ManagingEditors ShigeoKusuoka AkiraYamazaki UniversityofTokyo MeiseiUniversity Tokyo,JAPAN Tokyo,JAPAN Editors RobertAnderson Jean-MichelGrandmont NorioKikuchi UniversityofCalifornia, CREST-CNRS KeioUniversity Berkeley Malakoff,FRANCE Yokohama,JAPAN Berkeley,U.S.A. NorimichiHirano ToruMaruyama CharlesCastaing YokohamaNational KeioUniversity University UniversitéMontpellierII Tokyo,JAPAN Yokohama,JAPAN Montpellier,FRANCE HiroshiMatano LeonidHurwicz UniversityofTokyo FrankH.Clarke UniversityofMinnesota Tokyo,JAPAN UniversitédeLyonI Minneapolis,U.S.A. Villeurbanne,FRANCE TatsuroIchiishi KazuoNishimura TheOhioStateUniversity KyotoUniversity EgbertDierker Ohio,U.S.A. Kyoto,JAPAN UniversityofVienna AlexanderIoffe MarcelK.Richter Vienna,AUSTRIA IsraelInstituteof UniversityofMinnesota DarrellDuffie Technology Minneapolis,U.S.A. StanfordUniversity Haifa,ISRAEL YoichiroTakahashi Stanford,U.S.A. SeiichiIwamoto KyotoUniversity KyushuUniversity LawrenceC.Evans Kyoto,JAPAN Fukuoka,JAPAN UniversityofCalifornia, KazuyaKamiya MichelValadier Berkeley UniversityofTokyo UniversitéMontpellierII Berkeley,U.S.A. Tokyo,JAPAN Montpellier,FRANCE TakaoFujimoto KunioKawamata MakotoYano FukuokaUniversity KeioUniversity KyotoUniversity Fukuoka,JAPAN Tokyo,JAPAN Kyoto,JAPAN Aims and Scope. The project is to publish Advances in Mathematical Eco- nomicsonceayearundertheauspicesoftheResearchCenterforMathemati- calEconomics.Itisdesignedtobringtogetherthosemathematicianswhoare seriously interested in obtaining new challenging stimuli from economic the- ories and those economists who are seeking effective mathematical tools for theirresearch. The scope of Advances in Mathematical Economics includes, but is not limitedto,thefollowingfields: – Economictheoriesinvariousfieldsbasedonrigorousmathematicalreason- ing. – Mathematicalmethods(e.g.,analysis,algebra,geometry,probability)moti- vatedbyeconomictheories. – Mathematicalresultsofpotentialrelevancetoeconomictheory. – Historicalstudyofmathematicaleconomics. Authorsareaskedtodeveloptheiroriginalresultsasfullyaspossibleand also to give a clear-cut expository overview of the problem under discussion. Consequently,wewillalsoinvitearticleswhichmightbeconsideredtoolong forpublicationinjournals. S. Kusuoka, A. Yamazaki (Eds.) Advances in Mathematical Economics Volume 11 ShigeoKusuoka Professor GraduateSchoolofMathematicalSciences UniversityofTokyo 3-8-1Komaba,Meguro-ku Tokyo,153-0041Japan AkiraYamazaki Professor DepartmentofEconomics MeiseiUniversity Hino Tokyo,191-8506Japan ISBN978-4-431-77783-0 e-ISBN978-4-431-77784-7 Printedonacid-freepaper SpringerisapartofSpringerScience+BusinessMedia springer.com ©Springer2008 PrintedinJapan This work is subject to copyright. All rights are reserved, whether the whole or part ofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseof illustrations,recitation,broadcasting,reproductiononmicrofilmsorinotherways,and storageindatabanks.Theuseofregisterednames,trademarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. Camera-readycopypreparedfromtheauthors’LATEXfiles. PrintedandboundbyShinanoCo.Ltd.,Japan. SPIN:12219216 Table of Contents ResearchArticles T.Arai Optimalhedgingstrategiesonasymmetricfunctions 1 C.Castaing,C.Hess,M.Saadoune Tightnessconditionsandintegrabilityofthesequentialweak upperlimitofasequenceofmultifunctions 11 C.Hara Coreconvergenceineconomieswithbads 45 H.Komiya Adistanceandabinaryrelationrelatedtoincome comparisons 77 V.L.Levin Onpreferencerelationsthatadmitsmoothutilityfunctions 95 T.Matsuhisa,R.Ishikawa Rationalexpectationscanprecludetrades 105 R.J.Rossana TheLeChatelierPrincipleindynamicmodelsofthefirm 117 T.Shinotsuka Interdependentutilityfunctionsinanintergenerational context 147 SubjectIndex 157 InstructionsforAuthors 159 