Advances in Mathematical Economics Volume 17 PDF
Preview Advances in Mathematical Economics Volume 17
Shigeo Kusuoka Toru Maruyama Editors 17 Volume 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. Shigeo Kusuoka • Toru Maruyama Editors Advances in Mathematical Economics Volume 17 123 Editors 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) ISBN978-4-431-54323-7 978-4-431-54324-4(eBook) DOI10.1007/978-4-431-54324-4 SpringerTokyoHeidelbergNewYorkDordrechtLondon (cid:2)c SpringerJapan2013 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.CastaingandP.RaynauddeFitte LawoflargenumbersandErgodicTheoremforconvex weakstarcompactvaluedGelfand-integrablemappings 1 M.AliKhanandT.Mitra Discountedoptimalgrowthinatwo-sectorRSS model: afurthergeometricinvestigation 39 S.Kusuoka GaussianK-scheme:justificationforKLNVmethod 71 T.Suzuki Competitive equilibria of a large exchange economy ∞ onthecommodityspace(cid:2) 121 H.Tanaka Localconsistencyoftheiterativeleast-squaresestimator forthesemiparametricbinarychoicemodel 139 SubjectIndex 163 InstructionsforAuthors 167 v Adv.Math.Econ.17,1–37(2013) Law of large numbers and Ergodic Theorem for convex weak star compact valued Gelfand-integrable mappings C.Castaing1andP.RaynauddeFitte2 1 De´partementdeMathe´matiques,Universite´MontpellierII, Casecourrier051,34095MontpellierCedex5,France (e-mail:[email protected]) 2 LaboratoireRaphae¨lSalem,UMRCNRS6085, UFRSciences,Universite´deRouen,Avenuedel’Universite´, BP1276801SaintEtienneduRouvray,France (e-mail:[email protected]) Received:September13,2012 Revised:October9,2012 JELclassification:C01,C02 MathematicsSubjectClassification(2010):28B20,60F15,60B12 Abstract. Weproveseveralresultsintheintegrationofconvexweakstar(resp.norm compact)valuedrandomsetswithapplicationtoweakstarKuratowskiconvergencein thelawoflargenumbersforconvexnormcompact valuedGelfand-integrablemap- pings in the dual of a separable Banach space. We also establish several weak star Kuratowskiconvergenceinthelawoflargenumbersandergodictheoreminvolving thesubdifferentialoperatorsofLipschitzeanfunctionsdefinedonaseparableBanach space,andalsoprovideanapplicationtoaclosuretyperesultariseninevolutionin- clusions. Key words: Conditional expectation, Ergodic, Generalized directional derivative, Lawoflargenumbers,LocallyLipschitzean,Subdifferential 1. Introduction SeveralconvergenceproblemsinthedualofaseparableBanachspacehave been treated with Fatou Lemma in Mathematical Economics [2, 9, 16], S.KusuokaandT.Maruyama(eds.),AdvancesinMathematicalEconomics 1 Volume17,DOI:10.1007/978-4-431-54324-4 1, (cid:2)c SpringerJapan2013 2 C.CastaingandP.RaynauddeFitte martingales[8]andergodictheorem[11,19,48].Theaforementionedresults lead naturally to the law of large numbers in the dual space. At this point, the law of large numbers for Pettis-integrable functions in locally convex spaces has been studied in [12], in particular, almost sure convergence for the law of large numbers in the weak star dual space for some classes of Gelfand-integrable mappings is also available. Some related results for the lawoflargenumbersinvolvingthesubdifferentialofLipschitzeanfunctions havebeenstudiedin[42,43].Thereareaplethoreofresultsfortheconver- gence