微观金融学及其数学基础08

87789101111ØØØØØØØØØØ——3928.18.1.1201frequencyNnNnP=8-18-18-14,0402,0480.506812,0006,0180.501624,00012,0120.50052classical1measuretheory839323stochasticexperiment161616123outcomesamplepointwOsamplespaceeventbasiceventOsureevent∅impossibleevent811A)(AP1)(AP021)(=∑AP3AB∅)()()(BPAPBAP+=+)(APA018.1.28-2511933394O}6,5,4,3,2,1{8-2AAABABABABCAA2OjiCCji≠∀∅=,=∞=iiC1UO},2,1,{L=iCiOdecompositionpartition{=1C}{=2C}O}5,3,1{1=C}6,4,2{2=C∅=21CCC1C2OO}1{}2{}3{}4{}5{}6{finer3}4{}5{}6{}1{}2{}3{1}1{}2{}3{}4{}5{}6{∅1O3FF1O2F∈AF∈CA3L,2,1,=∈iCiFF∈∞=iiC1Uss811{=1C1-2}{=2C3-4}{=3C5-6}s88395O},,{321CCC=}{∅}{1C}{2C}{3C},{21CC},{31CC},{32CC64F64NsN2s{O∅}sss{OF,}measurablespaceFmeasurableset4{OF,}F),0[∞}{P=PsF+→RFP:310)(=∅P02)(APF∈∀A,0s03FL,2,1,=iCijiCCji≠∀∅=∩,∑∞=∞==11)()(iiiiCPCPUcountableadditivitymeasure{OFP}measurespace8.1.21OR=BF=L),(baA=abAL-=)(Lebesguemeasure2OR=BF=ffL),(baA=)()()(afbfALf-=-Lebesgue-Stieltjesmeasure53PO]1,0[:→FPPprobabilitymeasureprobability{OFP}probabilityspaceNOFPAAN⊂F∈A0)(=APNnegligibleF∈A0)(=APNAN⊂F∈Ninformationtransferprocess396812F∈A0)(=APAN⊂F∈NFPcomplete{OFP}completeprobabilityspacePˆ{}PFˆ,,ΩP}{Q=QPAQF∈∀⇔AAQAP0)(0)(8-20nullset0equivalentprobabilitymeasurePQ8-38-3{O,F,P}∅OAAF∈A1w.p.1a.s.a.e.{O,F}{})(,RRB-a.s.a.e.a.s.almostsurely{OFP}OwN∈wF∈N0)(=NP01withprobability1w.p.1Pa.e.almosteverywhereQPabsolutelycontinuous08.2.18.2.183971F∈BA,BA⊂)()()(APBPABP-=-F∈A)(1)(APAPC-=2F∈BA,BA⊂)(AP)(BP3F∈BA,P(AB))()()(ABPBPAP-+=4F∈}{nCCCn↑)(lim)(1nnnnCPCP∞→∞==U)()(CPCPn→F∈}{nCCCn↓)(lim)(1nnnnCPCP∞→∞==I)()(CPCPn→5BoolesinequalityF∈}{nC)(1U∞=iiCP∑∞=1)(iiCP8-1KnightF.1921——certaintyriskuncertainty100subjectiveprobabilityorprobabilitybeliefSavage1866stateofworldorstateofnature803988-1eventtree123sSt0t18-18.1.3randomvariable10{OF}},{BRORXOR→)(RB∈A)(1AX-FXFRsFF8399F{OPF,}},,{PBRstatespaceXY),(YXfYX+XY),max(YXF∈AAindicatorfunction⎩⎨⎧∉∈=AAA)(1}{iAFR∈=∑=iniAiiawawc,)(1)(18-3simplediscretecontinuousXX),[21xx{x1Xx2}{x1Xx2}1|{xAk})(2xAXk{x1Xx2}}{2