ABEL SYMPOSIA Edited by the Norwegian Mathematical Society ParticipantstotheAbelSymposium2005 Organizersoftheevent Photocredits:HeikoJunge,Scanpix Fred Espen Benth · Giulia Di Nunno Tom Lindstrøm · Bernt Øksendal Tusheng Zhang Editors Stochastic Analysis and Applications The Abel Symposium (cid:50)(cid:48)(cid:48)(cid:53) Proceedings of the Second Abel Symposium, Oslo, July (cid:50)(cid:57) − August (cid:52), (cid:50)(cid:48)(cid:48)(cid:53), held in honor of Kiyosi Itô Editors FredEspenBenth GiuliaDiNunno CentreofMathematicsforApplications(CMA) CentreofMathematicsforApplications(CMA) DepartmentofMathematics DepartmentofMathematics UniversityofOslo UniversityofOslo P.O.Box(cid:49)(cid:48)(cid:53)(cid:51)Blindern P.O.Box(cid:49)(cid:48)(cid:53)(cid:51)Blindern (cid:48)(cid:51)(cid:49)(cid:54)Oslo,Norway (cid:48)(cid:51)(cid:49)(cid:54)Oslo,Norway e-mail:[email protected] e-mail:[email protected] TomLindstrøm BerntØksendal CentreofMathematicsforApplications(CMA) CentreofMathematicsforApplications(CMA) DepartmentofMathematics DepartmentofMathematics UniversityofOslo UniversityofOslo P.O.Box(cid:49)(cid:48)(cid:53)(cid:51)Blindern P.O.Box(cid:49)(cid:48)(cid:53)(cid:51)Blindern (cid:48)(cid:51)(cid:49)(cid:54)Oslo,Norway (cid:48)(cid:51)(cid:49)(cid:54)Oslo,Norway e-mail:[email protected] e-mail:[email protected] TushengZhang DepartmentofMathematics UniversityofManchester OxfordRoad ManchesterM(cid:49)(cid:51)(cid:57)PL UnitedKingdom e-mail:[email protected] LibraryofCongressControlNumber:(cid:50)(cid:48)(cid:48)(cid:55)(cid:57)(cid:50)(cid:49)(cid:53)(cid:50)(cid:52) MathematicsSubjectClassification((cid:50)(cid:48)(cid:48)(cid:48)):(cid:48)(cid:48)B(cid:50)(cid:48),(cid:50)(cid:56)B(cid:88)(cid:88),(cid:50)(cid:56)C(cid:88)(cid:88)–(cid:50)(cid:56)C(cid:50)(cid:48),(cid:51)(cid:53)C(cid:88)(cid:88)–(cid:51)(cid:53)C(cid:50)(cid:48), (cid:51)(cid:53)K(cid:88)(cid:88)–(cid:51)(cid:53)C(cid:49)(cid:53),(cid:51)(cid:53)C(cid:53)(cid:53),(cid:51)(cid:53)Q(cid:88)(cid:88)–(cid:51)(cid:53)Q(cid:52)(cid:48),(cid:52)(cid:54)F(cid:88)(cid:88),(cid:52)(cid:54)N(cid:88)(cid:88)–(cid:52)(cid:54)N(cid:49)(cid:48),(cid:52)(cid:57)L(cid:88)(cid:88)–(cid:52)(cid:57)L(cid:50)(cid:53),(cid:54)(cid:48)A(cid:49)(cid:48), (cid:54)(cid:48)B(cid:88)(cid:88),(cid:54)(cid:48)E(cid:88)(cid:88),(cid:54)(cid:48)G(cid:88)(cid:88)–(cid:54)(cid:48)G(cid:48)(cid:55),(cid:54)(cid:48)G(cid:49)(cid:48),(cid:54)(cid:48)G(cid:49)(cid:53),(cid:54)(cid:48)G(cid:49)(cid:55),(cid:54)(cid:48)G(cid:49)(cid:56),(cid:54)(cid:48)G(cid:50)(cid:48),(cid:54)(cid:48)G(cid:51)(cid:53),(cid:54)(cid:48)G(cid:52)(cid:48), (cid:54)(cid:48)G(cid:52)(cid:54),(cid:54)(cid:48)G(cid:53)(cid:49),(cid:54)(cid:48)G(cid:53)(cid:55),(cid:54)(cid:48)G(cid:54)(cid:48),(cid:54)(cid:48)G(cid:55)(cid:48),(cid:54)(cid:48)H(cid:88)(cid:88)–(cid:54)(cid:48)H(cid:48)(cid:53),(cid:54)(cid:48)H(cid:48)(cid:55),(cid:54)(cid:48)H(cid:49)(cid:48),(cid:54)(cid:48)H(cid:49)(cid:53),(cid:54)(cid:48)H(cid:51)(cid:48),(cid:54)(cid:48)H(cid:51)(cid:53), (cid:54)(cid:48)H(cid:52)(cid:48),(cid:54)(cid:48)J(cid:88)(cid:88)–(cid:54)(cid:48)J(cid:50)(cid:53),(cid:54)(cid:48)J(cid:51)(cid:53),(cid:54)(cid:48)J(cid:54)(cid:48),(cid:54)(cid:48)J(cid:54)(cid:53),(cid:56)(cid:49)P(cid:88)(cid:88)–(cid:56)(cid:49)P(cid:49)(cid:53),(cid:56)(cid:49)S(cid:88)(cid:88)–(cid:56)(cid:49)S(cid:50)(cid:53),(cid:56