; TeX output 2003.10.30:1508 H`HZ2Dt G G cmr17Coherenqt7tandconvexriskmeasuresforbsoundedc9adlag proscesseslύtkXQ cmr12PratrickCheridito2K cmsy8 3K`y cmr10ORFEҭPrincetonUUUniversity/Princeton,UUNJ,USAdito@princeton.edu uFVreddyDelbaen2y iTDepartementUUfGurMathematik ETHUUZGurich :8092UUZGurich,Switzerland Wdelbaen@math.ethz.ch8oMicrhaelKuppSerğ2y "WDepartementUUfGurMathematikI ETHUUZGurich-.=8092UUZGurich,Switzerland4#kuppGer@math.ethz.chP{ SeptemrbSer20034lύ Ƃ+"V cmbx10Abstract@"|IfPtherandomfutureevolutionofvqaluesismoGdelledincontinuoustime,Ϛthena "|riskZmeasurecanbGeviewedasafunctionalonaspaceofcontinuous-timestoGchas-"|ticjproGcesses.pW*eextendthenotionsofcoherentandconvexriskmeasurestothe"|spaceofbGoundedcadlagprocessesthatareadaptedtoagivenltration.PThen,we"|prove representationresultsthatgeneralizeearlierresultsforone-andmulti-pGeriod"|riskUUmeasures,andwediscusssomeexamples.Ǎ"|KeyVfw9ords::Coherentriskmeasures,convexriskmeasures,coherentutilityfunction-"|als,concave}moneybasedutilityfunctionals,cadlag}proGcesses,representationtheorem."|Mathematics8Sub jectClassication(2000):㎲91B30,q60G07,52A07,46A55,46A20"A-N ff cmbx121iIntros3ductionqK`y 3 cmr10The5notionofacoheren!triskmeasurewasintroMducedin[ADEH1%z]and[ADEH2],LVwhereit w!as2RalsoshownthateverycoherentriskmeasureonthespaceofallrandomvdDariablesonaniteprobabilit!yspacecanbMerepresentedasasupremumoflinearfunctionals.{In[De1](seealso[De2]),theconceptofacoheren!triskmeasurewasextendedtogeneralprobabilityspaces,and|applicationstoriskmeasuremen!t,premiumcalculationandcapitalalloMcationproblemsw!erepresented.Itturnedoutthatthedenitionsandresultsof[ADEH1%z]and[ADEH2%z]ha!veadirectanaloginthesettingofageneralprobabilityspaceifonerestrictsriskmeasurestothespaceofbMoundedrandomvdDariables.OnthespaceofallrandomvdDariables,lcoheren!t riskmeasurescaningeneralonlyexistiftheyareallowedtotakethevdDalue/!", 3 cmsy101.7BIn[ADEHK1.;]and[ADEHK2],6Ptheresultsof[ADEH1%z]and[ADEH2]w!ere #2 ff ğ L͍Q -=q% cmsy6a%o cmr9SuppAortedTb9ytheSwissNationalScienceF:oundationandCreditSuisseJ=-=yaSuppAortedTb9yCreditSuisse w1 *H`H÷generalizedtom!ulti-pMeriodmodels.`In[FS1]and[FS2]themoregeneralconceptofa con!vexriskmeasurew!asintroMducedandrepresentationresultsof[De1]weregeneralizedtoJcon!vexriskmeasuresonthespaceofallbMoundedvdDariablesonageneralprobabilityspace.ThepurpMoseofthispaperisthestudyofriskmeasuresthattak!eintoaccountthefutureOev!olutionofvdDaluesoverawholetime-intervdDalratherthanatjustnitelymanytimes.OurfmainfoMcuswillbeonrepresen!tationresultsforsuchriskmeasures.WeemoMdelthefutureev!olutionofadiscountedvdDaluebyastoMchasticproMcess(.b> 3 cmmi10Xz2 cmmi8tʹ):ut2|{Y cmr8[0;T.:],andw!ecallsuchaproMcessXadiscountedvdDalueproMcess.TheuseofdiscountedvdDaluesiscommon&practiceinnance._HItjustmeansthatw!euseanum"Deraire(whichmayalsobMemoMdelledxasastoc!hasticprocess)andmeasureallvdDaluesinm!ultiplesofthenum"Deraire.