Table Of ContentDeGruyterGraduate
Schilling/Partzsch (cid:2) BrownianMotion
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René L. Schilling
Lothar Partzsch
Brownian Motion
An Introduction to Stochastic Processes
With a Chapter on Simulation by Björn Böttcher
De Gruyter
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MathematicsSubjectClassification2010:Primary:60-01,60J65;Secondary:60H05,60H10,
60J35,60G46,60J60,60J25.
ISBN978-3-11-027889-7
e-ISBN978-3-11-027898-9
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Preface
Brownianmotionisarguablythesinglemostimportantstochasticprocess.Historically
itwasthefirststochasticprocessincontinuoustimeandwithacontinuousstatespace,
andthusitinfluencedthestudyofGaussianprocesses,martingales,Markovprocesses,
diffusionsandrandomfractals.Itscentralpositionwithinmathematicsismatchedby
numerousapplicationsinscience,engineeringandmathematicalfinance.
ThepresentbookgrewoutofseveralcourseswhichwetaughtattheUniversityof
MarburgandTUDresden,anditdrawsonthelecturenotes[141]byoneofus.Many
studentsareinterestedinapplicationsofprobabilitytheoryanditisimportanttoteach
Brownian motion and stochastic calculus at an early stage of the curriculum. Such a
courseisverylikelythefirstencounterwithstochasticprocessesincontinuoustime,
followingdirectlyonanintroductorycourseonrigorous(i.e.measure-theoretic)prob-
abilitytheory.Typically,studentswouldbefamiliarwiththeclassicallimittheorems
ofprobabilitytheoryandbasicdiscrete-timemartingales,asitistreated,forexample,
byJacod&ProtterProbabilityEssentials[88],WilliamsProbabilitywithMartingales
[189],orinthemorevoluminoustextbooksbyBillingsley[11]andDurrett[50].
Generaltextbooksonprobabilitytheorycoverhowever,ifatall,Brownianmotion
only briefly. On the other hand, there is a quite substantial gap to more specialized
texts on Brownian motion which is notso easy to overcome for the novice. Ouraim
wastowriteabookwhichcanbeusedintheclassroomasanintroductiontoBrownian
motionandstochasticcalculus,andasafirstcourseincontinuous-timeandcontinuous-
stateMarkovprocesses.Wealsowantedtohaveatextwhichwouldbebothareadily
accessiblemathematicalback-upforcontemporaryapplications(suchasmathematical
finance)andafoundationtogeteasyaccesstoadvancedmonographs,e.g.Karatzas&
Shreve[99],Revuz&Yor[156]orRogers&Williams[161](forstochasticcalculus),
Marcus&Rosen[129](forGaussianprocesses),Peres&Mörters[133](forrandom
fractals), Chung [23] or Port & Stone [149] (for potential theory) or Blumenthal &
Getoor[13](forMarkovprocesses)tonamebutafew.
Thingsthereadersareexpectedtoknow:Ourpresentationisbasicallyself-con-
tained,startingfrom‘scratch’withcontinuous-timestochasticprocesses.Wedo,how-
ever,assumesomebasicmeasuretheory(asin[169])andafirstcourseonprobability
theoryanddiscrete-timemartingales(asin[88]or[189]).Some‘remedial’materialis
collectedintheappendix,butthisisreallyintendedasaback-up.
Howtoreadthis book:Ofcourse,nothingpreventsyoufromreadingitlinearly.
But there is more material here than one could cover in a one-semester course. De-
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vi Preface
pendingonyourneedsandlikings,thereareatleastthreepossibleselections:BMand
Itôcalculus,BManditssamplepathsandBMasaMarkovprocess.Thediagramon
pagexiwillgiveyousomeideashowthingsdependoneachotherandhowtoconstruct
yourown‘Browniansamplepath’throughthisbook.