in the law of large numbers for vector valued random variables and closed valued random sets in Banach spaces, see e.g. [14, 15, 31–37, 41] and the references therein. Here we provide new convergence (namely the weak star Kuratowski convergence) in the law of large numbers for con- vex weak star compact valued Gelfand-integrable mappings in a dual of a separable Banach space and, we also present some new versions of law of largenumbersandergodictheoreminvolvingthesubdifferentialoperatorof aLipschitzeanfunctiondefinedonaseparableBanachspace.Thepaperisor- ganizedasfollows.InSect.2wegivedefinitionsandpreliminariesonmea- surability properties for convex weak star compact valued mappings (alias multifunctions)inthedualofaseparableBanachspace.InSect.3wesumma- rize the propertiesofconditionalexpectationforconvexweak star compact valued Gelfand-integrablemappings,in particularwe presenta Jensen type inequalityforconvexweakstarcompactvaluedconditionalexpectationand aversionofdominatedLebesgueconvergencetheoremforconvexweakstar compact valued Gelfand-integrablemappings.In Sect.4 we presentseveral results ontheintegrationof convexweakstar (resp.convexnormcompact) valuedrandomsetswithapplicationtoweakstarKuratowskiconvergencein the law large numbersfor convexnormcompactvalued Gelfand-integrable mappings in the same vein as [14, 34, 35, 41] dealing with Wijsman and Moscoconvergenceinthelawoflargenumbersforclosedrandomintegrable sets in separableBanachspaces. InSect.5we providetwo weakstar Kura- towskiconvergenceresultsinthelawoflargenumbersandergodictheorem involvingthesubdifferentialoperatorsofLipschitzeanfunctionsdefinedona separableBanachspace,andalsoanapplicationtoaclosuretyperesultarisen inevolutioninclusions. 2. Notations andPreliminaries Let((cid:3),F,P)beacompleteprobabilityspace.LetEbeaseparableBanach ∗ space, E the topological dual of E, BE (resp. BE∗) the closed unit ball of E (resp. E∗), D1 = (ek)k∈N a dense sequence in BE. We denote by ∗ ∗ ∗ ∗ E (resp.E )thevectorspaceE endowedwith thetopologyσ(E ,E) of s b LawoflargenumbersandErgodicTheoremforconvexweakstar... 3 ∗ ∗ pointwise convergence,alias w topology(resp. the topologys associated twhiethtotphoelodguyalmn∗or=mσ||(.E||E∗,b∗)H, a),nwdhbeyreEHm∗∗isththeevleicnteoarrssppaacceeEof∗Eengdeonwereadtewdibthy D ,thatistheHausdorfflocallyconvextopologydefinedbythesequenceof 1 semi-norms P (x∗)=max{|(cid:6)e ,x∗(cid:7)|:k ≤n}, x∗ ∈E∗,n∈N. n k ∗ Recallthatthetopologym ismetrizable,forinstance,bythemetric (cid:2)∞ 1 dEm∗∗(x∗,y∗):= 2k|(cid:6)ek,x∗(cid:7)−(cid:6)ek,y∗(cid:7)|, x∗,y∗ ∈E∗. k=1 W∗eassumefromnowonthatdEm∗∗ isheldfix∗ed.Furth∗er,wehavem∗ ⊂w∗ ⊂ s . Ontheotherhand,therestrictionsofm andw to anyboundedsubset of E∗ coincideandthe BoreltribesB(Es∗) andB(Em∗∗) associated with Es∗ andEm∗∗ areequal,buttheconsiderationoftheBoreltribeB(Eb∗)associated ∗ ∗ with the topologyof E is irrelevant here. Noting that E is the countable b ∗ unionofclosedballs,wededucethatthespaceE isaLusinspace,aswell s as the metrizable topologicalspace Em∗∗. Let K∗ = cwk(Es∗) be the set of all nonemptyconvexweak star compactsubsetsin E∗. A K∗-valuedmulti- function(aliasmappingforshort)X : (cid:3) ⇒ E∗ isscalarlyF-measurableif, s ∀x ∈ E, the supportfunctionδ∗(x,X(.)) is F-measurable,henceits graph belongstoF ⊗B(Es∗).Indeed,let(fk)k∈N beasequenceinE whichsepa- ratesthepointsofE∗,thenwehavex∗ ∈X(ω)iff(cid:6)f ,x∗(cid:7) ≤ δ∗(f ,X(ω)) k k forallk ∈N.Consequently,foranyBorelsetG∈B(E∗),theset s X−G={ω∈(cid:3):X(ω)∩G(cid:14)=∅} isF-measurable,thatis,X−G ∈ F,thisisaconsequenceoftheProjection Theorem(seee.g.