Stochastic Calculus: An Introduction Through Theory and Exercises PDF
Preview Stochastic Calculus: An Introduction Through Theory and Exercises
Universitext Paolo Baldi Stochastic Calculus An Introduction Through Theory and Exercises Universitext Universitext Serieseditors SheldonAxler SanFranciscoStateUniversity CarlesCasacuberta UniversitatdeBarcelona AngusMacIntyre QueenMary,UniversityofLondon KennethRibet UniversityofCalifornia,Berkeley ClaudeSabbah Écolepolytechnique,CNRS,UniversitéParis-Saclay,Palaiseau EndreSüli UniversityofOxford WojborA.Woyczyn´ski CaseWesternReserveUniversity Universitext is a series of textbooksthat presents material from a wide variety of mathematicaldisciplinesatmaster’slevelandbeyond.Thebooks,oftenwellclass- testedbytheirauthor,mayhaveaninformal,personalevenexperimentalapproach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teachingcurricula,toverypolishedtexts. Thus as research topics trickle down into graduate-level teaching, first textbooks writtenfornew,cutting-edgecoursesmaymaketheirwayintoUniversitext. Moreinformationaboutthisseriesathttp://www.springer.com/series/223 Paolo Baldi Stochastic Calculus An Introduction Through Theory and Exercises 123 PaoloBaldi DipartimentodiMatematica UniversitaJdiRoma“TorVergata” Roma,Italy ISSN0172-5939 ISSN2191-6675 (electronic) Universitext ISBN978-3-319-62225-5 ISBN978-3-319-62226-2 (eBook) DOI10.1007/978-3-319-62226-2 LibraryofCongressControlNumber:2017948389 MathematicsSubjectClassification(2010):60H10,60H05,60H30,60G42,60G44 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To thememoryof GianniDi Masi Preface Courses in Stochastic Calculus have in the last two decades changed their target audience. Once this advanced part of mathematics was of interest mainly to postgraduates intending to pursue an academic research career, but now many professionalscannotdowithouttheabilitytomanipulatestochasticmodels. Theaim ofthisbookisto providea toolinthisdirection,startingfroma basic probability background (with measure theory, however). The intended audience should,moreover,haveseriousmathematicalbases. Theentirecontentofthisbookshouldprovidematerialforatwo-semesterclass. MyexperienceisthatChaps.2–9providethematerialforacourseof72h,including thetimedevotedtotheexercises. To be able to manipulate these notionsrequiresthe reader to acquire not only the elementsofthetheorybutalsotheabilitytoworkwiththemandtounderstandtheir applicationsandtheirconnectionswithotherbranchesofmathematics. Thefirstoftheseobjectivesistakencareof(oratleastIhavetriedto...)bythe developmentofalargesetofexerciseswhichareprovidedtogetherwithextensive solutions.Exercisesarehardwiredwiththetheoryandareintendedtoacquaintthe reader with the full meaning of the theoretical results. This set of exercises with theirsolutionispossiblythemostoriginalpartofthiswork. Asfortheapplications,thisbookdevelopstwokinds. The first is given by modeling applications. Actually there are very many situations(infinance,telecommunications,control,...)wherestochasticprocesses, andin particulardiffusions,are a naturalmodel.In Chap.13 we developfinancial applications,currentlyarapidlygrowingarea. Stochastic processes are also connected with other fields in pure mathematics and in particular with PDEs. Knowledge of diffusion processes contributes to a betterunderstandingofsomeaspectsofPDEproblemsand,conversely,thesolution of PDE problems can lead to the computation of quantities of interest related to diffusionprocesses.Thistwo-waytightconnectionbetweenprocessesandPDEsis developedinChap.10.Furtherinterestingconnectionsbetweendiffusionprocesses vii viii Preface andotherbranchesofmathematics(algebraicstructures,differentialgeometry,...) areunfortunatelynotpresentinthistext,primarilyforreasonsofspace. Thefirstgoalistomakethereaderfamiliarwiththebasicelementsofstochastic processes,suchasBrownianmotion,martingales,andMarkovprocesses,sothatit isnotsurprisingthatstochasticcalculusproperbeginsalmostin themiddleofthe book. Chapters2–3introducestochasticprocesses.Afterthedescriptionofthegeneral setting of a continuous time stochastic process that is given in Chap.2, Chap.3 introduces the prototype of diffusion processes, that is Brownian motion, and investigatesits,sometimessurprising,properties. Chapters 4 and 5 providethe main elements on conditioning,martingales, and their applications in the investigation of stochastic processes. Chapter 6 is about Markovprocesses. FromChap.7beginsstochasticcalculusproper.Chapters7and8areconcerned with stochastic integrals and Ito’s formula. Chapter 9 investigates stochastic dif- ferentialequations,Chap.10is aboutthe relationshipwith PDEs. Afterthedetour on numerical issues related to diffusion processes of Chap.11, further notions of stochasticcalculusareinvestigatedinChap.12(Girsanov’stheorem,representation theorems of martingales) and applications to finance are the object of the last chapter. The book is organized in a linear way, almost every section being necessary forthe understandingofthematerialthatfollows.Thefewsectionsandthesingle chapterthatcanbeavoidedaremarkedwithanasterisk. This book is based on courses that I gave first at the University of Pisa, then at Roma “Tor Vergata” and also at SMI (Scuola Matematica Interuniversitaria)in Perugia.Ithastakenadvantageoftheremarksandsuggestionsofmanycohortsof students and of colleagues who tried the preliminary notes in other universities. The list of the people I am indebted to is a long one, starting with the many studentsthathavesufferedunderthefirstversionsofthisbook.G.Lettawasvery helpfulin clarifying to me quite a few complicated situations. I am also indebted to C. Costantini, G. Nappo, M. Pratelli, B. Trivellato, and G. Di Masi for useful remarksontheearlierversions. I am also grateful for the list of misprints, inconsistencies, and plain mistakes pointed out to me by M. Gregoratti and G. Guatteri at Milano Politecnico and B.PacchiarottiatmyUniversityofRoma“TorVergata”.AndmainlyImustmention L.Caramellino,whoseclassnotesonmathematicalfinancewerethemainsourceof Chap.13. Roma,Italy PaoloBaldi Contents 1 ElementsofProbability.................................................... 