Equivalent probability measures

《Equivalent probability measures》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、Equivalentprobabilitymeasures1EquivalentprobabilitymeasuresDenition1.1.TwoprobabilitiesPandQ,denedonthesamespace(;F),areequivalent,andwewritePQ,if,forallA2FP(A)=0,Q(A)=0:(1)Inotherwords,Ais"impossible"underPifandonlyifAis"impossible"underQ.Obviously(1)isequivalenttoP(A)>0,Q(A)>0;thati

2、sthetwoprobabilitiessharethesame"possible"eventsbuttheycanassignthemadierentprobability.Letusconsideranitemeasurablespace,thatis=f!1;:::;!Ng(andF=P()1).Assumep=P(!)>0fori=1;:::;N.QPifandonlyifiiqi=Q(!i)>0fori=1;:::;N.qiLetusconsidertheratiosdi=pifori=1;:::NanddenetherandomvariableZas

3、Z(!i)=di.ThispositiverandomvariableiscalledtheRadon-NikodymdQderivativeofQwithrespecttoPandisdenotedas.dPdQGivendP,wecanderiveQfromP.Infact,ifwehaveP,thenwehavep1;:::;pNdQandsoqi=dP(!i)pi.Similarly,theRadon-NikodymderivativeofPwithrespecttoQisdened.Itistrivialtoshowthat1dP1dQ(!i)=di

4、=(!i):dQdPLetusnowconsiderarandomvariableXdenedonthegivenspaceanddenotexi=X(!i)fori=1;:::;N.TheexpectedvalueofXwithrespecttoPisgivenbyXNEP[X]=xpiii=11P()denotestheclassofallsubsetsof1andtheexpectationofXwithrespecttoQisXNXNXNQqidQPdQE[X]=xiqi=xipi=xi(!i)pi=EX:pidPdPi=1i=1i=1hiIn

5、particularQ(A)=EP1dQforallA2FandsimilarlyEP[X]=EQXdP:AdPdQAllweshowedisaparticularcaseofamoregeneralresultcalledtheRadon-Nikodymtheorem.Thistheorem,initsmoregeneralversion,dealswithabso-lutelycontinuousmeasures,butexpressedintermsofequivalentprobabilitiesstatesthatTheorem1.2.(Radon-Nikod

6、ym)Let(;F)onwhichPandQaredened.QdQisequivalenttoPifandonlyifthereexistsarandomvariableZ=>0P-dPa.s.suchthat,forallA2F,PdQQ(A)=E1AdPand"1#QdPQdQP(A)=E1A=E1A:dQdPNoticethat,since1=Q()=EP1dQ=EPdQ,necessarilyEPdQ=dPdPdPdQ1,otherwisedenesanitemeasure.dPRemark1.1.IfdQisbounded,t

7、henEP[X]<+1impliesEQ[X]<+1asdPwell.Remark1.2.GivenaprobabilityPandaP-a.spositiverandomvariableY,itispossibletodenetheprobabilityQPhavingtheRadon-NikodymderivativedQ=Y.dPEP[Y]Example1.1.Let=[0;1],FbethesetofallBorelsetsin[0;1]andPtheLebesguemeasure.LetdQ=1e!(thenormalizi