Advances in mathematical economics

139 Pages·2012·0.917 MB·English

by  Shigeo Kusuoka, Toru Maruyama (eds.)

ManagingEditors
ShigeoKusuoka ToruMaruyama
TheUniversityofTokyo KeioUniversity
Tokyo,JAPAN Tokyo,JAPAN
Editors
RobertAnderson Jean-MichelGrandmont KunioKawamata
UniversityofCalifornia, CREST-CNRS KeioUniversity
Berkeley Malakoff,FRANCE Tokyo,JAPAN
Berkeley,U.S.A.
HiroshiMatano
CharlesCastaing NorimichiHirano
TheUniversityofTokyo
Universite´MontpellierII YokohamaNational
Tokyo,JAPAN
Montpellier,FRANCE University
Yokohama,JAPAN
FrancisH.Clarke KazuoNishimura
Universite´deLyonI KyotoUniversity
Villeurbanne,FRANCE TatsuroIchiishi Kyoto,JAPAN
TheOhioStateUniversity
EgbertDierker Ohio,U.S.A. MarcelK.Richter
UniversityofVienna UniversityofMinnesota
Vienna,AUSTRIA AlexanderIoffe Minneapolis,U.S.A.
IsraelInstituteof
DarrellDuffie
Technology YoichiroTakahashi
StanfordUniversity
Haifa,ISRAEL TheUniversityofTokyo
Stanford,U.S.A.
Tokyo,JAPAN
LawrenceC.Evans
SeiichiIwamoto
UniversityofCalifornia, AkiraYamazaki
KyushuUniversity
Berkeley MeiseiUniversity
Fukuoka,JAPAN
Berkeley,U.S.A. Tokyo,JAPAN
TakaoFujimoto KazuyaKamiya MakotoYano
FukuokaUniversity TheUniversityofTokyo KyotoUniversity
Fukuoka,JAPAN Tokyo,JAPAN Kyoto,JAPAN
Aims and Scope. The project is to publish Advances in Mathematical
EconomicsonceayearundertheauspicesoftheResearchCenterforMath-
ematical Economics. It is designed to bring together those mathematicians
whoareseriouslyinterestedinobtainingnewchallengingstimulifromeco-
nomictheoriesandthoseeconomistswhoareseekingeffectivemathematical
toolsfortheirresearch.
The scope of Advances in Mathematical Economicsincludes, but is not
limitedto,thefollowingfields:
– Economictheoriesinvariousfieldsbasedonrigorousmathematicalrea-
soning.
– Mathematical methods (e.g., analysis, algebra, geometry, probability)
motivatedbyeconomictheories.
– Mathematicalresultsofpotentialrelevancetoeconomictheory.
– Historicalstudyofmathematicaleconomics.
Authorsareaskedtodeveloptheiroriginalresultsasfullyaspossibleand
alsotogiveaclear-cutexpositoryoverviewoftheproblemunderdiscussion.
Consequently,wewillalsoinvitearticleswhichmightbeconsideredtoolong
forpublicationinjournals.
S. Kusuoka, T. Maruyama
(Eds.)
