关于风险聚集凸序的两个充分条件.pdf 8页

'˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn关于风险聚集凸序的两个充分条件朱丹，尹传存曲阜师范大学统计学院，曲阜市　273165摘要：本文定义了高维空间的随机序称为弱相关序。证明了弱相关序可以推出具有相同边缘分布的多元相依风险和的止损序。进一步，讨论了随机序之间的关系及性质。关键词：概率论；相依风险；相关序；弱相关序；止损序，凸序中图分类号：O211TwosufficientconditionsforconvexorderingonriskaggregationZhuDan,YinChuancunSchoolofStatistics,QufuNormalUniversity,Qufu273165Abstract:Inthispaper,wedefinenewstochasticordersinhigherdimensionscalledweakcorrelationorders.Itisshownthatweakcorrelationordersimplystop-lossorderofsumsofmultivariatedependentriskswithsamemarginals.Moreover,somepropertiesandrelationsofstochasticordersarediscussed.Keywords:Probabilitytheory;Dependentrisk;Correlationorder;Weakcorrelationorder;Stop-lossorder;Convexorder0IntroductionCorrelationorderasanimportantstochasticorderrelationwasfirstintroducedbyJoe[1],DhaeneandGoovaerts[2]studiedthatthebivariatecasewithsamemarginals.Afterthat,thebivariatecasehasbeengeneralizedbyLuandZhang[3].Recallthatgiventworandomvectors(X1;


;Xn)and(Y1;


;Yn)withthesamemarginals,(X1;


;Xn)issaidtobelesscorrelatedthan(Y1;


;Yn);writtenas(X1;


;Xn)c(Y1;


;Yn);ifforeverypairofdisjointsubsetsA1andA2off1;2;


;ng;Cov(f(Xi;i2A1);g(Xj;j2A2))Cov(f(Yi;i2A1);g(Yj;j2A2));(0.1)wheneverfandgarenondecreasingfunctionsforwhichthecovariancesexist.ThemainresultofLuandZhang[3]showedthatthecorrelationorderimpliedstop-lossorderforportfoliosFoundations:ResearchFundfortheDoctoralProgramofHigherEducationofChina(20133705110002).AuthorIntroduction:ZhuDan(1988-),female,Ph.DCandidates，majorresearchdirection:RiskManagementandActuarialScience.E-mail:zhudanspring@163.com.Correspondenceauthor:YinChuancun(1963-),male,professor,majorresearchdirection:RiskManagementandActuarialScience.E-mail:ccyin@mail.qfnu.edu.cn-1- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn∑nofmultivariatedependentrisks,i.e.,(X1;X2;


;Xn)c(Y1;Y2;


;Yn)impliesi=1Xisl∑ni=1Yi.Stop-lossorderasaspecialcaseofconvexorderisthemostfrequentlyusedorderrelationforthecomparisonofrisks,writtenasXslY,foranytworandomvariablesXandY,ifandonlyiftheinequalityE[(Xd)+]E[(Yd)+]holdsforallreald;where()+denotesthepositivepartofthereal.Inaddition,XissaidtoprecedeYintheconvexordersense,defineXcxY,ifandonlyifXslYandE[X]=E[Y]:Formoredetailsandothercharacterizationsaboutstop-lossordercanbefoundinDenuitetal.[4],Dhaeneetal.[5],LandsmanandTsanakas[6],ShakedandShanthikumar[7].Rüschendorf[8]introducedanewdependenceorderamongriskscalledtheweaklyconditionalincreasinginsequenceorder,bydefinitiongiventworandomvectorsX=(X1;


;Xn)andY=(Y1;


;Yn)withthesamemarginals,(X1;


;Xn)issaidtobesmallerthan(Y1;


;Yn)intheweaklyconditionalincreasinginsequenceorder,writtenasXwcsY;ifforallt2R,all1in1andf~monotonicallynondecreasing,Cov(I(Xi>t);f~(X(i+1)))Cov(I(Yi>t);f~(Y(i+1)));(0.2)whereX(i+1)=(Xi+1;