[17,TheoremIII.23]andoftheequality X−G=proj {Gr(X)∩((cid:3)×G)}. (cid:3) Inparticularifu : (cid:3) → E∗ isa scalarlyF-measurablemapping,thatis,if s for everyx ∈ E, the scalar functionω (cid:17)→ (cid:6)x,u(ω)(cid:7) isF-measurable,then thefunctionf : (ω,x∗) (cid:17)→ ||x∗−u(ω)||E∗ isF ⊗B(Es∗)-measurable,and for every fixed ω ∈ (cid:3), f(ω,.) is lower sebmicontinuouson E∗, i.e. f is a s normalintegrand.Indeed,wehave ||x∗−u(ω)||E∗ =sup|(cid:6)ek,x∗−u(ω)(cid:7)|. b k∈N Aseachfunction(ω,x∗)(cid:17)→ (cid:6)e ,x∗−u(ω)(cid:7)isF ⊗B(E∗)-measurableand k s continuouson E∗ for each ω ∈ (cid:3), it follows that f is a normalintegrand. s 4 C.CastaingandP.RaynauddeFitte Consequently, the graph of u belongs to F ⊗ B(E∗). Let B be a sub-σ- s algebra of F. It is easy and classical to see that a mapping u : (cid:3) → E∗ s is (B,B(E∗)) measurable iff it is scalarly B-measurable. A mapping u : s (cid:3) → E∗ is said to be scalarly integrable (alias Gelfand integrable), if, for s everyx ∈ E,thescalarfunctionω (cid:17)→ (cid:6)x,u(ω)(cid:7)isF-measurableandinte- grable. We denote by G1E∗[E](F) the space of all Gelfand integrablemap- pings and by L1E∗[E](F) the subspace of all Gelfand integrable mappings u such that the function |u| : ω (cid:17)→ ||u(ω)||E∗ is integrable. The mea- surability of |u| follows easily from the above cobnsiderations.More gener- ally, by G1 ((cid:3),F,P) (or G1 (F) for short) we denote the space cwk(E∗) cwk(E∗) of all scalarly Fs -measurableand integsrablecwk(E∗)-valuedmappingsand s by L1 ((cid:3),F,P) (or L1 (F) for short) we denote the subspace cwk(E∗) cwk(E∗) s ∗ s ofallcwk(E )-valuedscalarlyintegrableandintegrablyboundedmappings s X, that is, such that the function |X| : ω → |X(ω)| is integrable, here |X(ω)| := supy∗∈X(ω)||y∗||E∗, bythe aboveconsideration,it iseasy to see that|X|isF-measurable. b For anyX ∈ L1cwk(E∗)(F), we denoteby SX1(F) the set ofall Gelfand- s integrable selections of X. The Aumann–Gelfand integral of X over a set A∈F isdefinedby (cid:3) (cid:3) E[1 X]= XdP:={ f dP:f ∈S1(F)}. A X A A We will consider on K∗, the Hausdorff dis∗tance dHm∗∗ associated with the dmiesttarinccdeEdm∗H∗b∗inasthsoecLiautseidnwmietthritzhaebnleosrpmacdeu(aEl|m|.∗|,|Edb∗Emo∗∗n)Eanb∗d,naalsmoetlhyeHausdorff dH∗(A,B)= sup |δ∗(x,A)−δ∗(x,B)| ∀A,B ∈K∗. b x∈BE ∗ Let (Xn)n∈N be a sequence of w -closed convex sets, the sequential ∗ ∗ weak upperlimitw -lsXn of(Xn)n∈N isdefinedby w∗-lsX ={x∗ ∈E∗ :x∗ =σ(E∗,E)- lim x∗; x∗ ∈X }. n j→∞ j j nj ∗ ∗ Similarlythesequentialweak lowerlimitw -liXnof(Xn)n∈Nisdefinedby w∗-liX ={x∗ ∈E∗ :x∗ =σ(E∗,E)- lim x∗; x∗ ∈X }. n n→∞ n n n ∗ Thesequence(Xn)n∈NweakstarKuratowski(w K forshort)convergestoa ∗ w -closedconvexsetX∞ ifthefollowingholds w∗-lsXn ⊂X∞ ⊂w∗-liXn a.s.
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