ΔAfewfoftheman!ypMossibleinterpretationsofadiscountedvdDaluedproMcessare:-ftheev!olutionofthediscountedmarketvdDalueofarm'sequity-ftheev!olutionofthediscountedaccountingvdDalueofarm'sequity-ftheev!olutionofthediscountedmarketvdDalueofapMortfolioofnancialsecurities-ftheev!olutionofthediscountedsurplusofaninsurancecompany(seefalsoSubsection2.1of[ADEHK2.;]).Since&onemigh!twanttobuildnewdiscountedvdDalueproMcessesfromoldonesbyaddingandQscalingthem,bitisnaturaltolettheclassofdiscoun!tedvdDalueproMcessesconsideredforriskmeasuremen!tbMeavectorspace.IInthispapMertheclassofdiscountedvdDalueproMcessesisxKthespaceR1xSofbMoundedcfadlagxKprocessesthatareadaptedtotheltrationofaltereddprobabilit!yspace( ;1F_;(Fztʹ)0tT+;PV)dthatsatisestheusualassumptions.StrictlyspMeaking,b!ytbounded,w!etmeanessentiallybMounded,andweidentifyindistinguishableproMcesses. .Inadoingsow!earereferringtotheprobabilitymeasurePV. .However,вthespace?RR1 ?Zsta!ysinvdDariantifwechangetoanequivdDalentprobabilitymeasure.Hence,ebyin!troMducing=PV,UrweonlyspMecifythesetofeventswithprobabilityzero.aFeormoMdellingpurpMoses,NtheW spaceR0 $ofallcfadlagW processesthatareadaptedtotheltration(Fztʹ)ismorein!terestingthanR1 .`!NotethatR0ֹisalsoinvdDariantunderchangetoanequivdDalentprobabilit!yBmeasure.mrThereasonwhyinthispapMerweworkwithR1Jisthatincontrastto(R0,Aiteasilylendsitselftotheapplicationofdualit!ytheorye,whic!hwillbMecrucialintheproMofSofTheorem3.3,dSthemainresultofthispaper.UInaforthcomingpaperw!ewillstudyrisklmeasuresonR0 ,anddiscussconditionsunderwhic!haconvexriskmeasureonR1can].bMeextendedtoR0.uWhereasw!erequirecoherentandconvexriskmeasuresonR1]6tobMereal-vdDalued,8w!ewillallowcoherentandconvexriskmeasuresonR0totakevdDaluesin( 1;11].AlthoughWw!econsidercontinuous-timediscountedvdDalueproMcesses,theriskmeasurestreatedinthispapMerarestaticasw!eonlymeasuretheriskofadiscountedvdDalueproMcessat+thebMeginningofthetimeperiod.nIn[ADEHK1.;],M [ADEHK2]+and[ES ]onecanndadiscussioncofdynamicriskmeasuresforrandomvdDariablesinadiscrete-timeframew!ork.DynamicSriskmeasuresforstoMc!hasticprocessesinacon!tinuous-timeSsetupisthesub jectoffongoingresearc!h.The]structureofthepapMerisafollo!ws:L{Section2containsnotationanddenitions.InSection3,w!estateresultsontherepresentationofcoherentandconvexriskmeasures w2 )H`H÷for\gbMoundedcfadlag\gprocessesandsk!etch\gthoseproofsthataresimplegeneralizationsof proMofs$in[De1]or[FS1].Weewillalsosho!whowresultsof[De1]and[FS1]canbMeextendedtorepresen!tationresultsforreal-vdDaluedcoherentandconvexriskmeasuresonthespacesRp̰:=S!e0u 3 cmex10 hX2 R0ʫjsup0tT0j3XztʝjFJ52Lp]!e Rιfor