;Xn)andY(i+1)=(Yi+1;


;Yn):Itisshowedthatmorepositivedependencewithrespecttothewcsorderingimpliedmoreriskwithrespecttothesupermod-ularordering(XsmY;iffEf(X)Ef(Y);forallsupermodularfunctionsfsuchthattheexpectationsexist)forn-dimensionalrandomvectors,i.e.,XwcsYimpliesXsmY.Notethat,letf(Xi;i2A1)=IfXi>tgandg(Xj;j2A2)=f~(X(i+1))in(0.1),(0.2)isanimme-diateconsequenceof(0.1),andbytheExample1.1ofthispaperthattheweaklyconditionalincreasinginsequenceorderisweakerthancorrelationorder,whilethestop-lossorderstillholdsbyMüller[9].Enlightenedbythis,wearecommittedtofindmoregeneralconditionsforthemultivariatecasewhichcanalsoimplystop-lossorder.Inthisshortnote,wegivetheconceptsofweakcorrelationordersinhigherdimensionsandshowthattheweakcorrelationordersimplystop-lossorderofmultivariatedependentriskswithsamemarginals.Theremainderofthepaperisorganizedasfollows.InSection2,weintroducesomeconceptsofstochasticordersincludingthenewdefinitionsanddiscussthepropertiesandstochasticorderrelations.ThemainresultsofthispaperarepresentedandprovedinSection3.1PreliminariesGivenFrechetspaceR(F1;


;Fn)ofalln-dimensionalrandomvectorsX=(X1;


;Xn),havingF1;


;Fnasmarginaldistributions,whereFi(x):=P(Xix);andthejointdistribu-tionfunctionisFX(x1;


;xn):=P(X1x1;


;Xnxn):ForallX2R(F1;


;Fn);we-2- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnhavethefollowinginequality:nMn(x1;


;xn)FX(x1;


;xn)Wn(x1;


;xn);forall(x1;


;xn)2R;∑nwhereWn(x1;


;xn):=minfF1(x1);


;Fn(xn)gandMn(x1;


;xn):=maxfi=1Fi(xi)n+1;0garecalledFrechetupperboundandFrechetlowerboundofR(F1;


;Fn);respectively.Throughouttheshortnote,itisassumedthatallrandomvariablesarerealrandomvariablesonthisspace.Comparingrandomvariablesistheessenceoftheactuarialprofession,inordertoacquiremoregeneralresults,wegivethenotionofweakcorrelationordersasfollows.Definition1.1.LetrandomvectorsX=(X1;:::;Xn)andY=(Y1;


;Yn)beelementsofR(F1;


;Fn);wesaythatXissmallerthanYinthetypeIweakcorrelationorder,writtenasXwcoIY;ifforallt;s2R,1kn1;theanyoffollowingequivalentconditionsholds:({})({})∑k∑k(i)CovIXi>t;IfXk+1>sgCovIYi>t;IfYk+1>sg;(1.3)i=1i=1({})({})∑k∑k(ii)CovIXit;IfXk+1sgCovIYit;IfYk+1sg;(1.4)i=1i=1whereIisanindicatorfunction.Definition1.2.RandomvectorsX=(X1;


;Xn)andY=(Y1;


;Yn)areelementsofR(F1;


;Fn);wesaythatXissmallerthanYinthetypeIIweakcorrelationorder,writtenasXwcoIIY;ifforallt;s2R,1kn1;theanyoffollowingequivalentconditionsholds:({})({})∑n∑n(i)CovIfXk>tg;IXi>sCovIfYk>tg;IYi>s;(1.5)i=k+1i=k+1({})({})∑n∑n(ii)CovIfXktg;IXisCovIfYktg;IYis:(1.6)i=k+1i=k+1∑nRemark1.1.Obviously,ifweletf~(X(i+1))=Ifj=i+1Xj>sg;then(1.3)isaspecialcaseof(1.2).LetrandomvectorX=(X1;


;Xn)andY=(Y1;


;Yn)beelementsofR(F1;