Whenever specialattentionisneededandtopointouttraps&pitfalls,wehaveused
the signinthemargin.Alsointhemargin,therearecross-referencestoexercisesat
Ex.N.N. theendofeachchapterwhichwethinkfit(andaresometimesneeded)atthatpoint.1
Theyarenotjustdrillproblemsbutcontainvariants,excursionsfromandextensions
ofthematerialpresentedinthetext.Theproofsofthecorematerialdonotseriously
dependonanyoftheproblems.
Writing an introductory text also meant that we had to omit many beautiful top-
ics. Often we had to stop at a point where we, hopefully, got you really interested...
Therefore, we close every chapter with a brief outlook on possible texts for further
reading.
Many people contributed towards the completion of this project: First of all the
studentswhoattendedourcoursesandhelped–oftenunwittingly–toshapethepre-
sentationofthematerial.WeprofitedalotfromcommentsbyNielsJacob(Swansea)
andPankiKim(SeoulNationalUniversity)whousedanearlydraftofthemanuscript
inoneofhiscourses.SpecialthanksgotoourcolleaguesandstudentsBjörnBöttcher,
KatharinaFischer,JulianHollender,FelixLindnerandMichaelSchwarzenbergerwho
readsubstantialpartsofthetext,oftenseveraltimesandatvariousstages.Theyfound
countless misprints, inconsistencies and errors which we would never have spotted.
Björn helped out with many illustrations and, more importantly, contributed Chap-
ter20onsimulation.FinallywethankourcolleaguesandfriendsatTUDresdenand
ourfamilieswhocontributedtothisworkinmanyuncreditedways.Wehopethatthey
approveoftheresult.
Dresden,February2012 RenéL.Schilling
LotharPartzsch
1 Forthereaders’conveniencethereisawebpagewhereadditionalmaterialandsolutionsareavail-
able.TheURLishttp://www.motapa.de/brownian_motion/index.html
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Dependence chart
Aswehavealreadymentionedinthepreface,thereareatleastthreepathsthroughthis
bookwhichhighlightdifferentaspectsofBrownianmotion:BrownianmotionandItô
calculus,BrownianmotionasaMarkovprocess,andBrownianmotionanditssample
paths. Below we suggest some fast tracks “C”, “M” and “S” for each route, and we
indicate how the other topics covered in this book depend on these fast tracks. This
shouldhelpyoutofindyourownpersonalsamplepath.Starred sections(inthegrey
ovals)containresultswhichcanbeusedwithoutproofandwithoutcompromisingtoo
muchonrigour.
Gettingstarted
ForallthreefasttracksyouneedtoreadChapters1and2first.Ifyouarenottoomuch
inahurry,youshouldchooseoneconstruction ofBrownianmotionfromChapter3.
Forthebeginnerwerecommendeither3.1or3.2.
Basicstochasticcalculus(C)
5.1–2 9.1 14.1–4 15 16.1–3
6.1–3(cid:2) 6.7(cid:2) 10.1(cid:2)
BasicMarkovprocesses(M)
5.1–2 6.1–3 6.4 7
4(cid:2) 6.7(cid:2)
Basicsamplepathproperties(S)
5.1–2 6.1–3 9.1+4 10.1–2 11.1 13
4(cid:2)
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xii Dependencechart
Dependencetothesections5.1–19.2
The following list shows which prerequisites are needed for each section. A star as
in4(cid:2)or6.7(cid:2)indicatesthatsomeresult(s)fromChapter4orSection6.7areusedwhich
may be used without proof and without compromising too much on rigour. Starred
sectionsarementionedonlywheretheyareactuallyneeded,whileotherprerequisites
arerepeatedtoindicatethefulllineofdependence.Forexample,
6.6:Mor Sor C,6.1–3
indicatesthattheprerequisitesforSection6.6arecoveredbyeither“M”or“S”or“C
ifyouadd6.1,6.2,6.3”.Sincewedonotrefertolatersectionswithhighernumbers,
you will only need those sections in “M”, “S”, or “C and 6.1, 6.2, 6.3” with section
numbersbelow6.6.Likewise,
17.1:C,16.4–5,14.6(cid:2)
meansthat17.1requires“C”plustheSections16.4and16.5.Someresultsfrom14.6
areused,buttheycanbequotedwithoutproof.
