Table Of ContentOptimality and Risk—
Modern Trends in Mathematical Finance
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Freddy Delbaen (cid:2) Miklós Rásonyi
Christophe Stricker
Editors
Optimality and Risk—
Modern Trends
in Mathematical Finance
The Kabanov Festschrift
FreddyDelbaen ChristopheStricker
DepartmentofMathematics LaboratoiredeMathématiques
ETHZürich UniversitéBesançon
Rämistr.101 16routedeGray
8092Zürich 25030Besançoncedex
Switzerland France
[email protected] [email protected]
MiklósRásonyi
Computer&Automation
InstituteoftheUngarian
AcademyofSciences
Kendeutca13-17
1111Budapest
Hungary
[email protected]
ISBN978-3-642-02607-2 e-ISBN978-3-642-02608-9
DOI10.1007/978-3-642-02608-9
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2009935004
MathematicsSubjectClassification(2000):91B28,60-06,91-06
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ThisvolumeisdedicatedtoYuriKabanov,
tocelebratehis60thbirthday
Preface
Youri (Yuri) Kabanov was born in a city Cherkassy, now Ukraine, formerly the
USSR,in 1948.Evenin highschoolhebecameinterestedinmathematicsandac-
tivelyandsuccessfullyparticipatedinlocalandnationwidemathematicalcompeti-
tionsforhighschoolstudents.Asaresult,withoutexamshewasadmittedtoaspe-
cialized mathematical boarding school at Kiev State University and he graduated
fromthisschoolwiththegoldmedal.
Inthesummerof1966YuricametoMoscowwiththehopesofstudyingtheoret-
icalphysicsandtoapplytotheMoscowInstituteofPhysicsandTechnology.This
institute, known worldwide as Phystech (sometimes called the Russian version of
MIT), was initially created in 1946 to produce high quality specialists in different
fieldsofPhysics.ButitwaslocatedinadistantsuburbofMoscow,andforastudent
whocamefromasmallprovincialcity,livinginadormitorytheremeantbeingcut
off from the life of the capital. One of his friends, who had a brother studying at
MoscowUniversity,decidedtoapplytotheDepartmentofMechanicsandMathe-
maticsofMoscowStateUniversity(theinternationallyrenownedMEKHMAT).He
convincedYurithatMEKHMATwouldbeabetterchoice.Afterpassingthedifficult
entranceexams,YuribecameastudentatMoscowUniversity.Whoknows,maybe
thisrandomeventtriggeredhislaterinterestinrandomprocesses.
Atthattime,knownastheGoldenYearsofMoscowMathematics(thetitleofthe
book,publishedbytheAmericanMath.Society),MEKHMAThadareputationas
one of the leading centers of mathematical research and math education. Students
weretaughtbysuchgreatmathematiciansasA.N.Kolmogorov,P.S.Aleksandrov,
I.M. Gelfand, B.V. Gnedenko as well as then young but well known people like
S.Novikov,Yu.Manin,D.Anosov,andYa.Sinai.
In the beginning Yuri was interested in many fields and attended a variety of
courses,butattheendofthesecondyearhehadchosenthesubfieldofprobability
theory.HisscientificadvisorwasAlbertN.Shiryaev,thenayoungprofessor,afor-
mer student of A. Kolmogorov. Shiryaev later became one of the most prominent
leadersofthetheoryofstochasticprocesses.
Yuri’sinterestsinhisstudentyearswerenotlimitedtomathematicsonly.Hewas
activelyinvolvedinsports,becomingamemberoftheprestigiousall-universityvol-
leyballteamandexcellingintrackandfieldandtheshotput.Havingaphenomenal
vii
viii Preface
memoryheknewmanyverses,anddelightedhisfriendsandgirlsbyrecitingfrom
heart verses from such poets as Baudelaire, Rimbaud, and Gumilyov, who were
rarelypublishedduringtheSovietyears.ThecitywhereYuriwasbornwasasmall
onebutratherold,foundedinthelate13thcentury,andYuriloveshistory,inwhich
heiswell-versed:todayheenjoyspresentingashorthistoryoftheRomanEmpire
usinghiscollectionofancientcoinsasanillustration.