)(cid:49)S(cid:52)(cid:48),(cid:56)(cid:49)T(cid:88)(cid:88)– (cid:56)(cid:49)T(cid:49)(cid:48),(cid:56)(cid:49)T(cid:52)(cid:53),(cid:57)(cid:49)B(cid:88)(cid:88)–(cid:57)(cid:49)B(cid:49)(cid:54),(cid:57)(cid:49)B(cid:50)(cid:52),(cid:57)(cid:49)B(cid:50)(cid:56),(cid:57)(cid:49)B(cid:55)(cid:48),(cid:57)(cid:49)B(cid:55)(cid:50),(cid:57)(cid:49)B(cid:56)(cid:52),(cid:57)(cid:51)E(cid:88)(cid:88)–(cid:57)(cid:51)E(cid:50)(cid:48) ISBN-(cid:49)(cid:48) (cid:51)(cid:45)(cid:53)(cid:52)(cid:48)(cid:45)(cid:55)(cid:48)(cid:56)(cid:52)(cid:54)(cid:45)(cid:52) SpringerBerlinHeidelbergNewYork ISBN-(cid:49)(cid:51) (cid:57)(cid:55)(cid:56)(cid:45)(cid:51)(cid:45)(cid:53)(cid:52)(cid:48)(cid:45)(cid:55)(cid:48)(cid:56)(cid:52)(cid:54)(cid:45)(cid:57) SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialisconcerned, specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproductionon microfilmorinanyotherway,andstorageindatabanks.Duplicationofthispublicationorpartsthereofis permittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember(cid:57),(cid:49)(cid:57)(cid:54)(cid:53),initscurrentversion, andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsareliableforprosecutionunderthe GermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:176)c Springer-VerlagBerlinHeidelberg(cid:50)(cid:48)(cid:48)(cid:55) Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsand regulationsandthereforefreeforgeneraluse. TypesettingbytheeditorsandSPiusingaSpringerLATEXmacropackage Coverdesign:WMXDesignGmbH,Heidelberg Printedonacid-freepaper SPIN:(cid:49)(cid:50)(cid:48)(cid:49)(cid:57)(cid:48)(cid:53)(cid:50) VA(cid:52)(cid:54)(cid:47)(cid:51)(cid:49)(cid:48)(cid:48)/SPi (cid:53)(cid:52)(cid:51)(cid:50)(cid:49)(cid:48) Preface to the Series TheNielsHenrikAbelMemorialFundwasestablishedbytheNorwegiangov- ernmentonJanuary1,2002.ThemainobjectiveistohonorthegreatNorwe- gian mathematician Niels Henrik Abel by awarding an international prize for outstanding scientific work in the field of mathematics. The prize shall con- tributetowardsraisingthestatusofmathematicsinsocietyandstimulatethe interest for science among school children and students. In keeping with this objective the board of the Abel fund has decided to finance one or two Abel Symposia each year. The topic may be selected broadly in the area of pure and applied mathematics. The Symposia should be at the highest interna- tionallevel,andservetobuildbridgesbetweenthenationalandinternational researchcommunities.TheNorwegianMathematicalSocietyisresponsiblefor the events. It has also been decided that the contributions from these Sym- posia should be presented in a series of proceedings, and Springer Verlag has enthusiastically agreed to publish the series. The board of the Niels Henrik AbelMemorialFundisconfidentthattheserieswillbeavaluablecontribution to the mathematical literature. Ragnar Winther Chairman of the board of the Niels Henrik Abel Memorial Fund Kiyosi Itoˆ, at his residence in Kyoto, Japan, April 2005 Photo Credit: Søren-Aksel Sørensen Preface Kiyosi Itoˆ, the founder of stochastic calculus, is one of the few central figures of the twentieth century mathematics who reshaped the mathematical world. Today stochastic calculus, also called Itˆo calculus, is a central research field in mathematics, with branches to several other mathematical disciplines and with