Advances in
Mathematical Economics
Volume 16
123
ShigeoKusuoka
Professor
GraduateSchoolofMathematicalSciences
TheUniversityofTokyo
3-8-1Komaba,Meguro-ku
Tokyo153-8914,Japan
ToruMaruyama
Professor
DepartmentofEconomics
KeioUniversity
2-15-45Mita,Minato-ku
Tokyo108-8345,Japan
ISSN1866-2226 1866-2234(electronic)
ISBN978-4-431-54113-4 978-4-431-54114-1(eBook)
DOI10.1007/978-4-431-54114-1
SpringerTokyoHeidelbergNewYorkDordrechtLondon
(cid:2)c SpringerJapan2012
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Table of Contents
ResearchArticles
C.CastaingandM.Lavie
SomeapplicationsofBirkhoff-Kingmanergodictheorem 1
Y.Hosoya
ElementaryformandproofoftheFrobeniustheoremfor
economists 39
S.KusuokaandT.Nakashima
Aremarkoncreditriskmodelsandcopula 53
N.YoshiharaandR.Veneziani
Profitsandexploitation:areappraisal 85
Notes
K.MiyazakiandS.Takekuma
Onthe equivalence betweenthe rejectivecoreand the
dividendequilibrium:anote 111
T.Fujita,N.Ishimura,andN.Kawai
Discrete stochastic calculus and its applications: an
expositorynote 119
SubjectIndex 133
InstructionsforAuthors 135
v

Adv.Math.Econ.16,1–38(2012)
Some applications of Birkhoff-Kingman
ergodic theorem
CharlesCastaing1andMarcLavie2
1 De´partmentdeMathematiques,Casecourrier051,Universite´MontpellierII,
34095MontpellierCedex5,France
(e-mail:[email protected])
2 LaboratoiredeMathe´matiquesapplique´es,Universite´dePauetdesPaysde
L’Adour,BP1155,64013,PaucedexFrance
(e-mail:[email protected])
Received:June30,2011
Revised:September27,2011
JELclassification:C01,C02
MathematicsSubjectClassification(2010):28B20
Abstract. Wepresentvariousconvergenceresultsformultivaluedergodictheorems
inBochner-Gelfand-Pettisintegration.
Key words: Conditional expectation, epiconvergence, ergodic, Bochner-Gelfand-
Pettis integration, Birkhoff-Kingman ergodic theorem, Mosco convergence, multi-
valuedconvergence,sliceconvergence
1. Introduction
Classical ergodic theorems for real valued random variables have been
recently extendedinto the contextof epiconvergencein [7, 17, 24, 25, 34].
UsingAbidresult[1]onthea.s.convergenceofsubadditivesuperstationary
process,Krupa[27]andSchurger[32]treatedtheErgodictheoremsforsub-
additivesuperstationaryfamiliesofconvexcompactrandomsets.Ghoussoub
and Steele [6] treated the a.s. norm convergencefor subadditive process in
anordercompleteBanachlatticeextendingtheKingman’stheoremforreal
S.KusuokaandT.Maruyama(eds.),AdvancesinMathematicalEconomics 1
Volume16,DOI:10.1007/978-4-431-54114-1 1,
(cid:2)c SpringerJapan2012
2 C.CastaingandM.Lavie
valuedsubadditiveprocess.Inthispaperwe presentvariousapplicationsof
the Birkhoff-Kingmanergodic theorem. The paper is organizedas follows.
In Sects.3–4 we state and summarize for references some results on the
conditionalexpectationofclosedconvexvaluedintegrablemultifunctionsin
separableBanach spacesandin their dualspaces. Main resultsare givenin
Sects.5–8.Forthesakeofcompletenessweprovideanepiconvergenceresult
forparametricergodictheoreminSect.5thatisastartingpointofthisstudy.
InSect.6wetreattheMoscoconvergenceforconvexweaklycompactvalued
ergodic theorem in Bochner integration and the weak star Kuratowskicon-
vergenceforconvexweaklystarcompactvaluedergodictheoreminGelfand
integrationandalso a scalarconvergenceresultforconvexweaklycompact
valued ergodic theorem in Pettis integration. An unusual convergence for
superadditive integrable process in Banach lattice is given in Sect.7 using
theintegrableselectiontheoremforthesequentialweakupperlimitofase-
quenceofmeasurableclosedconvexvaluedrandomsets.Somerelationships
witheconomicproblemsarealsodiscussed.InSect.8wepresenta conver-
gence theorem for convexweakly compact valued superadditiveprocess in
BochnerintegrationviaKomlo´stechniques.