AftergraduatingwithdistinctionfromMoscowUniversityin1971Yuribecame
a postgraduate student at the SteklovMathematicalInstitute.His first papers were
devoted to the integral representation of functionals of Wiener, Poisson, and other
processes with independent increments, and were published in the early 1970s in
Theory of Probability and its Applications. His next paper “A generalized Itô for-
mulaforanextendedstochasticintegralwithrespecttoPoissonrandommeasure”
waspublishedintheprestigiousRussianMathematicalSurveysin1974.Thisexten-
sion was obtained by means of a generalized Cameron-Martin expansion in series
of multiple Poisson integrals. Yuri also proved a generalized Itô formula for such
extendedstochasticintegrals.Asaresultofhisworkinthisnewandrapidlydevel-
opingfieldofstochasticprocesses,heobtainedhisPh.D.degreewiththedisserta-
tiontopicPointProcessesandExtendedStochasticIntegrals.Hissupervisorwashis
teacherA.N.Shiryaev.
As a graduate student he helped Shiryaev to organize the very popular an-
nual probability conferences in Bakuriani (Georgia). He recalls how he served as
a“bodyguard”foraforeignvisitor(atthetimerareatsuchconferences),F.Spritzer.
Astrikingfeatureoftheseconferenceswasamatchingofintensivescientificspirit
duringthetalksandeverydaysnow-skiinginthesurroundingmountains.Thelead-
ing figure at these conferences was Shiryaev, who instilled his love of skiing not
onlyinYuributinmanyotherparticipants.
After his postgraduate studies at the Steklov Institute, Yuri started working at
oneofthebestresearchinstitutesoftheRussianAcademyofSciences—intheCen-
tralMathematicalEconomicalInstitute(CEMI)inMoscow.HistiestothisInstitute
continued even after he moved to Besancon, France. At this Institute Yuri contin-
uedhisstudiesofstochasticprocessesandpublishedafewfundamentalpaperswith
R.LiptserandA.Shiryaevonabsolutecontinuityandsingularityofprobabilitydis-
tributions, and on the convergence of distributions of counting processes. He also
startedworkingonsomeproblemsofstochasticcontrolandproblemsofmathemat-
ical economics. He had strong ties with statisticians, initially working in a group
headedbyS.AivasianandlaterbecomingamemberoftheProbabilityLaboratory
headed by V. Arkin, who assembled a very strong group of probabilists, many of
whombecameco-authorsandfriendsofYuri.
Inthe1980s,undertheinfluenceofhisteacher,A.Shiryaev,Yuristartedworking
inthefieldoffunctionallimittheorems.Thesummaryofhisresults,onthecontigu-
ityofdistributionsofmultivariatepointprocessesandmanyothers,wascollectedin
hisD.Sc.thesis,whichhedefendedin1984.Attheendofthe1980sYuri,together
withhisstudentS.Pergamenshchikov,startedworkingintheareaofsingularlyper-
turbedstochasticequationsandpartialdifferentialequations.Theypublishedmore
thantenpapersonthesubject.
Preface ix
Fromthemiddleofthe1970sYuriactivelyparticipatedinteachingatavarietyof
MoscowandforeigninstitutionssuchastheMoscowInstituteofAdvancedStudies
forChemistryManagersandEngineers,theMoscowInstituteofAviationTechnol-
ogy (MATI), the Friedrich Schiller University in Jena, Germany, and the Moscow
InstituteofElectronicsandMathematics(MIEM).Healsoheldavisitingposition
atBilkentUniversityinAnkara,Turkey.
Intheearly1990sA.Shiryaev,togetherwithhisstudents,organizedaseminaron
financialmathematics,andthisareadefinedthemaindirectionsofhisstudiesthere-
after. In 1994 a renowned issue of Theory of Probability and its Applications was
almostentirelydevotedtofinancialmathematics.Thisissuecontainedfourpapers
whichYurihadparticipatedin.
