Stochastic Analysis and Applications: The Abel Symposium 2005

688 Pages·2007·4.82 MB·English

by  Benth F.E.

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
Itoˆ 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
DifferentialEquationsDrivenbyH¨olderContinuousFunctions
of Order Greater than 1/2
Yaozhong Hu, David Nualart ......................................399
On Asymptotics of Banach Space-valued
Itoˆ 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