;Fn)withvaluesinRn,FisthedistributionfunctionandFisthesurvivalfunctionofX,defineXXXY;iffF(x)F(x);foranyx2Rn;XY;iffF(x)F(x);foranyx2Rn;uoXYloXYXsmY;iffEf(X)Ef(Y);forallsupermodularfunctionsfsuchthattheexpectationsexist.Morediscussionaboutsupermodularorder(smorder)andorthantorders(uoorderandlo-3- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnorder),seeShakedandShanthikumar[10],DenuitandMesfioui[11]andKzldemirandPrivault[12].Thefollowingrelationsarewellknownbydefinitionsabove,LuandZhang[3]andMüller[9]:(i)XcY)XwcsY)XsmY)XuoYandXloY;(ii)XcY)XwcsY)XwcoIIY;(iii)XcY)XwcoIY:Specially,thesestochasticordersareequivalentinthebivariatecase.Itiswellknownthatcorrelatedorderandweaklyconditionalincreasinginsequenceordercandeducetheweakcorrelationorders,buttheconverseisnottrueincasen3,thefollowingexampleillustratesthispoint.Example1.1.LetXandYberandomvaluableswithdistributionsasP(X=0)=2,iii3P(X=1)=1,P(Y=0)=2andP(Y=1)=1,i=1;2;3.Inaddition,X=(X;X;X)i3i3i3123andY=(Y1;Y2;Y3)arerandomvectorswiththefollowingjointdistributions:22P(X1=0;X2=0;X3=0)=;P(X1=1;X2=0;X3=0)=;9921P(X1=0;X2=1;X3=0)=;P(X1=0;X2=0;X3=1)=;9911P(X1=1;X2=0;X3=1)=;P(X1=0;X2=1;X3=1)=;99and11P(Y1=1;Y2=0;Y3=0)=;P(Y1=0;Y2=1;Y3=0)=;331P(Y1=0;Y2=0;Y3=1)=:3Forallt;s2R;k=1;2,itiseasytogetthat({})({})∑k∑kCovIYi>t;IfYk+1>sgCovIXi>t;IfXk+1>sg;i=1i=1({})({})∑n∑nCovIfYk>tg;IYi>sCovIfXk>tg;IXi>s:i=k+1i=k+1However,fortwonondecreasingfunctionsf(u1)=u1andg(u2;u3)=I(u21;u31);wehaveCov(f(Y1);g(Y2;Y3))Cov(f(X1);g(X2;X3)):(1.7)ThisisbecauseCov(f(Y1);g(Y2;Y3))=E(f(Y1)
g(Y2;Y3))E(f(Y1))
E(g(Y2;Y3))=E(Y1
I(Y21;Y31))E(Y1)
P(Y21;Y31)1=0P(Y21;Y31))=0:(1.8)3-4- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnInthesameway,wecanget1Cov(f(X1);g(X2;X3))=:(1.9)27ThenweobtainYwcoIX;YcXandYwcoIIX;YcX:Letf1(u1;u2)=I(u11;u21);sof1isnondecreasing,thenwehaveCov(I(X11);f1(X2;X3))=Cov(I(X11);I(X21;X31))=P(X11;X21;X31)P(X11)
P(X21;X31)111=0
=:(1.10)3927Similarly,Cov(I(Y11);f1(Y2;Y3))=Cov(I(Y11);I(Y21;Y31))=0;(1.11)soYwcoIIXcannotdeduceYwcsX.Thenextpropertyisstraightforward,weomittedalltheminordetails.Property1.1.LettworandomvectorsX=(X1;


;Xn)andY=(Y1;


;Yn)beelementsofR(F1;


;Fn),foranyindependentrandomvectorZ=(Z1;


;Zm)whichisindependentofXandY,ifXwcoIY;wehave(X1;


;Xn;Z1;


;Zm)wcoI(Y1;


;Yn;Z1;


;Zm);(1.12)ifXwcoIIY;wecanget(Z1;


;Zm;X1;


;Xn)wcoII(Z1;


;Zm;Y1;


