Lévy Matters III: Lévy-Type Processes: Construction, Approximation and Sample Path Properties

215 Pages·2013·2.35 MB·English

by  Björn Böttcher

Lecture Notes in Mathematics 2099
Lévy Matters – A subseries on Lévy Processes
Björn Böttcher
René Schilling
Jian Wang
Lévy
Matters III
Lévy-Type Processes:
Construction, Approximation
and Sample Path Properties
Bernoulli Society
for Mathematical Statistics
and Probability
Lecture Notes in Mathematics 2099
Editors-in-Chief:
J.-M.Morel,Cachan
B.Teissier,Paris
AdvisoryBoard:
CamilloDeLellis(Zürich)
MariodiBernardo(Bristol)
AlessioFigalli(Austin)
DavarKhoshnevisan(SaltLakeCity)
IoannisKontoyiannis(Athens)
GaborLugosi(Barcelona)
MarkPodolskij(Heidelberg)
SylviaSerfaty(ParisandNY)
CatharinaStroppel(Bonn)
AnnaWienhard(Heidelberg)
Forfurthervolumes:
http://www.springer.com/series/304
“Lévy Matters” is a subseries of the Springer Lecture Notes in Mathematics, devoted to
the dissemination of important developments in the area of Stochastics that are rooted in
the theory of Lévy processes. Each volume will contain state-of-the-art theoretical results
aswellasapplications ofthisrapidly evolving field,withspecial emphasis onthecaseof
discontinuous paths. Contributions to this series by leading experts will present or survey
newandexcitingareasofrecenttheoreticaldevelopments,orwillfocusonsomeofthemore
promisingapplications inrelatedfields.Inthiswayeachvolumewillconstituteareference
textthatwillservePhDstudents,postdoctoralresearchersandseasonedresearchersalike.
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]
ManagingEditors
VickyFasen RobertStelzer
InstitutfürStochastik InstituteofMathematicalFinance
KarlsruherInstitutfürTechnologie UlmUniversity
76133Karlsruhe,Germany 89081Ulm,Germany
[email protected] [email protected]
ThevolumesinthissubseriesarepublishedundertheauspicesoftheBernoulliSociety.
BjoRrn BoRttcher (cid:129) René Schilling (cid:129) Jian Wang
Lévy Matters III
Lévy-Type Processes: Construction,
Approximation and Sample Path Properties
123
BjoRrnBoRttcher RenéSchilling
InstitutfuRrMathematischeStochastik InstitutfuRrMathematischeStochastik
TechnischeUniversitaRtDresden TechnischeUniversitaRtDresden
Dresden,Germany Dresden,Germany
JianWang
SchoolofMathematicsandComputer
Science
FujianNormalUniversity
Fuzhou,Fujian
People’sRepublicofChina
ISBN978-3-319-02683-1 ISBN978-3-319-02684-8(eBook)
DOI10.1007/978-3-319-02684-8
SpringerChamHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2013955905
MathematicsSubjectClassification(2010):60-02;60J25;60J35;60G17;35S05;60J75;60G48;60G51;
60H10;47D03;35S30
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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
nature and society like jumps, bursts and extremeshas led to a renaissance of the
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
ConferenceswereheldinAarhus(1999,2002),theThirdinParis(2003),theFourth
inManchester(2005)andtheFifthinCopenhagen(2007).
Toshowthebroadspectrumoftheseconferences,thefollowingtopicsaretaken
fromtheannouncementoftheCopenhagenconference:
(cid:129) StructuralresultsforLévyprocesses:distributionandpathproperties
(cid:129) Lévytrees,superprocessesandbranchingtheory
v
vi PrefacetotheSeriesLévyMatters
(cid:129) Fractalprocessesandfractalphenomena
(cid:129) Stableandinfinitelydivisibleprocessesanddistributions
(cid:129) Applicationsinfinance,physics,biosciencesandtelecommunications
(cid:129) Lévyprocessesonabstractstructures
(cid:129) Statistical,numericalandsimul ationaspectsofLévyprocesses
(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 article makes an invaluable comprehensivereference text. The intended
