Lévy matters II : recent progress in theory and applications: fractional Lévy fields, and scale functions

200 Pages·2012·1.355 MB·English

by  Serge Cohen, et al

Lecture Notes in Mathematics 2061
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304
Le´vyinStanford(PermissiongrantedbyG.L.Alexanderson)
“Le´vy Matters” is a subseries of the Springer Lecture Notes in Mathematics, devoted to the
disseminationofimportantdevelopmentsintheareaofStochasticsthatarerootedinthetheoryof
Le´vyprocesses.Eachvolumewillcontainstate-of-the-arttheoreticalresultsaswellasapplications
ofthisrapidlyevolvingfield,withspecialemphasisonthecaseofdiscontinuouspaths.Contributions
tothisseriesbyleadingexpertswillpresentorsurveynewandexcitingareasofrecenttheoretical
developments,orwillfocusonsomeofthemorepromisingapplicationsinrelatedfields.Inthisway
eachvolumewillconstitute areferencetextthatwillservePhDstudents,postdoctoral researchers
andseasonedresearchersalike.
Editors
OleE.Barndorff-Nielsen JeanJacod
ThieleCentreforAppliedMathematics InstitutdeMathe´matiquesdeJussieu
inNaturalScience CNRS-UMR7586
DepartmentofMathematicalSciences Universite´Paris6-PierreetMarieCurie
AarhusUniversity 75252ParisCedex05,France
8000AarhusC,Denmark [email protected]
[email protected]
JeanBertoin ClaudiaKlu¨ppelberg
Institutfu¨rMathematik ZentrumMathematik
Universita¨tZu¨rich TechnischeUniversita¨tMu¨nchen
8057Zu¨rich,Switzerland 85747GarchingbeiMu¨nchen,Germany
[email protected] [email protected]
ManagingEditors
VickyFasen RobertStelzer
DepartmentofMathematics InstituteofMathematicalFinance
ETHZu¨rich UlmUniversity
8092Zu¨rich,Switzerland 89081Ulm,Germany
[email protected] [email protected]
ThevolumesinthissubseriesarepublishedundertheauspicesoftheBernoulliSociety.
Serge Cohen • Alexey Kuznetsov
Andreas E. Kyprianou • Victor Rivero
Le´vy Matters II
Recent Progress in Theory and Applications:
Fractional Le´vy Fields, and Scale Functions
123
SergeCohen AndreasE.Kyprianou
Universite´PaulSabatier UniversityofBath
InstitutMathe´matiquedeToulouse DepartmentofMathematicalSciences
ToulouseCedex4,France Bath,UnitedKingdom
AlexeyKuznetsov VictorRivero
YorkUniversity CIMATA.C.
DepartmentofMathematics Guanajuato,Col.Valenciana,Mexico
andStatistics
Toronto,ON,Canada
ISBN978-3-642-31406-3 ISBN978-3-642-31407-0(eBook)
DOI10.1007/978-3-642-31407-0
SpringerHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2012945523
MathematicsSubjectClassification(2010):Primary:60G10,60G40,60G51,60G70,60J10,60J45
Secondary:91B28,91B70,91B84
(cid:2)c Springer-VerlagBerlinHeidelberg2012
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Preface
This second volume of the series “Le´vy Matters” consists of two surveys of two
topical areas, namely Fractional Le´vy Fields by Serge Cohen and the Theory of
Scale Functions for Spectrally Negative Le´vy Processes by Alexey Kuznetsov,
AndreasKyprianouandVictorRivero.
Roughly speaking, irregularity is a crucial aspect of random phenomena that
appearsin manydifferentcontexts.An importantissue in thisdirectionis to offer
tractablemathematicalmodelsthatencompassthevarietyofobservedbehavioursin
applications.FractionalLe´vyfieldsareconstructedbyintegrationofLe´vyrandom
measures; somehow they interpolate between Gaussian and stable random fields.
They exhibit a number of interesting features including local asymptotic self-
similarity and multi-fractional aspects. Calibration techniques and simulation of
fractionalLe´vyfieldsconstituteimportantelementsformanyapplications.
Areal-valuedLe´vyprocessisspectrallynegativewhenithasnopositivejumps.
In this situation, the distribution of several variables related to the first exit-time
froma boundedintervalcanbespecifiedintermsoftheso-calledscale functions;
thelatteralsoplayafundamentalroleinotheraspectsofthetheory.Scalefunctions
are characterized by their Laplace transform, but in general no explicit formula
is known, and therefore it is crucial in many applications to gather information
abouttheirasymptoticbehaviourandregularityandto provideefficientnumerical
methodstocomputethem.
Aarhus,Denmark OleE.Barndorff-Nielsen
Zu¨rich,Switzerland JeanBertoin
Paris,France JeanJacod
Munich,Germany ClaudiaKu¨ppelberg
v
(cid:127)
From the Preface to Le´vy Matters I
Overthepast10–15years,wehaveseenarevivalofgeneralLe´vyprocessestheory
aswellasaburstofnewapplications.Inthepast,BrownianmotionorthePoisson
process had been considered as appropriate models for most applications. Nowa-
days,theneedformorerealisticmodellingofirregularbehaviourofphenomenain
nature and society like jumps, bursts and extremeshas led to a renaissance of the
theory of generalLe´vy 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
ofLe´vymodelsinmodellingjumps,tails,dependenceandsamplepathbehaviour.
Le´vy processes or Le´vy-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 Le´vy jump
processes.
As in many other fields, the enormous power of modern computers has also