1 1.1 Probabilityspaces,randomvariables................................ 1 1.2 Variance,covariance,lawofar.v.................................... 3 1.3 Independence,productmeasure ..................................... 5 1.4 ProbabilitiesonRm................................................... 9 1.5 Convergenceofprobabilitiesandrandomvariables................ 11 1.6 Characteristicfunctions.............................................. 13 1.7 Gaussianlaws ........................................................ 15 1.8 Simulation............................................................ 21 1.9 Measure-theoreticarguments........................................ 24 Exercises ..................................................................... 25 2 StochasticProcesses........................................................ 31 2.1 Generalfacts.......................................................... 31 2.2 Kolmogorov’scontinuitytheorem................................... 35 2.3 Constructionofstochasticprocesses................................ 38 2.4 Next................................................................... 42 Exercises ..................................................................... 42 3 BrownianMotion........................................................... 45 3.1 Definitionandgeneralfacts.......................................... 45 3.2 Thelawofacontinuousprocess,Wienermeasure ................. 52 3.3 Regularityofthepaths............................................... 53 3.4 Asymptotics .......................................................... 57 3.5 Stoppingtimes........................................................ 63 3.6 Thestoppingtheorem................................................ 66 3.7 ThesimulationofBrownianmotion................................. 70 Exercises ..................................................................... 75 4 ConditionalProbability.................................................... 85 4.1 Conditioning.......................................................... 85 4.2 Conditionalexpectations............................................. 86 ix x Contents 4.3 Conditionallaws...................................................... 98 4.4 ConditionallawsofGaussianvectors............................... 100 4.5 TheaugmentedBrownianfiltration ................................. 102 Exercises ..................................................................... 104 5 Martingales.................................................................. 109 5.1 Definitionsandgeneralfacts......................................... 109 5.2 Discretetimemartingales............................................ 111 5.3 Discretetimemartingales:a.s.convergence........................ 114 5.4 Doob’sinequality;Lpconvergence,thep>1case................. 118 5.5 UniformintegrabilityandconvergenceinL1 ....................... 121 5.6 Continuoustimemartingales ........................................ 124 5.7 Complements:theLaplacetransform............................... 133 Exercises ..................................................................... 139 6 MarkovProcesses........................................................... 151 6.1 Definitionsandgeneralfacts......................................... 151 6.2 TheFellerandstrongMarkovproperties ........................... 160 6.3 Semigroups,generators,diffusions.................................. 169 Exercises ..................................................................... 175 7 TheStochasticIntegral .................................................... 181 7.1 Introduction........................................................... 181 7.2 Elementaryprocesses ................................................ 182 7.3 Thestochasticintegral ............................................... 185 7.4 Themartingaleproperty ............................................. 192 7.5 ThestochasticintegralinM2 ....................................... 200 loc 7.6 Localmartingales .................................................... 205 Exercises ..................................................................... 209 8 StochasticCalculus......................................................... 215 8.1 Ito’sformula.......................................................... 215 8.2 Application:exponentialmartingales ............................... 224 8.3 Application:Lpestimates............................................ 229 8.4 Themultidimensionalstochasticintegral ........................... 231 8.5 (cid:2)A case study: recurrence of multidimensional Brownianmotion..................................................... 243 Exercises ..................................................................... 246 9 StochasticDifferentialEquations......................................... 255 9.1 Definitions............................................................ 255 9.2 Examples ............................................................. 257 9.3 Anaprioriestimate .................................................. 260 9.4 ExistenceforLipschitzcontinuouscoefficients .................... 265 9.5 LocalizationandexistenceforlocallyLipschitzcoefficients ...... 270 9.6 Uniquenessinlaw.................................................... 273 9.7 TheMarkovproperty................................................. 275
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