5.1: CorMorS 9.2: SorCorM,9.1 16.3: C
5.2: CorMorS 9.3: SorCorM,9.1 16.4: C
5.3: CorMorS 9.4: SorCorM,9.1 16.5: C,16.4
6.1: MorSorC 10.1: SorCorM 16.6: C
6.2: MorSorC,6.1 10.2: SorCorM 17.1: C,16.4–5,14.6(cid:2)
6.3: MorSorC,6.1–2 10.3: SorCorM 17.2: C,16.4–5
6.4: MorSorC,6.1–3 11.1: S,10.3(cid:2)orC,10.3(cid:2)or 17.3: C,16.4–5,17.1
6.5: MorSorC,6.1–3 M,10.3(cid:2) 17.4: C,16.4–5
6.6: MorSorC,6.1–3 11.2: SorC,11.1or 17.5: C,14.5–6,16.4–5,
6.7: MorSorC,6.1–3 M,11.1 17.2
7.1: M,4.2(cid:2)orC,6.1,4.2(cid:2) 12.1: S 17.6: C,16.4–5,17.2
7.2: MorC,6.1,7.1 12.2: S,12.1 17.7: C,16.4–5
7.3: MorC,6.1,7.1–2 12.3: S,12.1–2,4(cid:2) 18.1: C
7.4: MorC,6.1,7.1–3 13: SorCorM 18.2: C,18.1
7.5: MorC,6.1,7.1–4 14.1: CorM 18.3: C,16.4–5,18.1–2
8.1: MorC,6.1,7.1–3 14.2: C,6.7(cid:2)or 18.4: C,6.1,16.4–5,18.1–3
8.2: M,8.1orC,6.1, M,14.1,6.7(cid:2) 18.5: C,16.4–5,18.1–3
7.1–3,8.1 14.3: CorM,14.1–2 18.6: C,16.4–5,18.1–3,
8.3: M,8.1–2orC,6.1, 14.4: CorM,14.1–3,9.1(cid:2) 10.1(cid:2),17.7(cid:2)
7.1–4,8.1–2 14.5: C,6.7(cid:2) 19.1: MorC,6.1,7
8.4: M,6.7(cid:2),8.1–3(cid:2)or 14.6: C,14.5 19.2: C,6.1,7,16.4–5,18,
C,6.1–4,7,6.9(cid:2), 15: CorM,14.1–4 19.1
8.1–3(cid:2) 16.1: C
9.1: SorCorM 16.2: C
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Contents
Preface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Dependencechart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Indexofnotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1 RobertBrown’snewthing. . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 BrownianmotionasaGaussianprocess . . . . . . . . . . . . . . . . . . 7
2.1 Thefinitedimensionaldistributions . . . . . . . . . . . . . . . . . . 7
2.2 InvariancepropertiesofBrownianmotion . . . . . . . . . . . . . . . 12
2.3 BrownianMotioninRd . . . . . . . . . . . . . . . . . . . . . . . . 15
3 ConstructionsofBrownianmotion . . . . . . . . . . . . . . . . . . . . . 21
3.1 TheLévy–Ciesielskiconstruction . . . . . . . . . . . . . . . . . . . 21
3.2 Lévy’soriginalargument . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Wiener’sconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.4 Donsker’sconstruction . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 TheBachelier–Kolmogorovpointofview . . . . . . . . . . . . . . . 37
4 Thecanonicalmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.1 Wienermeasure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Kolmogorov’sconstruction . . . . . . . . . . . . . . . . . . . . . . . 44
5 Brownianmotionasamartingale . . . . . . . . . . . . . . . . . . . . . 48
5.1 Some‘Brownian’martingales . . . . . . . . . . . . . . . . . . . . . 48
5.2 Stoppingandsampling . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.3 TheexponentialWaldidentity . . . . . . . . . . . . . . . . . . . . . 57
6 BrownianmotionasaMarkovprocess . . . . . . . . . . . . . . . . . . . 62
6.1 TheMarkovproperty . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.2 ThestrongMarkovproperty . . . . . . . . . . . . . . . . . . . . . . 65
6.3 DesiréAndré’sreflectionprinciple . . . . . . . . . . . . . . . . . . . 68
