfinitelydivisibledistributions.But,asfundamental as all those results are, this book contains more: new methods, like martingales which,althoughnotgivena name,areusedina fundamentalway;andalsoanew wayoflookingatprocesses,whichisthe“pathwise”way:hewascertainlythefirst tounderstandtheimportanceoflookingatanddescribingthepathsofastochastic process,insteadofconsideringthateverythingisencapsulatedintothedistribution oftheprocesses. This is of course not the end of the story. Paul Lévy undertook a very deep analysisofBrownianmotion,culminatingin hisbook“Processusstochastiqueset mouvementbrownien” in 1948, completed by a second edition in 1965. This is a remarkable achievement, in the spirit of path properties, and again it contains so many deep results: the Lévy modulus of continuity, the Hausdorff dimension of thepath,themultiplepointsandtheLévycharacterisationtheorem.Heintroduced local time and proved the arc-sine law. He was also the first to consider genuine stochasticintegrals,with thearea formula.In thistopicagain,hisideashavebeen the origin of a huge amount of subsequent work, which is still going on. It also laid some of the basis for the fine study of Markov processes, like the local time again, or the new concept of instantaneous state. He also initiated the topic of multi-parameterstochasticprocesses,introducinginparticularthemulti-parameter Brownianmotion. As should be quite clear, the account given here does not describe the whole of Paul Lévy’s mathematical achievements, and one can consult for many more details the first paper (byMichel Loève)publishedin the first issue of the Annals of Probability(1973).Italso doesnotaccountforthe humanityand gentlenessof thepersonPaulLévy.ButI wouldlike toendthis shortexpositionofPaulLévy’s work by hopingthat this series will contribute to fulfilling the program,which he initiated. Paris,France JeanJacod Preface ThisfifthvolumeoftheseriesLévyMattersconsistsofthreechapters,eachdevoted to an importantaspect of Lévy processes and their applications. They all concern distributions of certain functionals of Lévy processes, which appear naturally in differentsettings. Historically, processes with independent increments have been considered by Paul Lévy to reveal the fine structure of infinitely divisible distributions; the paradigmbeingthataprobabilitymeasure,say(cid:2),isinfinitelydivisibleifandonlyif thereisaLévyprocessfXtgsuchthatX1hasthelaw(cid:2).Inturn,thenotionofinfinite divisibilityforprobabilitymeasuresarisesnaturallyinthecontextoflimittheorems forsumsoftriangulararrays.Well-knownspecialcasesofinfinitelydivisiblelaws include stable distributions, which describe the weak limits of certain properly rescaledrandomwalkswithheavy-tailedstepdistributions,andmoregenerallyself- decomposabledistributions,which in turn arise similarly for sums of independent variables. The so-called Generalized Gamma Convolutions,or distributionsin the Thorin class, lie somewhat in between the two former. The first chapter of this volume, by Makoto Maejima, surveys representations of the main sub-classes of infinitelydivisibledistributionsintermsofmappingsofcertainLévyprocessesvia stochasticintegration Z fXg7! f.t/dX; t t I wheref issomespecificdeterministicfunctionoversomeintervalI.Animportant motivationforstudyingsuchmappingsstemsfromfreeprobability,andmorespecif- ically from the role of free cumulants in this area. The study of the compositions and the iterations of such mappings, and of their limits, then sheds light on the nestedstructureofthosesubclasses.Agreatvarietyofclassicalandnot-yetclassical examplesofinfinitelydivisibledistributionsarethenanalyzedfromthisperspective. Overall, this chapter can be seen as a companion to the contribution by K. Sato “Fractionalintegralsandextensionsofself-decomposability,”whichappearedinthe ix
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