Probability and Its Applications PublishedinassociationwiththeAppliedProbabilityTrust Editors:J.Gani, C.C.Heyde,P.Jagers,T.G.Kurtz Probability and Its Applications Anderson:Continuous-TimeMarkovChains(1991) Azencott/Dacunha-Castelle:SeriesofIrregularObservations(1986) Bass:DiffusionsandEllipticOperators(1997) Bass:ProbabilisticTechniquesinAnalysis(1995) Chen:Eigenvalues,Inequalities,andErgodicTheory(2005) Choi:ARMAModelIdentification(1992) Costa/Fragoso/Marques:Discrete-TimeMarkovJumpLinearSystems Daley/Vere-Jones:AnIntroductionoftheTheoryofPointProcesses Volume I: Elementary Theory and Methods, (2nd ed. 2003. Corr. 2nd printing 2005) DelaPeña/Giné:Decoupling:FromDependencetoIndependence(1999) DelMoral:Feynman-KacFormulae:GenealogicalandInteractingParticleSystems withApplications(2004) Durrett:ProbabilityModelsforDNASequenceEvolution(2002) Galambos/Simonelli:Bonferroni-typeInequalitieswithApplications(1996) Gani(Editor):TheCraftofProbabilisticModelling(1986) Grandell:AspectsofRiskTheory(1991) Gut:StoppedRandomWalks(1988) Guyon:RandomFieldsonaNetwork(1995) Kallenberg:FoundationsofModernProbability(2nded.2002) Kallenberg:ProbabilisticSymmetriesandInvariancePrinciples(2005) Last/Brandt:MarkedPointProcessesontheRealLine(1995) Leadbetter/Lindgren/Rootzén:ExtremesandRelatedPropertiesofRandom SequencesandProcesses(1983) Molchanov:TheoryandRandomSets(2005) Nualart:TheMalliavinCalculusandRelatedTopics(2nded.2006) Rachev/Rüschendorf:MassTransportationProblemsVolumeI:Theory(1998) Rachev/Rüschendorf:MassTransportationProblemsVolumeII:Applications(1998) Resnick:ExtremeValues,RegularVariationandPointProcesses(1987) Shedler:RegenerationandNetworksofQueues(1986) Silvestrov:LimitTheoremsforRandomlyStoppedStochasticProcesses(2004) Thorisson:Coupling,Stationarity,andRegeneration(2000) Todorovic:AnIntroductiontoStochasticProcessesandTheirApplications(1992) David Nualart The Malliavin Calculus and Related Topics ABA C DavidNualart DepartmentofMathematics,UniversityofKansas,405SnowHall,1460JayhawkBlvd,Lawrence, Kansas66045-7523,USA SeriesEditors J.Gani C.C.Heyde StochasticAnalysisGroup,CMA StochasticAnalysisGroup,CMA AustralianNationalUniversity AustralianNationalUniversity CanberraACT0200 CanberraACT0200 Australia Australia P.Jagers T.G.Kurtz MathematicalStatistics DepartmentofMathematics ChalmersUniversityofTechnology UniversityofWisconsim SE-41296Göteborg 480LincolnDrive Sweden Madison,WI53706 USA LibraryofCongressControlNumber:2005935446 MathematicsSubjectClassification(2000):60H07,60H10,60H15,60-02 ISBN-10 3-540-28328-5SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-28328-7SpringerBerlinHeidelbergNewYork ISBN0-387-94432-X1steditionSpringerNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2006 PrintedinTheNetherlands Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorandTechBooksusingaSpringerLLLAAATTTEEEXmacropackage Coverdesign:ErichKirchner,Heidelberg Printedonacid-freepaper SPIN:11535058 41/TechBooks 543210 To my wife Maria Pilar Preface to the second edition There have been ten years since the publication of the first edition of this book. Since then, new applications and developments of the Malliavin cal- culus have appeared. In preparing this second edition we have taken into account some of these new applications, and in this spirit, the book has two additional chapters that deal with the following two topics: Fractional Brownian motion and Mathematical Finance. The presentation of the Malliavin calculus has been slightly modified at some points, where we have taken advantage of the material from the lecturesgiveninSaintFlourin1995(seereference[248]).Themainchanges and additional material are the following: In Chapter 1, the derivative and divergence operators are introduced in the framework of an isonormal Gaussian process associated with a general Hilbert space H. The case where H is an L2-space is trated in detail after- wards (white noise case). The Sobolev spaces Ds,p, with s is an arbitrary real number, are introduced following Watanabe’s work. Chapter2includesageneralestimateforthedensityofaone-dimensional random variable, with application to stochastic integrals. Also, the com- position of tempered distributions with nondegenerate random vectors is discussed following Watanabe’s ideas. This provides an alternative proof of the smoothness of densities for nondegenerate random vectors. Some properties of the support of the law are also presented. In Chapter 3, following the work by Alo`s and Nualart [10], we have included some recent developments on the Skorohod integral and the asso- ciated change-of-variables formula for processes with are differentiable in future times. Also, the section on substitution formulas has been rewritten viii Preface to the second edition and an Itoˆ-Ventzell formula has been