;Yn):(1.13)2MainResultsandProofsInthissection,wewillgivethemainresultsofthispaper.Thatthecorrelationorderimpliesstop-lossorderforportfoliosofmultivariatedependentriskshasbeeninvestigatedbyLuandZhang[3],ZhangandWeng[13].Inthenexttheorems,wewillshowthatthepropertystillholdsinweakcorrelationorders.Theorem2.1.LetX=(X1;


;Xn)andY=(Y1;


;Yn)beelementsofR(F1;


;Fn);ifXwcoIY,then∑n∑nXislYi:i=1i=1ToproveTheorem2.1,weneedthefollowingtwolemmas.Forsimplicity,letrandomvari-ablesmarkedwithasterisksbeindependent,andeveryrandomvariablemarkedwithasteriskhasthesamedistributionwithitsoriginal.-5- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnLemma2.1.Foranyrandomvector(X1;X2)andforalld2R,thefollowingequalityholds∫1E(X1+X2d)+E(X1+X2d)+=Cov(IfX1>xg;IfX2>dxg)dx:1(2.14)Proof.Foranyx1;x22R;wehave∫1(x1+x2d)+=Ifx1>x;x2>dxgdx;1sothatE(X1+X2d)+E(X1+X2d)+∫1=[E(IfX1>x;X2>dxg)E(IfX1>x;X2>dxg)]dx1∫1=[P(X1>x;X2>dx)P(X1>x;X2>dx)]dx1∫1=[E(IfX1>xg;IfX2>dxg)E(IfX1>xg)
E(IfX2>dxg)]dx1∫1=Cov(IfX1>xg;IfX2>dxg)dx:(2.15)1Lemma2.2.(ShakedandShanthikumar[7])IfrandomvariablesX1;Y1satisfythatX1slY1,Z1andZ2havethesamedistributionandareindependentofX1andY1,thenX1+Z1slY1+Z2.ProofofTheorem2.1.Forbivariatecase,byLemma2.1wehaveE(X1+X2d)+E(Y1+Y2d)+∫1=[Cov(IfX1>xg;IfX2>dxg)Cov(IfY1>xg;IfY2>dxg)]dx10:(2.16)Assumethatitistrueforn1,wewillprovethatitisalsotrueforninthefollowing.DefinesymbolsX=X+X+


+X;Y=Y+Y+


+YandX=(n1)12n1(n1)12n1(n1)(X+X+


+X);Y=(Y+Y+


+Y);fromLemma2.1wehave12n1(n1)12n1E(X(n1)+Xnd)+E(X(n1)+Xnd)+∫1=Cov(IfX(n1)>xg;IfXn>dxg)dx1∫1Cov(IfY(n1)>xg;IfYn>dxg)dx1=E(Y(n1)+Ynd)+E(Y(n1)+Ynd)+;(2.17)sothatE(X(n1)+Xnd)+E(Y(n1)+Ynd)+E(X(n1)+Xnd)+E(Y(n1)+Ynd)+:(2.18)-6- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnSinceXandYhavethesamedistribution,byinductionandLemma2:2,weobtainE(X+nn(n1)Xd)E(Y+Yd)0:Hence,wefinishtheproof.n+(n1)n+∑n∑nThenexttheoremshowsthati=1Xisli=1Yistillholdsinwco-IIorder.Theorem2.2.LetX=(X1;:::;Xn)andY=(Y1;


;Yn)beelementsofR(F1;


;Fn),ifXwcoIIY,then∑n∑nXislYi:i=1i=1Proof.TheprooffollowsimmediatelybythesamemethodasTheorem2.1andLemma2:1,Lemma2:2.Remark2.1.Infact,sinceXwcoIIY;itfollowsfromTheorem2.1that∑n∑nXni+1slYni+1:i=1i=1Remark2.2.IfY=(Y1


;Yn)isarandomvectorinR(F1;


;Fn);suchthatXwcoIYorXwcoIIYforallX=(X1;


;Xn)2R(F1;


