Lévy Matters V: Functionals of Lévy Processes

242 Pages·2015·2.43 MB·English

by  Lars Nørvang Andersen

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
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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