Brownian motion : an introduction to stochastic processes

395 Pages·2012·2.08 MB·English

by  René L Schilling

DeGruyterGraduate
Schilling/Partzsch (cid:2) BrownianMotion

René L. Schilling
Lothar Partzsch
Brownian Motion
An Introduction to Stochastic Processes
With a Chapter on Simulation by Björn Böttcher
De Gruyter
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-
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
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
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
Contents ix
15 Stochasticintegrals:beyondL2 . . . . . . . . . . . . . . . . . . . . . . 227
T
16 Itô’sformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
16.1 Itôprocessesandstochasticdifferentials . . . . . . . . . . . . . . . . 233
16.2 TheheuristicsbehindItô’sformula . . . . . . . . . . . . . . . . . . . 235
16.3 ProofofItô’sformula(Theorem16.1) . . . . . . . . . . . . . . . . . 236
16.4 Itô’sformulaforstochasticdifferentials . . . . . . . . . . . . . . . . 239
16.5 Itô’sformulaforBrownianmotioninRd . . . . . . . . . . . . . . . 242
16.6 Tanaka’sformulaandlocaltime . . . . . . . . . . . . . . . . . . . . 243
17 ApplicationsofItô’sformula . . . . . . . . . . . . . . . . . . . . . . . . 248
17.1 Doléans–Dadeexponentials. . . . . . . . . . . . . . . . . . . . . . . 248
17.2 Lévy’scharacterizationofBrownianmotion . . . . . . . . . . . . . . 253
17.3 Girsanov’stheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
17.4 Martingalerepresentation–1 . . . . . . . . . . . . . . . . . . . . . . 258
17.5 Martingalerepresentation–2 . . . . . . . . . . . . . . . . . . . . . . 261
17.6 Martingalesastime-changedBrownianmotion . . . . . . . . . . . . 263
17.7 Burkholder–Davis–Gundyinequalities . . . . . . . . . . . . . . . . . 266
18 Stochasticdifferentialequations . . . . . . . . . . . . . . . . . . . . . . 272
18.1 TheheuristicsofSDEs . . . . . . . . . . . . . . . . . . . . . . . . . 273
18.2 Someexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
18.3 Existenceanduniquenessofsolutions . . . . . . . . . . . . . . . . . 280
18.4 SolutionsasMarkovprocesses . . . . . . . . . . . . . . . . . . . . . 285
18.5 Localizationprocedures . . . . . . . . . . . . . . . . . . . . . . . . . 286
18.6 Dependenceontheinitialvalues . . . . . . . . . . . . . . . . . . . . 289
19 Ondiffusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
19.1 Kolmogorov’stheory . . . . . . . . . . . . . . . . . . . . . . . . . . 300
19.2 Itô’stheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
20 SimulationofBrownianmotionbyBjörnBöttcher . . . . . . . . . . . . . 312
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
20.2 Normaldistribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
20.3 Brownianmotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
20.4 MultivariateBrownianmotion . . . . . . . . . . . . . . . . . . . . . 321
20.5 Stochasticdifferentialequations . . . . . . . . . . . . . . . . . . . . 323
20.6 MonteCarlomethod . . . . . . . . . . . . . . . . . . . . . . . . . . 328
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
A.1 Kolmogorov’sexistencetheorem . . . . . . . . . . . . . . . . . . . . 329
A.2 Apropertyofconditionalexpectations . . . . . . . . . . . . . . . . . 333