changed the view of Le´vy processes. Simulation methods for paths of Le´vy
processes and realizations of their functionals have been developed. Monte Carlo
simulationmakesitpossibletodeterminethedistributionoffunctionalsofsample
pathsofLe´vyprocessestoahighlevelofaccuracy.
ThisdevelopmentofLe´vyprocesseswasaccompaniedandtriggeredbyaseries
ofConferencesonLe´vyProcesses:TheoryandApplications.TheFirstandSecond
ConferenceswereheldinAarhus(1999,2002),theThirdinParis(2003),theFourth
inManchester(2005)andtheFifthinCopenhagen(2007).
Toshowthebroadspectrumoftheseconferences,thefollowingtopicsaretaken
fromtheannouncementoftheCopenhagenconference:
(cid:129) StructuralresultsforL e´vyprocesses:distributionandpathproperties
(cid:129) L e´vytrees,superprocessesandbranchingtheory
(cid:129) Fractalprocessesandfractalphenomena
(cid:129) Stableandinfinitelydivisibleprocessesanddistributions
vii
viii FromthePrefacetoLe´vyMattersI
(cid:129) Applicationsinfinance,physics,biosciencesandtelecommunications
(cid:129) L e´vyprocessesonabstractstructures
(cid:129) Statistical,numericalandsimulationaspectsofL e´vyprocesses
(cid:129) L e´vyandstablerandomfields
AttheConferenceonLe´vyProcesses:TheoryandApplicationsinCopenhagen
theideawasborntostartaseriesofLectureNotesonLe´vyprocessestobearwitness
oftheexcitingrecentadvancesintheareaofLe´vyprocessesandtheirapplications.
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 article makes an invaluable comprehensivereference text. The intended
audiencearePhDandpostdoctoralstudents,orresearchers,whowanttolearnabout
recent advances in the theory of Le´vy processes and to get an overview of new
applicationsindifferentfields.
Now,withthefieldinfullflourishandwith futureinterestdefinitelyincreasing
itseemedreasonabletostartaseriesofLectureNotesinthisarea,whoseindividual
volumeswillappearovertimeunderthecommonname“Le´vyMatters,”intunewith
thedevelopmentsinthefield.“Le´vyMatters”appearsasasubseriesoftheSpringer
Lecture Notes in Mathematics, thus ensuring wide dissemination of the scientific
material.Themainlyexpositoryarticlesshouldreflectthebroadnessoftheareaof
Le´vyprocesses.
We take thepossibilityto acknowledgetheverypositivecollaborationwith the
relevantSpringerstaffandtheeditorsoftheLNseriesandthe(anonymous)referees
ofthearticles.
We hope that the readers of “Le´vy Matters” enjoy learning about the high
potentialof Le´vyprocessesin theoryandapplications.Researcherswith ideasfor
contributions to further volumes in the Le´vy Matters series are invited to contact
anyoftheeditorswithproposalsorsuggestions.
Aarhus,Denmark OleE.Barndorff-Nielsen
Zu¨rich,Switzerland JeanBertoin
Paris,France JeanJacod
Munich,Germany ClaudiaKu¨ppelberg
Contents
FractionalLe´vyFields........................................................... 1
SergeCohen
1 Introduction.................................................................... 1
2 RandomMeasures............................................................. 3
2.1 PoissonRandomMeasure .............................................. 3
2.2 Le´vyRandomMeasure................................................. 4
2.3 RealStableRandomMeasure.......................................... 6
2.4 ComplexIsotropicRandomMeasure.................................. 8
3 StableFields................................................................... 13
3.1 GaussianFields.......................................................... 13
3.2 Non-GaussianFields.................................................... 16
4 Le´vyFields.................................................................... 20
4.1 MovingAverageFractionalLe´vyFields............................... 20
4.2 RealHarmonizableFractionalLe´vyFields............................ 27
4.3 AComparisonofLe´vyFields.......................................... 38
4.4 RealHarmonizableMultifractionalLe´vyFields ...................... 38
5 Statistics ....................................................................... 40
5.1 EstimationforRealHarmonizableFractionalLe´vyFields ........... 40
5.2 IdentificationofmafLf.................................................. 45
6 Simulation..................................................................... 53
6.1 RateofAlmostSureConvergenceforShotNoiseSeries............. 55
6.2 StochasticIntegralsRevisited .......................................... 56
6.3 GeneralizedShotNoiseSeries ......................................... 58
6.4 NormalApproximation................................................. 64
6.5 Summary ................................................................ 69
6.6 Examples................................................................ 69
6.7 SimulationofHarmonizableFields.................................... 79
Appendix........................................................................... 82
References......................................................................... 94
ix