;Fn);thenYiscomonotonic.Remark2.3.LetliandibetheessentialinfimumandessentialsupremumofarandomvariableXi,X=(X1;


;Xn)isafixedrandomvectorinR(F1;


;Fn)whichsatisfies∑n∑ni=1(1Fi(li))1ori=1Fi(i)1;ifXwcoIYorXwcoIIYforallY=(Y1;


;Yn)2R(F1;


;Fn);thenXismutuallyexclusive.MoredetailsaboutcomonotonicityandmutualexclusivitycanbefoundinDhaeneandDenuit[14],CheungandLo[15,16],MesfiouiandDenuit[17]andPuccettiandWang[18].Corollary2.1.LetrandomvectorsX=(X1;


;Xn)andY=(Y1;


;Yn)beelementsof∑n∑nR(F1;


;Fn);randomvariableZisindependentofi=1Xiandi=1Yi,ifXwcoIYorXwcoIIYholds,then∑n∑nXi+ZslYi+Z:i=1i=1Proof.TheproofcanbeobtainedimmediatelybyLemma2.2,Theorem2.1andTheorem2.2.参考文献（References）[1]Joe,H.,1990.Multivariateconcordance.JournalofMultivariateAnalysis,35,12-30.[2]Dhaene,J.,Goovaerts,M.J.,1996.Dependencyofrisksandstop-lossorder.ASTINBul-letin26,201-212.-7- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn[3]Lu,T.Y.,Zhang,Y.,2004.Generalizedcorrelationorderandstop-lossorder.Insurance:MathematicsandEconomics35,69-76.[4]Denuit,M.,Dhaene,J.,Goovaerts,M.,Kaas,R.,2005.ActuarialTheoryforDependentRisks:Measures,OrdersandModels.JohnWileyandSons,Chichester.[5]Dhaene,J.,Vanduffel,S.,Tang,Q.,Goovaerts,M.,Kaas,R.,Vyncke,D.,2006.Riskmeasuresandcomonotonicity:areview.StochasticModels22,573-606.[6]Landsman,Z.,Tsanakas,A.,2006.Stochasticorderingofbivariateellipticaldistributions.StatisticsandProbabilityLetters76(5),488-494.[7]Shaked,M.,Shanthikumar,J.G.,2007.StochasticOrders.Springer,NewYork.[8]Rüschendorf,L.,2004.Comparisonofmultivariaterisksandpositivedependence.JournalofAppliedProbability41,391-406.[9]Müller,A.,1997.Stop-lossorderforportfoliosofdependentrisks.Insurance:MathematicsandEconomics21,219-223.[10]Shaked,M.,Shanthikumar,J.G.,1994.StochasticOrdersandtheirApplications.AcademicPress,London.[11]Denuit,M.,Mesfioui,M.,2013.Asufficientconditionofcrossingtypeforthebivariateorthantconvexorder.StatisticsandProbabilityLetters83(83),157-162.[12]Kzldemir,B.,Privault,N.,2015.SupermodularorderingofPoissonarrays.StatisticsandProbabilityLetters98,136-143.[13]Zhang,Y.,Weng,C.,2006.Onthecorrelationorder.StatisticsandProbabilityLetters76,1410-1416.[14]Dhaene,J.,Denuit,M.,1999.Thesafestdependencystructureamongrisks.Insurance:MathematicsandEconomics25,11-21.[15]Cheung,K.C.,Lo,A.,2013.Characterizationsofcounter-monotonicityanduppercomono-tonicityby(tail)convexorder.Insurance:MathematicsandEconomics53(2),334-342.[16]Cheung,K.C.,Lo,A.,2014.Characterizingmutualexclusivityasthestrongestnegativemultivariatedependencestructure.Insurance:MathematicsandEconomics55,180-190.[17]Mesfioui,M.,Denuit,M.M.,2015.Comonotonicity,orthantconvexorderandsumsofrandomvariables.StatisticsandProbabilityLetters96,356-364.[18]Puccetti,G.,Wang,R.D.,2015.Extremaldependenceconcepts.StatisticalScience30(4),485-517.-8-'