2. Notations andpreliminaries
Throughoutthispaper((cid:2),F,P)isacompleteprobabilityspace,(Fn)n∈Nis
anincreasingsequenceofsub-σ-algebrasofF suchthatF istheσ-algebra
generatedby∪n∈NFn,EisaseparableBanachspaceandE∗isitstopological
dual. Let BE (resp. BE∗) be the closed unit ball of E (resp. E∗) and 2E
thecollectionofallsubsetsofE.Letcc(E)(resp.cwk(E))(resp.Lwk(E))
(resp.Rwk(E)) bethesetofnonemptyclosedclosedconvex(resp.convex
weaklycompact)(resp.closedconvex weaklylocallycompactsubsetsofE
whichcontainnolines)(resp.ball-weaklycompactclosedconvex)subsetsof
E,hereaclosedconvexsubsetinEisball-weaklycompact ifitsintersection
with any closed ball in E is weakly compact. For A ∈ cc(E), the distance
andthesupportfunctionassociatedwithAaredefinedrespectivelyby
d(x,A)=inf{(cid:6)x−y(cid:6) :y ∈A},(x ∈E)
δ∗(x∗,A)=sup{(cid:7)x∗,y(cid:8) :y ∈A},(x∗ ∈E∗).
Wealsodefine
|A|=sup{||x|| :x ∈A}.
A sequence (Kn)n∈N in cwk(E) scalarly converges to K∞ ∈ cwk(E) if
limn→∞δ∗(x∗,Kn) = δ∗(x∗,K∞),∀x∗ ∈ E∗. LetB be a closed bounded
SomeapplicationsofBirkhoff-Kingmanergodictheorem 3
convexsubsetofEandletCbetheclosedconvexsubsetofE.Thenthegap
[4]D(B,C)betweenB andC isdenotedby
D(B,C) =inf{||x−y||:x ∈B,y ∈C}.
ByHahnBanachtheoremweknowthat
D(B,C)= sup {−δ∗(x∗,C)−δ∗(−x∗,B)}.
x∗∈BE∗
Givenasub-σ-algebraB in(cid:2),amappingX : (cid:2) → 2E isB-measurableif
foreveryopensetU inE theset
X−U :={ω∈(cid:2):X(ω)∩U (cid:13)=∅}
isamemberofB.Afunctionf : (cid:2) → E isaB-measurableselectionofX
iff(ω)∈X(ω)forallω ∈(cid:2).ACastaingrepresentationofXisasequence
(fn)n∈N ofB-measurableselectionsofXsuchthat
X(ω)=cl{f (ω),n∈N} ∀ω∈(cid:2)
n
where the closure is taken with respect to the topology of associated with
thenorminE.Itisknownthatanonemptyclosed-valuedmultifunctionX :
(cid:2) → c(E) is B-measurableiff it admits a Castaing representation.If B is
complete,theB-measurabilityisequivalenttothemeasurabilityinthesense
ofgraph,namelythegraphofXisamemberofB⊗B(E),hereB(E)denotes
the Borel tribe on E. A cc(E)-valued B-measurable X : (cid:2) → cc(E) is
integrableif the set S1(B) of all B-measurableand integrableselections of
X
X is nonempty.We denote by L1(B) the space of E-valued B-measurable
E
andBochner-integrablefunctionsdefinedon(cid:2)andL1 (B)thespaceof
cwk(E)
allB-measurablemultifunctionsX : (cid:2) → cwk(E)suchthat|X| ∈ L1(B).
R
We refer to [16] for the theory of Measurable Multifunctions and Convex
Analysis,andto[18,29]forRealAnalysisandProbability.
3. Multivalued conditional expectation theorem
Givenasub-σ-algebraBofF andanintegrableF-measurablecc(E)-valued
multifunctionX:(cid:2)⇒E,HiaiandUmegaki[20]showedtheexistenceofa
B-measurablecc(E)-valuedintegrablemultifunctiondenotedbyEBX such
that
S1 (B)=cl{EBf :f ∈S1(F)}
EBX X
theclosurebeingtakeninL1((cid:2),A,P);EBXisthemultivaluedconditional
E
expectationofXrelativetoB.IfX ∈ L1 (F)andthestrongdualE∗ is
cwk(E) b