risk measures and stochastic orders using integrals of distorted quantile functions

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1、RiskMeasuresandStochasticOrdersUsingIntegralsofDistortedQuantileFunctionsMiguelMendesandIgnacioCascosAbstractWeconstructnewriskfunctionalsandstochasticorderingsbasedonintegralsusingdiscontinuousdistortionfunctions.Weprovearesultonthesubadditivityoftheriskfunctio

2、nal.Astothestochasticorderwegiveacharacterizationofthisorderwhichallowsustoconstructspectralriskmeasuresthatareconsistentwiththatorder.1IntroductionInthischapterwecallintegralofdistortedquantilefunctiontoallintegralsoftheformtqX(ϕ(s))ds(1)0whereqXisthepseudo-in

3、verse(commonlyknownasquantilefunction)ofthedistributionfunctionFX(x):=P(X≤x)foragivenprobabilitymeasurePonameasurespace(Ω,F)andϕ:[0,1]→[0,1]isanon-decreasingfunctionsatisfyingϕ(0)=0andϕ(1)=1.Thisfunctionisinterpretedasapseudo-inverseofagivendistortionfunctionψ:[

4、0,1]→[0,1]whichgeneratesadistortedprobabilityP∗:=ψ◦P.Thedistortionfunctionwillbeassumedtopossessdiscontinuitiesandtothebestofourknowledgethishadnotbeenconsideredpreviously.M.Mendes()FEUPandCMUP,UniversidadedoPorto,Portugale-mail:mmendes@fc.up.ptI.CascosDepartme

5、ntofStatistics,UniversidadCarlosIIIdeMadrid,Spaine-mail:ignacio.cascos@uc3m.esP.E.Oliveiraetal.(eds.),RecentDevelopmentsinModelingandApplications261inStatistics,StudiesinTheoreticalandAppliedStatistics,DOI10.1007/978-3-642-32419-226,©Springer-VerlagBerlinHeidelb

6、erg2013262M.MendesandI.CascosIntegralsofthisformappearnaturallyinthefollowingclassicalconstruction.LetX1,...,Xnbei.i.d.randomvariables.ItcanbeeasilyseenthattheexpectationofX1:n:=min{X1,...,Xn}canbewrittenas11/nE(X1:n)=qX1−(1−s)ds01/nnwhereϕ(s):=1−(1−s)isthein

7、versefunctionofψ(u):=1−(1−u)andψissuchthatP(X1:n≤x)=ψ(P(X≤x))forallx∈R.Itsrelationwithriskmeasuresisalsostraightforward.IfoneseesXasarandomvariablerepresentingthenetworthafterdiscountingofacertainportfolioandinterpretsX1:nasaworstcasescenarioovernsimulations,the

8、riskofX1:ncalculatedbymeansoftheExpectedShortfallisgivenbythefollowingintegral:t11/nESt(X1:n)=−qX1−(1−s)ds.t0InSect.2wewillmakeuseofintegralsofform(1)inamoregenera