6.4 Transienceandrecurrence . . . . . . . . . . . . . . . . . . . . . . . 73
6.5 Lévy’striplelaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.6 Anarc-sinelaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.7 Somemeasurabilityissues . . . . . . . . . . . . . . . . . . . . . . . 80
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viii Contents
7 Brownianmotionandtransitionsemigroups . . . . . . . . . . . . . . . 86
7.1 Thesemigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.2 Thegenerator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.3 Theresolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
7.4 TheHille-Yosidatheoremandpositivity . . . . . . . . . . . . . . . . 100
7.5 Dynkin’scharacteristicoperator . . . . . . . . . . . . . . . . . . . . 103
8 ThePDEconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.1 Theheatequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8.2 Theinhomogeneousinitialvalueproblem . . . . . . . . . . . . . . . 117
8.3 TheFeynman–Kacformula . . . . . . . . . . . . . . . . . . . . . . . 119
8.4 TheDirichletproblem. . . . . . . . . . . . . . . . . . . . . . . . . . 123
9 ThevariationofBrownianpaths . . . . . . . . . . . . . . . . . . . . . . 137
9.1 Thequadraticvariation . . . . . . . . . . . . . . . . . . . . . . . . . 138
9.2 Almostsureconvergenceofthevariationsums . . . . . . . . . . . . 140
9.3 Almostsuredivergenceofthevariationsums . . . . . . . . . . . . . 143
9.4 Lévy’scharacterizationofBrownianmotion . . . . . . . . . . . . . . 146
10 RegularityofBrownianpaths . . . . . . . . . . . . . . . . . . . . . . . . 152
10.1 Höldercontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
10.2 Non-differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . 155
10.3 Lévy’smodulusofcontinuity . . . . . . . . . . . . . . . . . . . . . . 157
11 ThegrowthofBrownianpaths . . . . . . . . . . . . . . . . . . . . . . . 164
11.1 Khintchine’sLawoftheIteratedLogarithm . . . . . . . . . . . . . . 164
11.2 Chung’s‘other’LawoftheIteratedLogarithm . . . . . . . . . . . . 168
12 Strassen’sFunctionalLawoftheIteratedLogarithm . . . . . . . . . . 173
12.1 TheCameron–Martinformula . . . . . . . . . . . . . . . . . . . . . 174
12.2 Largedeviations(Schilder’stheorem) . . . . . . . . . . . . . . . . . 181
12.3 TheproofofStrassen’stheorem . . . . . . . . . . . . . . . . . . . . 186
13 Skorokhodrepresentation . . . . . . . . . . . . . . . . . . . . . . . . . . 193
14 Stochasticintegrals:L2-Theory . . . . . . . . . . . . . . . . . . . . . . 203
14.1 Discretestochasticintegrals . . . . . . . . . . . . . . . . . . . . . . 203
14.2 Simpleintegrands . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
14.3 ExtensionofthestochasticintegraltoL2 . . . . . . . . . . . . . . . 211
T
14.4 EvaluatingItôintegrals . . . . . . . . . . . . . . . . . . . . . . . . . 215
14.5 WhatistheclosureofE ? . . . . . . . . . . . . . . . . . . . . . . . 219
T
14.6 Thestochasticintegralformartingales . . . . . . . . . . . . . . . . . 222
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