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1、ARTICLESRISKTHEORYWITHTHEGAMMAPROCESSBYFRANt~OISDUFRESNE,HANSU.GERBERANDELIASS.W.SHIULavalUniversity,UniversityofLausanneandUniversityofManitobaABSTRACTTheaggregateclaimsprocessismodelledbyaprocesswithindependent,stationaryandnonnegativeincrements.SuchaprocessiseithercompoundPoissonorelseapro
2、cesswithaninfinitenumberofclaimsineachtimeinterval,forexampleagammaprocess.Itisshownhowclassicalrisktheory,andinparticularruintheory,canbeadaptedtothismodel.Adetailedanalysisisgivenforthegammaprocess,forwhichtabulatedvaluesoftheprobabilityofruinareprovided.KEYWORDSAggregateclaims;compoundPois
3、sonprocess;gammaprocess;infinitedivisi-bility;risktheory;ruinprobability;simulation;stabledistributions;inverseGaussiandistribution.
4、.INTRODUCTIONInclassicalcollectiverisktheory,theaggregateclaimsprocessisassumedtobecompoundPoisson(PANJERandWILLMOT,1984).Hereweshallexamineamoregeneralmodelfor
5、theaggregateclaimsprocess:processeswithindepen-dent,stationaryandnonnegativeincrements.Suchaprocessiseithercom-poundPoissonorelseaprocesswithaninfinitenumberofclaimsinanytimeinterval.Themostprominentprocesswiththisintriguingpropertyisthegammaprocess.Sincetheprocessunderconsiderationiseitherac
6、ompoundPoissonprocessoralimitofcompoundPoissonprocesses,itspropertiescanbederivedfromthebasicpropertiesofthecompoundPoissonprocess.ThegeneralresultsarederivedinSection2(fortheaggregateclaimsprocess)andSection6(fortheprobabilityofruin).ThegammaprocessisexaminedindetailinSections3,4and5(forthea
7、ggregateclaimsprocess)andSections7and8(fortheprobabilityofruin).ASTINBULLETIN,Vol.21,No.2178FRAN(~OISDUFRESNE,HANSU.GERBERANDELIASS.W.SHIU2.PROCESSESWITHINDEPENDENT,STATIONARYANDNONNEGATIVEINCREMENTSLetQ(x)beanonnegativeandnonincreasingfunctionofx,x>0,withtheproperties:Q(x)~0asx--.andoo(2.1)i
8、Q(x)dx0,letN(t;x)denotethenumberofclaimswithanamountgreat