Advances in Mathematical Economics Volume 16 PDF
Preview Advances in Mathematical Economics Volume 16
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 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewhole orpartofthematerial isconcerned, specifically therights oftranslation, reprinting, reuseof illustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway, andtransmissionorinformationstorageandretrieval,electronicadaptation,computersoftware, orbysimilarordissimilarmethodologynowknownorhereafterdeveloped.Exemptedfromthis legalreservationarebriefexcerptsinconnectionwithreviewsorscholarlyanalysisormaterial suppliedspecificallyforthepurposeofbeingenteredandexecutedonacomputersystem,for exclusive usebythepurchaserofthework.Duplicationofthispublication orpartsthereofis permitted only under the provisions of the Copyright Law of the Publisher’s location, in its currentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Permissionsfor usemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violationsareliable toprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthis publication does notimply, even inthe absence ofaspecific statement, that suchnames are exemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateof publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsi- bilityforanyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,express orimplied,withrespecttothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) 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 4 C.CastaingandM.Lavie separable,thenEBX ∈ L1 (B)withS1 (B)= {EBf : f ∈S1(F)}. cwk(E) EBX X A unified approachfor generalconditionalexpectationof cc(E)-valuedin- tegrablemultifunctionsisgivenin[33]allowingtorecoverboththecc(E)- valuedconditionalexpectationofcc(E)-valuedintegrablemultifunctionsin thesenseof[20]andthecwk(E)-valuedconditionalexpectationofcwk(E)- valued integrably bounded multifunctions given in [5]. For more informa- tion onmultivaluedconditionalexpectationandrelatedsubjectswe referto [2, 9, 16, 20, 24, 33]. In the contextof this paper we summarize a specific versionofconditionalexpectationinaseparableBanachspace. Proposition3.1. AssumethatthestrongdualE∗isseparable.LetBbeasub- b σ-algebraofF andanintegrableF-measurablecc(E)-valuedmultifunction X : (cid:2) ⇒ E.AssumefurtherthereisaF-measurableball-weaklycompact cc(E)-valued multifunction K : (cid:2) ⇒ E such that X(ω) ⊂ K(ω) for all ω ∈ (cid:2). Then there is a unique(forthe equalitya.s.) B-measurablecc(E)- valuedmultifunctionY satisfyingtheproperty (cid:2) (cid:2) (∗) ∀v ∈L∞E∗(B), δ∗(v(ω),Y(ω))dP(ω) = δ∗(v(ω),X(ω))dP(ω). (cid:2) (cid:2) EBX :=Y istheconditionalexpectationofX. Proof. TheproofisanadaptationoftheoneofTheoremVIII.35in[16].Let u beanintegrableselectionofX.Foreveryn∈N,let 0 X (ω)=X(ω)∩(u (ω)+nB ) ∀n∈N ∀ω ∈(cid:2). n 0 E AsX(ω)⊂K(ω)forallω∈(cid:2),weget X (ω)=X(ω)∩(u (ω)+nB )⊂K(ω)∩(u (ω)+nB ) ∀n∈N ∀ω ∈(cid:2). n 0 E 0 E As K(ω) is ball-weakly compact, it is immediate that X ∈ L1 (F). n cwk(E) so that, by virtue of ([5] or ([33], Remarks of Theorem 3), the conditional expectationEBX ∈L1 (B).Itfollowsthat n cwk(E) (cid:2) (cid:2) (∗∗) δ∗(v(ω),EBX (ω))P(dω)= δ∗(v(ω),X (ω))P(dω) n n (cid:2) (cid:2) ∀n∈N,∀v ∈L∞E∗(B).Nowlet Y(ω)=cl(∪n∈NEBXn(ω)) ∀ω∈(cid:2).
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