Since his 1995 nomination as a professor at the University of Besancon, Yuri
hasrevivedaforgottenlocaltraditionbydevotinghimselfmainlytomathematical
finance: Louis Bachelier, the founder of mathematical finance, was a professor in
Besanconfrom1927untilhisretirementin1937.
Aninitialseriesofhisworksisrelatedtolargefinancialmarkets;totheoptional
decomposition theorem which is an essential tool for solving the superreplication
problem;andtoageneralmodelofbondmarkets,whichnecessitatedtheintroduc-
tionofanewtypeofstochasticintegralwithrespecttoprocesseswithvaluesinthe
spaceofcontinuousfunctions.
Hesoonbecameinterestedinfinancialmarketswithtransactioncosts,afieldnot
too actively studied at the time because of its difficulty. In 1997 he and M. Safar-
ianshowedthatLeland’sapproximationstrategydidnotasymptoticallyhedgecall
options, contrary to what was asserted in the literature. Then he launched himself
intomodelingmarketswithseveral(finitelymany)basisassets.Infact,allmodels
studiedatthattimecomprisedtwoassetsonly:arisklessandariskyone.Onlythe
model of Jouini and Kallal allowed for multiple assets, but under very restrictive
conditions:directtransfersbeingforbidden,allexchangeswererequiredtopassby
thebankaccount.Inhispaper“Hedgingandliquidationundertransactioncostsin
currencymarkets”(FinanceandStochastics,1999)heproposedageometricmodel
formarketswithtransactioncostsandshowedthattheobjectanalogoustomartin-
gale measures is vector measures whose density processes take values in the dual
of the solvency cone. This kind of model, adopted by all researchers in this field,
has greatly contributed to progress, allowed for a satisfactory theory of arbitrage
andforattackingimportantproblemsoffinance(inparticular,thesuperhedgingof
contingentclaims)inthepresenceofproportionaltransactioncosts.Yuri,working
withvariousco-authors,hasobtainedcriteriafortheabsenceofarbitrageandsuper-
replicationtheorems:indiscretetimewithcompleteorincompleteinformation;and
incontinuoustimeinincreasinglygeneralsettings.Hehasjustcompletedhislatest
book,withM.Safarian:“Marketswithtransactioncosts,mathematicaltheory”.
Yuri has also continued to be interested in stochastic differential equations and
in problems of stochastic control. In his book, written together with S. Pergamen-
shchikov, “Two-scale stochastic systems, asymptotic analysis and control”, he es-
tablished a stochastic version of Tikhonov’s theorem, obtained an asymptotic ex-
pansionforthissystemandshowedalargedeviationlimittheorem.
x Preface
Apartfromhisactivitiesasafirst-classresearcher,Yurihasdoneagreatdealfor
thedisseminationofmathematics.In1997heactivelyparticipatedinthefounding
ofFinanceandStochastics,whichinonlyafewyearshasbecomeoneoftheleading
journals in mathematical finance. The three Bachelier Colloquia he has organized
(in2000inBesancon,in2005andin2008inMétabief)wereextremelysuccessful,
collectingresearchersfromeverycontinentoftheworld.
Yuriisnowintheprimeofhisresearchandcareer.Heisnotonlyoneofthemost
notable figures in financial mathematics and stochastic processes but also a very
friendlypersonwithnewandfreshideas,readytosharethemwithothers.Almost
30peoplearecountedamonghisco-authors.Heisafatherofanowadultsonand
youngdaughter,andalovingandcaringhusband.HiswifeMargarita,orRitaasher
friends call her, is a gifted photographer, and those visiting his or her website can
seethisforthemselves.
Thereisnodoubtthathisfriends,hisco-authors,andmanyotherswhohavehad
thechancetomeethimpersonallyorjusttoreadhispapers,willjoinusinwishing
Yurigoodhealth,happinessandmanyyearstocomeofcreativeactivity.
E.Presman,I.Sonin,andCh.Stricker