audience are Ph.D. and postdoctoral students, or researchers, who want to learn
about recent advances in the theory of Lévy processes and to get an overview of
newapplicationsindifferentfields.
Now,withthefieldinfullflourishandwith futureinterestdefinitelyincreasing
itseemedreasonabletostartaseriesofLectureNotesinthisarea,whoseindividual
volumeswillappearovertimeunderthecommonname“LévyMatters,”intunewith
thedevelopmentsinthefield.“LévyMatters”appearsasasubseriesoftheSpringer
Lecture Notes in Mathematics, thus ensuring wide dissemination of the scientific
material.Themainlyexpositoryarticlesshouldreflectthebroadnessoftheareaof
Lévyprocesses.
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 JeanBertoinandJeanJacod
Munich,Germany ClaudiaKlüppelberg
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
PolytechniqueinParisandwassoonappointedasprofessorofmathematicsinthe
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,provinginparticularthecharacterization
theoremand thefirst “Lévy’stheorem”aboutconvergence.Thisnaturallyledhim
to study more deeply the convergencein law with its metric and also to consider
sums of independent variables, a hot topic at the time: Paul Lévy proved a form
of the 0-1 law, as well as many other results, for series of independentvariables.
He also introducedstable and quasi-stabledistributionsand unravelledtheir weak
and/orstrongdomainsofattractions,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–Khintchineformula 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
analysis of Brownian motion, culminating in his book Processus stochastiques et
mouvement Brownien 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,themultiplepointsandtheLévycharacterizationtheorem.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 (by Michel Loève)publishedin the first issue of the Annals
ofProbability(1973).It also doesnotaccountfor the 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
BehindeverydecentMarkovprocessthereisafamilyofLévyprocesses.Indeed,let
.Xt/t>0beaMarkovprocesswithstatespaceRd andassume,forthemoment,that
thelimit
1(cid:2)Exei(cid:2)(cid:2).Xt(cid:3)x/
lim Dq.x;(cid:2)/ 8x;(cid:2) 2Rd (F)
t!0 t
existssuchthatthefunction(cid:2) 7!q.x;(cid:2)/iscontinuous.Wewillseebelowthatthis
isenoughtoguaranteethatq.x;(cid:3)/is,foreachx 2 Rd,thecharacteristicexponent
ofaLévyprocess;assuch,itenjoysaLévy–Khintchinerepresentation
Z
(cid:2) (cid:3)
1
q.x;(cid:2)/D(cid:2)il.x/(cid:3)(cid:2)C (cid:2)(cid:3)Q.x/(cid:2)C 1(cid:2)eiy(cid:2)(cid:2)Ci(cid:2)(cid:3)y1 .jyj/ N.x;dy/
Œ(cid:3)1;1(cid:3)
2 Rdnf0g
where.l.x/;Q.x/;N.x;dy//isforeveryfixedx 2Rd aLévytriplet.Thefunction
q W Rd (cid:4) Rd ! C is called the symbol of the process. The processes which
admit a symbol behave locally like a Lévy process, and their infinitesimal gener-
ators resemble the generators of Lévy processes with variable, i.e. x-dependent,
coefficients—thisjustifiesthenameLévy-typeprocesses.Thisguidesustothemain
topicsofthepresenttract:
Characterization: ForwhichMarkovprocessesdoesthelimit(F)exist?
Construction: Is there a Lévy-type process with a given symbol
q.x;(cid:2)/? Is there a 1-to-1 correspondence between
symbolsandprocesses?
Samplepaths: Canweusethesymbolq.x;(cid:2)/inordertodescribethe
samplepathbehaviouroftheprocess?
Approximation: Is it possible to use q.x;(cid:2)/ to approximate and to
simulatetheprocess?
Let us put this point of view into perspective by considering first of all some
d-dimensionalLévyprocess.Xt/t>0.Beinga(strong)Markovprocess,.Xt/t>0can
ix