some classes of multivariate risk measures

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1、SomeclassesofmultivariateriskmeasuresMartaCardinandElisaPaganiAbstract.Inactuarialliteraturethepropertiesofriskmeasuresorinsurancepremiumprin-cipleshavebeenextensivelystudied.Weproposeanewkindofstop-losstransformandarelatedorderinthemultivariatesettingandsomeequivalentconditions.I

2、nourworkthereisacharacterisationofsomeparticularclassesofmultivariateandbivariateriskmeasuresandanewrepresentationresultinamultivariateframework.Keywords:riskmeasures,distortionfunction,concordancemeasures,stochasticorders1IntroductionInactuarialsciencesitisfairlycommontocomparetw

3、orandomvariablesthatarerisksbystochasticorderingsdefinedusinginequalitiesonexpectationsoftherandomvariablestransformedbymeasurablefunctions.Bycharacterisingtheconsideredsetoffunctionssomeparticularstochasticorderingsmaybeobtainedsuchasstochasticdominanceorstop-lossorder.Thesestocha

4、sticorderrelationsofintegralformmaybeextendedtocoveralsothecaseofrandomvectors.Themaincontributionofthispaperconcernstheconstructionofamathematicalframeworkfortherepresentationofsomeclassesofmultivariateriskmeasures;inparticularwestudytheextensiontothemultivariatecaseofdistortedri

5、skmeasuresandweproposeanewkindofvectorriskmeasure.Moreover,weintroducetheproductstop-losstransformofarandomvectortoderiveamultivariateproductstop-lossorder.2MultivariatecaseWeconsideronlynon-negativerandomvectors.Let!bethespaceofthestatesofnature,Fbetheσ-fieldandPbetheprobabilityme

6、asureonF.OurrandomvectoristhefunctionX:!→RnsuchthatX(ω)representsthepayoffobtained+ifstateωoccurs.Wealsospecifysomenotations:FX(x):Rn→[0,1]isthedistributionfunctionofX,SX(x):Rn→[0,1]isitssurvivalortailfunction,and(X(ω)−a)+=max(X(ω)−a,0)componentwise.M.Corazzaetal.(eds.),Mathematic

7、alandStatisticalMethodsforActuarialSciencesandFinance©Springer-VerlagItalia201064M.CardinandE.PaganiAriskmeasure,orapremiumprinciple,isthefunctionalR:X→R˜,whereXisasetofnon-negativerandomvectorsandR˜istheextendedrealline.InwhatfollowswepresentsomedesirablepropertiesPforriskmeasure

8、s,thatareourproposaltogeneraliset