many areas of application, for example physics, engineering and biology. Perhapsthemostspectacularfieldofapplicationsatpresentiseconomicsand finance.Indeed,theNobelPrizeinEconomicsin1997wasawardedtoRobert MertonandMyronScholesfortheirderivationofthecelebratedBlack–Scholes option pricing formula using stochastic calculus. The Abel Symposium 2005 took place in Oslo, July 29th – August 4th 2005, and was organized as a tribute to Kiyosi Itoˆ and his works on the occasion of his 90th birthday. Distinguished researchers from all over the world were invited to present thenewestdevelopmentswithintheexcitingandfastgrowingfieldofstochas- tic calculus. We were happy that so many took part to this event. They were, in alphabetical order, (cid:127) Luigi Accardi, Universita’ di Roma “Tor Vergata”, Italy (cid:127) Sergio Albeverio, University of Bonn, Germany (cid:127) Ole E. Barndorff-Nielsen, University of Aarhus, Denmark (cid:127) Giuseppe Da Prato, Scuola Normale Superiore di Pisa, Italy (cid:127) Eugene B. Dynkin, Cornell University, USA (cid:127) David Elworthy, University of Warwick, UK (cid:127) Hans Fo¨llmer, Humboldt University, Germany (cid:127) Masatoshi Fukushima, Kansai University, Japan (cid:127) Takeyuki Hida, Meijo University, Nagoya, Japan (cid:127) Yaozhong Hu, University of Kansas, USA (cid:127) Ioannis Karatzas, Columbia University, USA (cid:127) Claudia Klu¨ppelberg, Technical University of Munich, Germany (cid:127) Torbjo¨rn Kolsrud, KTH, Sweden (cid:127) Paul Malliavin, University of Paris VI, France VIII Preface (cid:127) HenryP.McKean,CourantInstituteofMathematicalScience,NewYork, USA (cid:127) Shige Peng, Shandong University, China (cid:127) Yuri A. Rozanov, CNR, Milano, Italy (cid:127) Paavo Salminen, ˚Abo Akademy University, Finland (cid:127) Marta Sanz-Sol´e, Univeristy of Barcelona, Spain (cid:127) Martin Schweizer, ETH Zu¨rich, Switzerland (cid:127) Michael Sørensen, Univeristy of Copenhagen, Denmark (cid:127) Esko Valkeila, Helsinki University of Technology, Finland (cid:127) Srinivasa Varadhan, Courant Institute of Mathematical Science, New York, USA (cid:127) Shinzo Watanabe, Ritsumeikan University, Japan (cid:127) Tusheng Zhang, University of Manchester, England (cid:127) Xianyin Zhou, Chinese University of Hong Kong In addition there were many other international experts both attending and presenting valuable contributions to the conference. We are grateful to all for making this symposium so successful. The present volume combines both papers from the invited speakers and contributions by the presenting lecturers. We are happy that so many sent their papers for publication in this proceedings, making it a valuable account of the research frontiers in stochastic analysis. Our gratitude is also directed to all the referees that put time and effort in reading the manuscripts. A special feature of this volume is given by the Memoirs that Kiyosi Itoˆ himself wrote for this occasion. We all thank him for these valuable pages which mean so much to both young and established researchers in the field. WealsothanktheAbelFoundation,throughtheNorwegianMathematical Society,andtheCentreofMathematicsforApplications(CMA)attheUniver- sityofOslo,fortheirfinancialsupportandfortheirhelpwiththepreparation andorganizationofthesymposium.OurspecialthanksgotoIngaB˚ardshaug Eide and Helge Galdal for their help