Introduction to Quasi-Monte Carlo Integration and Applications PDF
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Compact Textbooks in Mathematics Gunther Leobacher Friedrich Pillichshammer Introduction to Quasi-Monte Carlo Integration and Applications Compact Textbooks in Mathematics Forfurthervolumes: http://www.springer.com/series/11225 CompactTextbooksinMathematics This textbook series presents concise introductions to current topics in math- ematics and mainly addresses advanced undergraduatesand master students. Theconceptistooffersmallbookscoveringsubjectmatterequivalentto2-or 3-hourlecturesorseminarswhicharealsosuitableforself-study.Thebookspro- videstudentsandteacherswithnewperspectivesandnovelapproaches.They featureexamplesandexercisestoillustratekeyconceptsandapplicationsofthe theoreticalcontents.Theseriesalsoincludestextbooksspecificallyspeakingto theneedsofstudentsfromotherdisciplinessuchasphysics,computerscience, engineering,lifesciences,finance. Gunther Leobacher Friedrich Pillichshammer Introduction to Quasi-Monte Carlo Integration and Applications GuntherLeobacher FriedrichPillichshammer InstituteofFinancialMathematics UniversityofLinz UniversityofLinz Linz,Austria Linz,Austria ISBN978-3-319-03424-9 ISBN978-3-319-03425-6(eBook) DOI10.1007/978-3-319-03425-6 SpringerChamHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2014945104 2010MathematicsSubjectClassification:11K06,11K38,11K45,65C05,65D30 (cid:2)c SpringerInternationalPublishingSwitzerland2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. While the advice and information in this book are believed to be true and accurate at the date of publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with respecttothematerialcontainedherein. Coverdesign:deblik,Berlin Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To CharlotteandGisi Preface While Fubini’s theoremstates that the integralof a functionon the s-dimensional unit cube can be computedsimply by computingiterated integrals, the attemptof doing so for a function for which those integrals cannot be given by closed-form formulasisinmostcasesdoomedtofail,ifs,say,isgreaterthan10.Thereasonfor thisisthatbyiterationofaone-dimensionalintegrationrulethenumberoffunction evaluationneededforthecorrespondingproductrulegrowsexponentiallyins. Thisconstraintonthepracticalcomputationofintegralsledtothedevelopment of probabilistic methods.Here, the integralis interpretedas the expectedvalue of the integrand evaluated at a random variable that is uniformly distributed on the s-dimensional unit cube. These methods were first applied by E. Fermi, S. Ulam andJ.vonNeumann,thelatteralsobeingtheoriginatorofthename“MonteCarlo simulation”. In contrast to Monte Carlo integration rules, which sample the integrand at randompoints,so-calledquasi-MonteCarlorulesusedeterministicsamplepoints. The relationship between Monte Carlo and quasi-Monte Carlo correspondsto the relationshipbetweentwonotionsof“uniformdistribution”inmathematics.Thefirst istheprobabilisticnotionofarandomvariableforwhichtheprobabilityoftaking values in a given subset of the unit cube is precisely the volume of that set. The secondnotionisthatofasequenceofpoints,forwhichtheproportionofpointsof thesequencelyinginagivens-dimensionalsub-intervaloftheunitcubeequalsthe volumeofthesub-interval. While the notion of uniformlydistributed sequences and examples thereof had beencoinedearlier,thebirthofthetheoryof“UniformDistributionModuloOne” ismarkedbyH.Weyl’sseminalpaper“U¨berdieGleichverteilungmod.Eins”,first publishedintheyear1916.Itwasalreadyknownbythenthat,atleastinprinciple, uniformly distributed sequences could be used to integrate Riemann-integrable functions. However, under the weak assumptions the convergence of the sample meantotheintegralcanbearbitrarilyslow,makingthe“method”impractical. Asthestartingpointfortheanalysisofquasi-MonteCarlomethodsfornumerical integration one can consider the establishment of the Koksma-Hlawka inequality, which was shown by J.F. Koksma in 1942 for the one-dimensional case and by E. Hlawka in 1961 for arbitrary dimensions. Since then the Koksma-Hlawka inequality is the prototypicalerror estimate for quasi-Monte Carlo integration. Its mainfeatureisthatitboundstheintegrationerrorbytheproductoftwoterms,the vii viii Preface variationof the functionandthe star discrepancyof the underlyingsample nodes. Thesecondnotionisrelatedtothatofuniformdistributionofasequence,butwhile thelatterisanasymptoticquality,thestardiscrepancyallowstoassessthequalityof uniformityofafinitenumberofpoints.Knowinghowwellthepointscanbechosen with respect to that measure means – thanks to the Koksma-Hlawka inequality – knowingthe possible convergenceof the integrationerror.This is where