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1、MA313ProbabilityforFinanceandEconomics§2RandomVariablesandExpectationsGrahamBrightwellOctober2005References:GrimmettandStirzaker,ProbabilityandRandomProcesses,Chapters2–4,isgoodforaccuratedescription,butdoesnottreattheexpectationingeneral.AtreatmentveryclosetotheonehereisinChapters3and4ofRosenth
2、al’sAFirstLookatRigorousProbabilityTheory.AnothergoodsourceisWilliams,ProbabilitywithMartingales,Chapters3and6.Youmightalsowanttoreadpp28-42ofBinghamandKiesel,Risk-NeutralValuation:PricingandHedgingofFinancialDerivatives.ThereismoreinthenotesthanIplantocoverinthelectures.Thedefaultpositionisthat
3、anythingnotcoveredinlecturesisnotexaminable.1MeasurablefunctionsWearegoingtorestrictourattentionprettymuchentirelytoreal-valuedfunctions,butinafewplacesitwillpaytoallowourfunctionstotakethevalues+∞and?∞.Weneedtotreatthesesymbolswithcaution,andnotjustasordinarynumbers.LetR?bethesetR∪{+∞,?∞}.Weare
4、notgoingtoattempttode?neafullarithmeticonR?,butnaturallyweset+∞>x>?∞foranyrealnumberx.Thenotation[0,∞)meansthesetofnon-negativereals,while[0,∞]=[0,∞)∪{+∞}?R?.De?nition1.1.LetFbeaσ-?eldofsubsetsof?.Afunctionf:?→R?isF-measurableifallthesets{ω∈?:f(ω)≤a},fora∈R?,areinF.Roughlyspeaking,afunctionfisF-
5、measurableiftheσ-?eldFis“richenough”tocontainalltheimportantinformationaboutthevalueoff.Example1.IfF=P(?),everyfunctionfrom?toR?isF-measurable.Example2.IfF={?,?},onlytheconstantfunctionsareF-measurable.(Checkthisasanexercise.)Example3.Suppose?=[0,1),andF=B,thefamilyofBorelsets.Consider?rstthefun
6、ctionf:?→Rde?nedbyf(ω)=ω.Then{ω∈?:f(ω)≤a}=[0,a],for0≤a<1,andtheclosedinterval[0,a]isaBorelset.Ifa<0,then{ω∈?:f(ω)≤a}=?,whereasifa≥1,then{ω∈?:f(ω)≤a}=[0,1),inbothcasesBorelsets.ThereforefisB-measurable.1Moregenerally,letf:[0,1)→Rbeanycontinuousfunctionand,forany?xedrealnumbera,considerSa={ω∈?:f(ω
7、)>a}.Supposeω0∈Sa,sothatε=f(ω0)?a>0.Sincefiscontinuous,thereissomeδ>0suchthat,wheneverω∈?and
8、ω?ω0
9、<δ,wehave
10、f(ω)?f(ω0)
11、<ε,whichimpliesf(ω)>a.Thismeansthattheinterval(ω0?δ,ω0+δ)∩?iscontainedinSa.SuchasetS,withthepropertythat,