with all practical matters before and during the conference. Last but not least we are indebted to Sergio Albeverio for his scientific advice in the organization of the program. Oslo, November 2006 Fred Espen Benth Giulia Di Nunno Tom Lindstrøm Bernt Øksendal Tusheng Zhang Contents Memoirs of My Research on Stochastic Analysis Kiyosi Itˆo....................................................... 1 Itˆo Calculus and Quantum White Noise Calculus Luigi Accardi, Andreas Boukas..................................... 7 Homogenization of Diffusions on the Lattice Zd with Periodic Drift Coefficients, Applying a Logarithmic Sobolev Inequality or a Weak Poincar´e Inequality Sergio Albeverio, M. Simonetta Bernabei, Michael Ro¨ckner, Minoru W. Yoshida .............................................. 53 Theory and Applications of Infinite Dimensional Oscillatory Integrals Sergio Albeverio, Sonia Mazzucchi.................................. 73 Ambit Processes; with Applications to Turbulence and Tumour Growth Ole E. Barndorff-Nielsen, Ju¨rgen Schmiegel ......................... 93 A Stochastic Control Approach to a Robust Utility Maximization Problem Giuliana Bordigoni, Anis Matoussi, Martin Schweizer ................125 Extending Markov Processes in Weak Duality by Poisson Point Processes of Excursions Zhen-Qing Chen, Masatoshi Fukushima, Jiangang Ying ...............153 Hedging with Options in Models with Jumps Rama Cont, Peter Tankov, Ekaterina Voltchkova.....................197 X Contents Power Variation Analysis of Some Integral Long-Memory Processes Jos´e Manuel Corcuera ............................................219 Kolmogorov Equations for Stochastic PDE’s with Multiplicative Noise Giuseppe Da Prato ...............................................235 Stochastic Integrals and Adjoint Derivatives Giulia Di Nunno, Yuri A. Rozanov.................................265 An Application of Probability to Nonlinear Analysis Eugene B. Dynkin ...............................................309 The Space of Stochastic Differential Equations K. David Elworthy ...............................................327 Extremes of supOU Processes Vicky Fasen, Claudia Klu¨ppelberg ..................................339 Gaussian Bridges Dario Gasbarra, Tommi Sottinen, Esko Valkeila .....................361 Some of the Recent Topics on Stochastic Analysis Takeyuki Hida ...................................................383 DifferentialEquationsDrivenbyHo¨lderContinuousFunctions of Order Greater than 1/2 Yaozhong Hu, David Nualart ......................................399 On Asymptotics of Banach Space-valued Itˆo Functionals of Brownian Rough Paths Yuzuru Inahama, Hiroshi Kawabi ..................................415 Continuous-Time Markowitz’s Problems in an Incomplete Market, with No-Shorting Portfolios Hanqing Jin, Xun Yu Zhou........................................435 Quantum and Classical Conserved Quantities: Martingales, Conservation Laws and Constants of Motion Torbjørn Kolsrud ................................................461 Different Lattice Approximations for Hoˆegh-Krohn’s Quantum Field Model Song Liang......................................................493 Itˆo Atlas, its Application to Mathematical Finance and to Exponentiation of Infinite Dimensional Lie Algebras Paul Malliavin...................................................501