concepts fromDiscrepancyTheoryenterthegame. Fromtheearly1960sonseveralpeople,amongtheseN.M.Korobov,E.Hlawka, I.M. Sobol’, J. Halton, H. Faure, H. Niederreiter and C.P. Xing provided con- structions of point sets and sequences with excellent distribution properties, i.e., with low star discrepancy or related/alternative quality measures. The point sets and sequences constructed in this way are therefore suitable sample points for quasi-Monte Carlo rules. However, a certain disadvantageous dependence of the discrepancy bounds on the dimension led to the belief that quasi-Monte Carlo rulescanonlybeappliedinverymoderatedimensions.Contrarytotheseopinions, quasi-MonteCarlorulesare nowadaysusedfornumericalintegrationof functions in hundreds or even thousands of dimensions, and since recently there is also a stream of research which studies infinite-dimensional integration. The motivation forthisparadigmchangeliesinresultsofnumericalexperimentspublishedin1995 byS.H.PaskovandJ.F.Traub,whostudiedquasi-MonteCarlorulesforfunctions in360dimensionscomingfromMathematicalFinance.But,despitetheirapparent effectivityevenforthose veryhigh-dimensionalproblems,the questionofexactly whyquasi-MonteCarlorulesshouldgivethesegoodresultsisstillnotcompletely resolved. In 2010, at the MCQMC meeting in Warsaw, I.H. Sloan spoke in this context about “The unreasonable effectiveness of quasi-Monte Carlo”. Although in the meantime some partial answers come from the study of weighted function spacesandfromtractabilitytheory,thequestforanexplanationofthisunreasonable effectivenessofquasi-MonteCarloisstillaveryactivepartofresearch. Assuggestedbyitstitle,thisbookisanintroductorytexttoquasi-MonteCarlo methods and some of their applications, and it aims at giving a comprehensible treatmentofthesubjectwithdetailedexplanationsofthebasicconcepts.Originating froma2-honesemesterundergraduatecourse,itshouldbeaccessibletostudentsin mathematicsorcomputersciencewithbasicknowledgeofalgebra,calculus,linear algebra,andprobabilitytheory.Althoughthemainfocusisonthetheorybehindthe conceptsofquasi-MonteCarlo,severalpracticalapplicationswithanemphasison financialproblemsarediscussed. Thetopicsofthebookroughlyretracethehistoryofquasi-MonteCarlomethods as sketched above, but do so using up-to-date concepts and notations. Thus we start with the classical multi-dimensional integration problem and its first high- dimensionalalternative,Monte Carlo integration.Chapter 2 is devotedto uniform distributionofsequencesandseveralconceptsofdiscrepancy.Wegiveadiscrepancy estimateforoneoftheoldestspecimensoflowdiscrepancysequences,theHalton sequence. In Chap.3 we introduce the modern framework of reproducing kernel Hilbertspaces forobtainingboundson the integrationerrorfor functionsin those spaces. The Koksma-Hlawka inequality, though not in its most general form, Preface ix appearsasaspecialcaseofthattheory.Thenexttwochaptersaremostlydevotedto constructionsoflow-discrepancypointsetsandsequences,namelylatticepointsets, .t;m;s/-netsand.t;s/-sequences.Thechapteronlatticerulesincludesasectionon integrationinweightedKorobovspaces.Theconceptofweightedspaceshassome bearingontheissueofeffectivenessofquasi-MonteCarlomethodsforveryhigh- dimensionalproblems.Chapter6concludesthetheoreticalpartbyprovidingmore informationaboutthecurseofdimensionalityandtractabilityofdiscrepancy. The last two chapters constitute the application part of the book. Chapter 7 gives a very condensed introduction to concepts from Mathematical Finance, in particular derivative pricing. We introduce some models and derivatives that can serve as specimens for trying out the simulation methods provided in Chap.8. This last chapter coverssome of the basics of simulation, like generation of non- uniform random variables and generation of Brownian paths. The emphasis is on (fast-)orthogonaltransformsfor speedingupconvergence.Severalexamplesserve toillustratethemethods. The compilationofa textbookdemandsa greatdealnotonlyfromtheauthors, but also from their families, colleagues, and students, some of the common time of which has to be diverted to the project. We want to thank all of them for their supportandunderstanding. We appreciatevaluablecomments,suggestionsandimprovementsfromseveral colleagues which we would like to mention here: Josef Dick, Aicke Hinrichs, PeterKritzer,GerhardLarcher,HaraldNiederreiter,KlausRitterandWolfgangCh. Schmid. We hope that the book will turn out to be useful for teaching, self-study, and asareference,andthatitwillencouragemanypeopletostudyquasi-MonteCarlo methodsand/orapplythemtoproblemsfromMathematicalFinanceorotherareas. Linz,Austria GuntherLeobacher October2013 FriedrichPillichshammer
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