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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、notgoingtoattempttodefineafullarithmeticonR∗,butnaturallyweset+∞>x>−∞foranyrealnumberx.Thenotation[0,∞)meansthesetofnon-negativereals,while[0,∞]=[0,∞)∪{+∞}⊆R∗.Definition1.1.LetFbeaσ-fieldofsubsetsofΩ.Afunctionf:Ω→R∗isF-measurableifallthesets{ω∈Ω:f(ω)≤a},fora∈R∗,areinF.Roughlyspeaking,afunctionfisF-
5、measurableiftheσ-fieldFis“richenough”tocontainalltheimportantinformationaboutthevalueoff.Example1.IfF=P(Ω),everyfunctionfromΩtoR∗isF-measurable.Example2.IfF={∅,Ω},onlytheconstantfunctionsareF-measurable.(Checkthisasanexercise.)Example3.SupposeΩ=[0,1),andF=B,thefamilyofBorelsets.Considerfirstthefun
6、ctionf:Ω→Rdefinedbyf(ω)=ω.Then{ω∈Ω:f(ω)≤a}=[0,a],for0≤a<1,andtheclosedinterval[0,a]isaBorelset.Ifa<0,then{ω∈Ω:f(ω)≤a}=∅,whereasifa≥1,then{ω∈Ω:f(ω)≤a}=[0,1),inbothcasesBorelsets.ThereforefisB-measurable.1Moregenerally,letf:[0,1)→Rbeanycontinuousfunctionand,foranyfixedrealnumbera,considerSa={ω∈Ω:f(ω
7、)>a}.Supposeω0∈Sa,sothatε=f(ω0)−a>0.Sincefiscontinuous,thereissomeδ>0suchthat,wheneverω∈Ωand
8、ω−ω0
9、<δ,wehave
10、f(ω)−f(ω0)
11、<ε,whichimpliesf(ω)>a.Thismeansthattheinterval(ω0−δ,ω0+δ)∩ΩiscontainedinSa.SuchasetS,withthepropertythat,