Adv.Math.Econ.11,1–10(2008) Optimal hedging strategies on asymmetric functions ∗ TakujiArai DepartmentofEconomics,KeioUniversity,2-15-45Mita,Minato-ku,Tokyo108-8345, Japan (e-mail:[email protected]) Received:June26,2007 Revised:November7,2007 JELclassification:G10 MathematicsSubjectClassification(2000):91B28,52A41,60H05 Abstract. We treat in this paper optimal hedging problems for contingent claims in anincompletefinancialmarket,whichproblemsarebasedonasymmetricfunctions.In summary,weconsidertheproblem min E[f(H−GT(ϑ))], ϑ∈Θ where H is a contingent claim, Θ, which is a suitable set of predictable processes, representsthecollectionofalladmissiblestrategies,GT(ϑ)isaportfoliovalueatthe maturityT inducedbyanadmissiblestrategyϑ,and f : R → R+ isadifferentiable strictlyconvexfunctionwith f(0) = 0.Inparticular,undertheassumptionthatthere existtwopositiveconstantsc0 andC1 suchthat,forany x ∈ R beingfarawayfrom 0sufficiently,c0|x|p ≤ f(x),and|f(cid:5)(x)| ≤ C1|x|p−1,where1 < p < ∞,weshall provetheuniqueexistenceofasolutionandshalldiscussitsmathematicalproperty. Keywords: mathematical finance, incomplete market, convex function, semimartin- gale,stochasticintegral ∗ TheauthorwouldliketothankJanKallsenandShigeoKusuokafortheirvaluable commentsanddiscussion,andisverygratefultoananonymousrefereeforhelpful comments,whichhasgreatlyimprovedthepaper.Thefinancialsupportoftheauthor hasbeenpartiallygrantedbyGrant-in-AidforYoungScientists(B)No.16740062 andScientificResearch(C)No.19540144fromtheMinistryofEducation,Culture, Sports,ScienceandTechnologyofJapan. 2 T.Arai 1. Introduction Let (Ω,F,P) be a complete probability space. We fix T > 0, and suppose thatF={Ft}t∈[0,T] isafiltrationsatisfyingtheso-calledusualcondition,that is, F is right-continuous and F contains all null sets of F. In addition, we 0 assumethatF istrivialandF =F.LetX beanF-adaptedRd-valuedRCLL 0 T semimartingaleon(Ω,F,P).Xisnotassumedtobecontinuous.Moreover,Θ denotes some sub(cid:1)space of Rd-valued X-integrable predictable processes. We t define G (ϑ):= ϑ dX for any t ∈ [0,T] and any ϑ ∈ Θ, and G := t s s T 0 {G (ϑ)|ϑ ∈Θ}.NotethatΘ andG isassumedtobelinearspaces. T T Consideranincompletefinancialmarketwhichconsistsofonerisklessasset andd riskyassetswhosefluctuationisdescribedbythesemimartingale X.We regardthefixedT >0asthematurityofourmarket.Supposethattheinterest rate of our market is given by 0, namely, the price of the riskless asset is 1 at all times. Furthermore, we consider the set Θ of predictable processes as thecollectionofalladmissiblestrategies.Thus,wecalleachelementofΘ an admissiblestrategy.Let H beacontingentclaimwhichisakindofpay-offat T. Mathematically, H is an F -measurable random variable. We assume an T investorwhointendstohedgethecontingentclaim H withasuitablestrategy which belongs to Θ. Suppose that the initial endowment of the investor is 0, and the investor attempts to construct her portfolio to approach, in some rational sense, the contingent claim as much as possible at the maturity. The mean-variance hedging is well-known as one of strong candidates for such optimalhedgingstrategies.However,itdependsonlyonthesizeofthehedging error,whichisthedifferencebetweenthevalueofthecontingentclaimandthe portfoliovalueatthematurity.Ingeneral,investorsareinterestedwhethertheir hedgingerrorispositiveornegative.Hence,itisimportanttowidenthewidth ofproblemswhichwecantreat. Throughoutthispaper,weshallmake,inthelightoftheabovematters,a newattackonthefollowingminimizationproblem: inf E[f(H −G (ϑ))], (1) T ϑ∈Θ where f :R →R+ isadifferentiablestrictlyconvexfunctionwith f(0)=0, R+ =[0,∞),andΘisdefinedsothat f(H−GT(ϑ))maybeintegrableforany ϑ ∈Θ.Thecaseof f(x)= x2 and=|x|p for1< p <∞arecorresponding to the mean-variance hedging and the p-optimal hedging undertaken by Arai [1],respectively.Indeed,optimalhedgingstrategiesdependingonlyonthesize of the hedging error are corresponding to the case where f is symmetric, in whichproblem(1)becomeanormminimizationproblemwhichisinaneasy tohandlemathematically.Ontheotherhand,inordertoreflectthesignofthe hedging error, we have to treat asymmetric functions. Hence, our aim in this Hedgingonasymmetricfunctions 3 paperistoextendthemean-variancehedgingorthe p-optimalhedgingtothe asymmetriccase. Remark1. Wecanrewrite(1)asfollows: inf E[f(H −x)], (2) x∈GT since the operator G (·) : Θ → G is an injection under the no-arbitrage T T condition.Throughoutthispaper,weregard(2)astheprimalproblem. Letussketchouttheproblem(2).LetxH ∈ G befixed.Weassumethat T E[f(cid:5)(H −xH)x] = 0 for any x ∈ G . The convexity of f implies that, for T anyx ∈G , T E[f(H −x)] ≥ E[f(H −xH)+ f(cid:5)(H −xH)(H −x −(H −xH))] = E[f(H −xH)]+E[f(cid:5)(H −xH)(xH −x)] = E[f(H −xH)]. Thefollowingtheoremisbasedontheabovefact. Theorem1. Suppose that there exists an xH ∈ G satisfying E[f(cid:5)(H − T xH)x]=0foranyx ∈G .Then,xH istheuniquesolutionto(2). T Proof. We have only to prove the uniqueness. Suppose that there exist two solutions x and x .Remarkthat E[f(H −x )] = E[f(H −x )].Denoting 0 1 0 1 xα :=αx1+(1−α)x0foranyα ∈(0,1),H−xα =α(H−x1)+(1−α)(H−x0). Since f isconvex,wehave f(H −xα)≤αf(H −x1)+(1−α)f(H −x0). Now,weset Aα :={f(H −xα)<αf(H −x1)+(1−α)f(H −x0)},which satisfies P(Aα) > 0, since x0 (cid:8)= x1 and the strict convexity of f. Then, we obtainthat E[f(H −xα)]= E[f(H −xα)1Aα + f(H −xα)1Acα] < E[αf(H −x )+(1−α)f(H −x )] 1 0 = E[f(H −x )], 1 whichiscontradiction.Asaresult,thesolutionexistsuniquely. Next,weconsideradualproblemundertheassumptionofTheorem1.We definetheconvexdual (cid:2)f of f as (cid:2)f(y):=supx∈R[xy−f(x)],andtheorthogonal complementofG as T G⊥ :={y|(H −xH)y− (cid:2)f(y)isintegrableand E[xy]=0foranyx ∈G }, T wherexH istheuniquesolutionto(2).Then,wehavethefollowing: 4 T.Arai Theorem2. Underthesameassumptionastheprevioustheorem,wehave inf E[f(H −x)]= sup E[Hy− (cid:2)f(y)]. (3) x∈GT y∈G⊥ Proof. Letting I(y) := (f(cid:5))−1(y),wehave (cid:2)f(y) = yI(y)− f(I(y))forany yE[∈f(RH.S−inxceH)(cid:2)f](≥y)E≥[(xHy−−xfH(x))yf−or(cid:2)afn(yy)x],,wyh∈erRe,wienftxa∈kGeTaEra[nfd(oHm−vaxri)a]b=le y so that the right hand side should be integrable. Thus, for any y ∈ G⊥, infx∈GT E[f(H−x)]≥supy∈G⊥ E[(H−xH)y− (cid:2)f(y)]=supy∈G⊥ E[Hy− (cid:2)f(y)]. Weprovethereverseinequality.TheassumptioninTheorem1guarantees that f(cid:5)(H −xH)∈G⊥.Thus,wehave inf E[f(H −x)] = E[f(H −xH)] x∈GT = E[(H −xH)f(cid:5)(H −xH)− (cid:2)f(f(cid:5)(H −xH))] ≤ sup E[(H −xH)y− (cid:2)f(y)]= sup E[Hy− (cid:2)f(y)]. y∈G⊥ y∈G⊥ Consequently,Theorem2follows. NotethattheseresultsareobtainedundertheassumptioninTheorem1.In general, it is very difficult to check whether a concrete model given satisfies the assumption or not. Thus, we shall focus on a sufficient condition for the assumption. In order to achieve this goal, it might be important how we set theunderlyingmarketanddefinethesetΘ.TheclosednessofG mightbea T significantkeyword. InSect.2,wedefineadmissiblestrategiesandconfirmthatthespaceofall their stochastic integrals is closed. Moreover, under the setting introduced in Sect.2,weproveinSect.3theuniqueexistenceofasolutionxH totheproblem (2)undertheconditionwhichtherearetwopositiveconstantsc andC such 0 1 that,foranyx ∈Rwhoseabsolutevalueissufficientlarge, c |x|p ≤ f(x), and |f(cid:5)(x)|≤C |x|p−1. 0 1 In addition, we mention that E[f(cid:5)(H −xH)x] = 0 for any x ∈ G . For all T unexplained notation, we refer to Dellacherie and Meyer [4] and Cˇerný and Kallsen[3]. 2. Setup In this section, we address our standing assumptions and define admissible strategies.Throughoutthissection,let1< p <∞befixedarbitrarily.