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Actuarial Theory for Dependent Risks Measures, Orders and Models.pdf

Actuarial Theory for Dependent …

上传者: nwguru 2012-12-16 评分 0 0 0 0 0 0 暂无简介 简介 举报

简介:本文档为《Actuarial Theory for Dependent Risks Measures, Orders and Modelspdf》,可适用于财会税务领域,主题内容包含ActuarialTheoryforDependentRisksMeasures,OrdersandModelsMDenuitUniversiteC符等。

ActuarialTheoryforDependentRisksMeasures,OrdersandModelsMDenuitUniversiteCatholiquedeLouvain,BelgiumJDhaeneKatholiekeUniversiteitLeuven,BelgiumandUniversiteitvanAmsterdam,TheNetherlandsMGoovaertsKatholiekeUniversiteitLeuven,BelgiumandUniversiteitvanAmsterdam,TheNetherlandsRKaasUniversiteitvanAmsterdam,TheNetherlandsCopyrightJohnWileySonsLtd,TheAtrium,SouthernGate,Chichester,WestSussexPOSQ,EnglandTelephone()Email(forordersandcustomerserviceenquiries):csbookswileycoukVisitourHomePageonwwwwileycomAllRightsReservedNopartofthispublicationmaybereproduced,storedinaretrievalsystemortransmittedinanyformorbyanymeans,electronic,mechanical,photocopying,recording,scanningorotherwise,exceptunderthetermsoftheCopyright,DesignsandPatentsActorunderthetermsofalicenceissuedbytheCopyrightLicensingAgencyLtd,TottenhamCourtRoad,LondonWTLP,UK,withoutthepermissioninwritingofthePublisherRequeststothePublishershouldbeaddressedtothePermissionsDepartment,JohnWileyandSonsLtd,TheAtrium,SouthernGate,Chichester,WestSussexPOSQ,England,oremailedtopermreqwileycouk,orfaxedto()DesignationsusedbycompaniestodistinguishtheirproductsareoftenclaimedastrademarksAllbrandnamesandproductnamesusedinthisbookaretradenames,servicemarks,trademarksorregisteredtrademarksoftheirrespectiveownersThePublisherisnotassociatedwithanyproductorvendormentionedinthisbookThispublicationisdesignedtoprovideaccurateandauthoritativeinformationinregardtothesubjectmattercoveredItissoldontheunderstandingthatthePublisherisnotengagedinrenderingprofessionalservicesIfprofessionaladviceorotherexpertassistanceisrequired,theservicesofacompetentprofessionalshouldbesoughtOtherWileyEditorialOfficesJohnWileySons,Inc,RiverStreet,Hoboken,NJ,USAJosseyBass,MarketStreet,SanFrancisco,CA,USAWileyVCHVerlagGmbH,Boschstr,DWeinheim,GermanyJohnWileySonsAustraliaLtd,McDougallStreet,Milton,Queensland,AustraliaJohnWileySons(Asia)PteLtd,ClementiLoop#,JinXingDistripark,SingaporeJohnWileySonsCanadaLtd,WorcesterRoad,Etobicoke,Ontario,CanadaMWLWileyalsopublishesitsbooksinavarietyofelectronicformatsSomecontentthatappearsinprintmaynotbeavailableinelectronicbooksLibraryofCongressCataloginginPublicationDataActuarialtheoryfordependentrisksMDenuit…etalpcmIncludesbibliographicalreferencesandindexISBNX(acidfreepaper)Risk(Insurance)MathematicalmodelsIDenuit,M(Michel)HGA′′dcBritishLibraryCataloguinginPublicationDataAcataloguerecordforthisbookisavailablefromtheBritishLibraryISBN(HB)ISBNX(HB)TypesetinptTimesbyIntegraSoftwareServicesPvtLtd,Pondicherry,IndiaPrintedandboundinGreatBritainbyAntonyRoweLtd,Chippenham,WiltshireThisbookisprintedonacidfreepaperresponsiblymanufacturedfromsustainableforestryinwhichatleasttwotreesareplantedforeachoneusedforpaperproductionContentsForewordxiiiPrefacexvPARTITHECONCEPTOFRISKModellingRisksIntroductionTheProbabilisticDescriptionofRisksProbabilityspaceExperimentanduniverseRandomeventsSigmaalgebraProbabilitymeasureIndependenceforEventsandConditionalProbabilitiesIndependenteventsConditionalprobabilityRandomVariablesandRandomVectorsRandomvariablesRandomvectorsRisksandlossesDistributionFunctionsUnivariatedistributionfunctionsMultivariatedistributionfunctionsTailfunctionsSupportDiscreterandomvariablesContinuousrandomvariablesGeneralrandomvariablesQuantilefunctionsIndependenceforrandomvariablesMathematicalExpectationConstructionRiemann–StieltjesintegralviCONTENTSLawoflargenumbersAlternativerepresentationsforthemathematicalexpectationinthecontinuouscaseAlternativerepresentationsforthemathematicalexpectationinthediscretecaseStochasticTaylorexpansionVarianceandcovarianceTransformsStoplosstransformHazardrateMeanexcessfunctionStationaryrenewaldistributionLaplacetransformMomentgeneratingfunctionConditionalDistributionsConditionaldensitiesConditionalindependenceConditionalvarianceandcovarianceThemultivariatenormaldistributionThefamilyoftheellipticaldistributionsComonotonicityDefinitionComonotonicityandFréchetupperboundMutualExclusivityDefinitionFréchetlowerboundExistenceofFréchetlowerboundsinFréchetspacesFréchetlowerboundsandmaximaMutualexclusivityandFréchetlowerboundExercisesMeasuringRiskIntroductionRiskMeasuresDefinitionPremiumcalculationprinciplesDesirablepropertiesCoherentriskmeasuresCoherentandscenariobasedriskmeasuresEconomiccapitalExpectedriskadjustedcapitalValueatRiskDefinitionPropertiesVaRbasedeconomiccapitalVaRandthecapitalassetpricingmodelCONTENTSviiTailValueatRiskDefinitionSomerelatedriskmeasuresPropertiesTVaRbasedeconomiccapitalRiskMeasuresBasedonExpectedUtilityTheoryBriefintroductiontoexpectedutilitytheoryZeroUtilityPremiumsEsscherriskmeasureRiskMeasuresBasedonDistortedExpectationTheoryBriefintroductiontodistortedexpectationtheoryWangriskmeasuresSomeparticularcasesofWangriskmeasuresExercisesAppendix:ConvexityandConcavityDefinitionEquivalentconditionsPropertiesConvexsequencesLogconvexfunctionsComparingRisksIntroductionStochasticOrderRelationsPartialordersamongdistributionfunctionsDesirablepropertiesforstochasticorderingsIntegralstochasticorderingsStochasticDominanceStochasticdominanceandriskmeasuresStochasticdominanceandchoiceunderriskComparingclaimfrequenciesSomepropertiesofstochasticdominanceStochasticdominanceandnotionsofageingStochasticincreasingnessOrderingmixturesOrderingcompoundsumsSufficientconditionsConditionalstochasticdominanceI:HazardrateorderConditionalstochasticdominanceII:LikelihoodratioorderComparingshortfallswithstochasticdominance:DispersiveorderMixedstochasticdominance:LaplacetransformorderMultivariateextensionsConvexandStopLossOrdersConvexandstoplossordersandstoplosspremiumsConvexandstoplossordersandchoiceunderriskComparingclaimfrequenciesviiiCONTENTSSomecharacterizationsforconvexandstoplossordersSomepropertiesoftheconvexandstoplossordersConvexorderingandnotionsofageingStochastic(increasing)convexityOrderingmixturesOrderingcompoundsumsRiskreshapingcontractsandLorenzorderMajorizationConditionalstoplossorder:MeanexcessorderComparingshortfallwiththestoplossorder:RightspreadorderMultivariateextensionsExercisesPARTIIDEPENDENCEBETWEENRISKSModellingDependenceIntroductionSklar’sRepresentationTheoremCopulasSklar’stheoremforcontinuousmarginalsConditionaldistributionsderivedfromcopulasProbabilitydensityfunctionsassociatedwithcopulasCopulaswithsingularcomponentsSklar’srepresentationinthegeneralcaseFamiliesofBivariateCopulasClayton’scopulaFrank’scopulaThenormalcopulaTheStudentcopulaBuildingmultivariatedistributionswithgivenmarginalsfromcopulasPropertiesofCopulasSurvivalcopulasDualandcocopulasFunctionalinvarianceTaildependenceTheArchimedeanFamilyofCopulasDefinitionFrailtymodelsProbabilitydensityfunctionassociatedwithArchimedeancopulasPropertiesofArchimedeancopulasSimulationfromGivenMarginalsandCopulaGeneralmethodExploitingSklar’sdecompositionSimulationfromArchimedeancopulasCONTENTSixMultivariateCopulasDefinitionSklar’srepresentationtheoremFunctionalinvarianceExamplesofmultivariatecopulasMultivariateArchimedeancopulasLoss–AlaeModellingwithArchimedeanCopulas:ACaseStudyLossesandtheirassociatedALAEsPresentationoftheISOdatasetFittingparametriccopulamodelstodataSelectingthegeneratorforArchimedeancopulamodelsApplicationtoloss–ALAEmodellingExercisesMeasuringDependenceIntroductionConcordanceMeasuresDefinitionPearson’scorrelationcoefficientKendall’srankcorrelationcoefficientSpearman’srankcorrelationcoefficientRelationshipsbetweenKendall’sandSpearman’srankcorrelationcoefficientsOtherdependencemeasuresConstraintsonconcordancemeasuresinbivariatediscretedataDependenceStructuresPositivedependencenotionsPositivequadrantdependenceConditionalincreasingnessinsequenceMultivariatetotalpositivityoforderExercisesComparingDependenceIntroductionComparingDependenceintheBivariateCaseUsingtheCorrelationOrderDefinitionRelationshipwithorthantordersRelationshipwithpositivequadrantdependenceCharacterizationsintermsofsupermodularfunctionsExtremalelementsRelationshipwithconvexandstoplossordersCorrelationorderandcopulasCorrelationorderandcorrelationcoefficientsOrderingArchimedeancopulasOrderingcompoundsumsCorrelationorderanddiversificationbenefitxCONTENTSComparingDependenceintheMultivariateCaseUsingtheSupermodularOrderDefinitionSmoothsupermodularfunctionsRestrictiontodistributionswithidenticalmarginalsAcompanionorder:ThesymmetricsupermodularorderRelationshipsbetweensupermodulartypeordersSupermodularorderanddependencemeasuresExtremaldependencestructuresinthesupermodularsenseSupermodular,stoplossandconvexordersOrderingcompoundsumsOrderingrandomvectorswithcommonvaluesStochasticanalysisofduplicatesinlifeinsuranceportfoliosPositiveOrthantDependenceOrderDefinitionPositiveorthantdependenceorderandcorrelationcoefficientsExercisesPARTIIIAPPLICATIONSTOINSURANCEMATHEMATICSDependenceinCredibilityModelsBasedonGeneralizedLinearModelsIntroductionPoissonCredibilityModelsforClaimFrequenciesPoissonstaticcredibilitymodelPoissondynamiccredibilitymodelsAssociationDependencebymixtureandcommonmixturemodelsDependenceinthePoissonstaticcredibilitymodelDependenceinthePoissondynamiccredibilitymodelsMoreResultsfortheStaticCredibilityModelGeneralizedlinearmodelsandgeneralizedadditivemodelsSomeexamplesofinteresttoactuariesCredibilitytheoryandgeneralizedlinearmixedmodelsExhaustivesummaryofpastclaimsAposterioridistributionoftherandomeffectsPredictivedistributionsLinearcredibilitypremiumMoreResultsfortheDynamicCredibilityModelsDynamiccredibilitymodelsandgeneralizedlinearmixedmodelsDependenceinGLMMbasedcredibilitymodelsAposterioridistributionoftherandomeffectsSupermodularcomparisonsPredictivedistributionsOntheDependenceInducedbyBonus–MalusScalesExperienceratinginmotorinsuranceMarkovmodelsforbonus–malussystemscalesPositivedependenceinbonus–malusscalesCONTENTSxiCredibilityTheoryandTimeSeriesforNonNormalDataTheclassicalactuarialpointofviewTimeseriesmodelsbuiltfromcopulasMarkovmodelsforrandomeffectsDependenceinducedbyautoregressivecopulamodelsindynamicfrequencycredibilitymodelsExercisesStochasticBoundsonFunctionsofDependentRisksIntroductionComparingRisksWithFixedDependenceStructureTheproblemOrderingrandomvectorswithfixeddependencestructurewithstochasticdominanceOrderingrandomvectorswithfixeddependencestructurewithconvexorderStopLossBoundsonFunctionsofDependentRisksKnownmarginalsUnknownmarginalsStochasticBoundsonFunctionsofDependentRisksStochasticboundsonthesumoftworisksStochasticboundsonthesumofseveralrisksImprovementoftheboundsonsumsofrisksunderpositivedependenceStochasticboundsonfunctionsoftworisksImprovementsoftheboundsonfunctionsofrisksunderpositivequadrantdependenceStochasticboundsonfunctionsofseveralrisksImprovementoftheboundsonfunctionsofrisksunderpositiveorthantdependenceThecaseofpartiallyspecifiedmarginalsSomeFinancialApplicationsStochasticboundsonpresentvaluesStochasticannuitiesLifeinsuranceExercisesIntegralOrderingsandProbabilityMetricsIntroductionIntegralStochasticOrderingsDefinitionPropertiesIntegralProbabilityMetricsProbabilitymetricsSimpleprobabilitymetricsIntegralprobabilitymetricsxiiCONTENTSIdealmetricsMinimalmetricIntegralordersandmetricsTotalVariationDistanceDefinitionTotalvariationdistanceandintegralmetricsComonotonicityandtotalvariationdistanceMaximalcouplingandtotalvariationdistanceKolmogorovDistanceDefinitionStochasticdominance,KolmogorovandtotalvariationdistancesKolmogorovdistanceundersinglecrossingconditionforprobabilitydensityfunctionsWassersteinDistanceDefinitionPropertiesComonotonicityandWassersteindistanceStopLossDistanceDefinitionStoplossorder,stoplossandWassersteindistancesComputationofthestoplossdistanceunderstochasticdominanceordangerousnessorderIntegratedStopLossDistanceDefinitionPropertiesIntegratedstoplossdistanceandpositivequadrantdependenceIntegratedstoplossdistanceandcumulativedependenceDistanceBetweentheIndividualandCollectiveModelsinRiskTheoryIndividualmodelCollectivemodelDistancebetweencompoundsumsDistancebetweentheindividualandcollectivemodelsQuasihomogeneousportfoliosCorrelatedrisksintheindividualmodelCompoundPoissonApproximationforaPortfolioofDependentRisksPoissonapproximationDependenceinthequasihomogeneousindividualmodelExercisesReferencesIndexForewordDependenceisbeginningtoplayanincreasinglyimportantroleintheworldofrisk,withitsstrongembedmentinareaslikeinsurance,financialactivities,safetyengineering,etcWhileindependencecanbedefinedinonlyoneway,dependencecanbeformulatedinanunlimitednumberofwaysTherefore,theassumptionofindependenceprevailsasitmakesthetechnicaltreatmenteasyandtransparentNevertheless,inapplicationsdependenceistherule,independencetheexceptionDependencequicklyleadstoanintricateandafarlessconvenientdevelopmentTheauthorshaveacceptedthechallengetooffertheirreadershipasurveyoftherapidlyexpandingtopicofdependenceinrisktheoryTheyhavebroughttogetherthemostsignificantresultsondependenceavailableuptonowThebreadthofcoverageprovidesanalmostfullscalepictureoftheimpactofdependenceinrisktheory,inparticularinactuarialscienceNevertheless,thetreatmentisnotencyclopaedicIntheirtreatmentofrisk,theemphasisismoreontheideasthanonthemathematicaldevelopment,moreonconcretecasesthanonthemostgeneralsituation,moreonactuarialapplicationsthanonabstracttheoreticalconstructionsThefirstthreechaptersprovideindepthexplorationsofrisk:afterdealingwiththeconceptofrisk,itsmeasurementiscoveredviaaplethoraofdifferentriskmeasuresitsrelativepositionwithrespecttootherrisksisthentreatedusingdifferentformsofstochasticorderingsThenextthreechaptersgiveasimilartreatmentofdependenceassuch:modellingofdependenceisfollowedbyitsmeasurementanditsrelativepositionwithinotherdependenceconceptsWhileillustrationscomemainlyfromtheactuarialworld,thesefirsttwopartsofthebookhavemuchbroaderapplicabilitytheymakethebookalsousefulforotherareasofriskanalysislikereliabilityandengineeringThelastthreechaptersshowastrongerfocusonapplicationstoinsurance:credibilitytheoryisfollowedbyathoroughstudyofboundsfordependentrisksthetextendswithatreatmentofriskcomparisonbyusingintegralorderingsandprobabilitymetricsAnassetofthebookisthatawealthofadditionalmaterialiscoveredinexercisesthataccompanyeachchapterThissuccincttextprovidesathoroughtreatmentofdependencewithinariskcontextanddevelopsacoherenttheoreticalandempiricalframeworkTheauthorsillustratehowthistheorycanbeusedinavarietyofactuarialareasincludingamongothers:valueatrisk,ALAEmodelling,bonusmalusscales,annuities,portfolioconstruction,etcJozefLTeugelsKatholiekeUniversiteitLeuven,BelgiumPrefaceTraditionally,insurancehasbeenbuiltontheassumptionofindependence,andthelawoflargenumbershasgovernedthedeterminatio

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