added, following [248]. This for- mula allows us to solve anticipating stochastic differential equations in Stratonovich sense with random initial condition. There have been only minor changes in Chapter 4, and two additional chapters have been included. Chapter 5 deals with the stochastic calculus with respect to the fractional Brownian motion. The fractional Brownian motion is a self-similar Gaussian process with stationary increments and variance t2H. The parameter H ∈ (0,1) is called the Hurst parameter. The main purpose of this chapter is to use the the Malliavin Calculus techniques to develop a stochastic calculus with respect to the fractional Brownian motion. Finally, Chapter 6 contains some applications of Malliavin Calculus in Mathematical Finance. The integration-by-parts formula is used to com- pute“greeks”,sensitivityparametersoftheoptionpricewithrespecttothe underlyingparameters ofthe model.Wealsodiscusstheapplication of the Clark-Ocone formula in hedging derivatives and the additional expected logarithmic utility for insider traders. August 20, 2005 David Nualart Preface The origin of this book lies in an invitation to give a series of lectures on Malliavin calculus at the Probability Seminar of Venezuela, in April 1985. The contents of these lectures were published in Spanish in [245]. Later these notes were completed and improved in two courses on Malliavin cal- culus given at the University of California at Irvine in 1986 and at E´cole PolytechniqueF´´ed´eraledeLausannein1989.Thecontentsofthesecourses correspond to the material presented in Chapters 1 and 2 of this book. Chapter 3 deals with the anticipating stochastic calculus and it was de- veloped from our collaboration with Moshe Zakai and Etienne Pardoux. The series of lectures given at the Eighth Chilean Winter School in Prob- ability and Statistics, at Santiago de Chile, in July 1989, allowed us to write a pedagogical approach to the anticipating calculus which is the ba- sisofChapter3.Chapter4dealswiththenonlineartransformationsofthe WienermeasureandtheirapplicationstothestudyoftheMarkovproperty for solutions to stochastic differential equations with boundary conditions. The presentation of this chapter was inspired by the lectures given at the Fourth Workshop on Stochastic Analysis in Oslo, in July 1992. I take the opportunity to thank these institutions for their hospitality, and in par- ticular I would like to thank Enrique Caban˜˜a, Mario Wschebor, Joaqu´n Ortega,Su¨¨leymanU¨stu¨nel,BerntØksendal,RenzoCairoli,Ren´´eCarmona, and Rolando Rebolledo for their invitations to lecture on these topics. We assume that the reader has some familiarity with the Itˆo stochastic calculus and martingale theory. In Section 1.1.3 an introduction to the Itoˆ calculusisprovided,butwesuggestthereadercompletethisoutlineofthe classical Itˆo calculus with a review of any of the excellent presentations of x Preface this theory that are available (for instance, the books by Revuz and Yor [292] and Karatzas and Shreve [164]). Inthepresentationofthestochasticcalculusofvariations(usuallycalled the Malliavin calculus) we have chosen the framework of an arbitrary cen- teredGaussianfamily,andhavetriedtofocusourattentiononthenotions and results that depend only on the covariance operator (or the associated Hilbertspace).Wehavefollowedsomeoftheideasandnotationsdeveloped by Watanabe in [343] for the case of an abstract Wiener space. In addition toWatanabe’sbookandthesurveyonthestochasticcalculusofvariations writtenbyIkedaandWatanabein[144]wewouldliketomentionthebook by Denis Bell [22] (which contains a survey of the different approaches to theMalliavincalculus),andthelecturenotesbyDanOconein[270].Read- ers interested in the Malliavin calculus for jump processes can consult the book by Bichteler, Gravereaux, and Jacod [35]. The objective of this book is to introduce the reader to the Sobolev dif- ferential calculus for functionals of a Gaussian process. This is called the analysis on the Wiener space, and is developed in Chapter 1. The other chapters are devoted to different applications of this theory to problems such as the smoothness of probability laws (Chapter 2), the anticipating stochastic calculus (Chapter 3), and the shifts of the underlying Gaussian process (Chapter 4). Chapter 1, together with selected parts of the sub- sequent chapters, might constitute the basis for a graduate course on this subject. I would like to express my gratitude to the people who have read the several versions of the manuscript, and who have encouraged me to com- pletethework,particularlyIwouldliketothankJohnWalsh,GiuseppeDa Prato, Moshe Zakai, and Peter Imkeller. My special thanks go to Michael Ro¨ckner for his careful reading of the first two chapters of the manuscript. March 17, 1995 David Nualart