xX}{1xX}{xXxXxFR∈∈xxAXAkk,})(|{F,}{xXprobabilitydistributionfunctionXD=)(xDP{Xx}-∞→x0)(→xD∞→x1)(→xD8-2X1x?x±1)(d)(d)2121(11111xxP?xx?xPD==+-PBillingsley1968400P1?x+1dP1x)(xD?x-1x8-2dP8-21∫∫∞+∞-Ω==1d)(dPxD8-4∫∞-===xssxxPxxxd)(d)(dd)(d)(dDddensityfunction8-4∫∫∞+∞-∞+∞-==1d)()(dxxxdD8-58.1.4almostsurely1withprobability1almosteverywhere11}{≥nnX{OPF,}XF∈N0)(=NPN∉w)}({wnX)(wX1)}()(lim:{==∞→}{nX1X.s.aXXn→XXn→w.p.121}{≥nnX{O,FP}X0e0)|(|lim=-∞→eXXPnn}{nXXXXPn→nXXn03{Xn}n1{OPF,}}{nXL,2,1,=nnDXDDx)()(limxxnnDD=∞→}{nXXXXnD→4{Xn}n1{OPF,}X0)(lim=-∞→rnnXXEnXrmomentX1=r2=rmeansquareconvergence12238-3'rr'rr18-3weakconvergence4028.1.5n{Xx}{Yy}XP{)(}xxXD=YP{)(}yyYD=X{Yx}{Xy{}=Yx,}y)(VXx)(VYyVxyXY},{),(yYxXPyxXY≤≤=D8-610),(),(),(=-∞=-∞=-∞-∞yxDDD21),(=∞∞DyxyxyxXYXY∂∂∂=),(),(2Dd0XYXPxX()(=DXPx()=),(),,∞=∞xyxDX∫∞∞-=),()(dyyxxXYXddX)(xXD)(yYD),(yxXYD)(),(xxXXYDD=∞y∞X{Yx,X{}=∞}x)(),(yyYXYDD=∞8403828.1.3P8.2.11234561/61/61/61/121/121/331612151214613612611)(×+×+×+×+×+×=XE∑==niAii1)(1)(wawc∑==niiiAPE1)()(ac8-7X{OPF,}X}{nX}{nX)(lim)(nnXEXE∞→=)(XEXX1989p41404{}⎪⎩⎪⎨⎧∈-=⎭⎬⎫⎩⎨⎧+∈=nXnnkkXkkXnnnnn)(,12,,1,0,21)(2,2)()(wnX)(wnX)(1w+nX0)()(wwnXX-n21XXnn=∞→lim∑-=⎜⎝⎛=12022)(nnknnnkPkXE(+⎟⎠⎞+21nXnPkX)n8-8XXnk2nkX21+12,,1,0,2-==nnnkkXLXnnX=X)(nXEn)(lim)(nnXEXE∞→=nk2n∑=Δniiixf1)(x)(ifx⎜⎝⎛nkP2⎟⎠⎞+nkX21PXnixΔPFF0X{OPF,}PX8.2.1X{OPF,}XP∞∫Ω)(d)(wwPXO,)d()()(d)()(∈==∫∫ΩΩ=XXX+XX⎩⎨⎧∞=+,0)(0),()(⎩⎨⎧∞--=-,00)(),()(∫∫∫Ω-Ω+Ω-=dddPXPXPX)()()(-+-=XEXEXEF∈A∫∫∫===ΩAAAAPXPXPXXE)d()()(d)()(d)()(1)1(~8-5∫∫∫∞+∞-∞+∞-Ω===d)()(d)(d)()(xxxxxPXXEdDww8-118.2.21linearity1∫∫∫ΩΩΩ+=+ddd)(PYPXPYX1997p19319974062∫∫ΩΩ=ddPXaPaXa)()()(YbEXaEbYaXE+=+3A∅=B∫∫∫+=+BABAPXPXPXddd∑∫∫=∪nBBnnnPXPXdd2monotonicity1XYZ∫APXd∫APYd∫APZd2X0∫APXd03X00d=∫ΩPX0=X4|d|∫APX∫APXd||modulusinequality5f)d(∫ΩPXf∫Ωd)(PXffJensensinequality)]([XEf)]([XfE3absoluteintegrability1∞∫AdPX||∫AXdP2||XYYX4X{OPF,})(xD)(xgX)(xg)(xg)(xD∫∫∞+∞-∞+∞-==d)()()(d)()]([xxxgxxgxgEdD8-1284078.2.3821Levi’smonotoneconvergencetheorem}{nX{OPF,}nX1X∫∫=∞→PXPXnnddlim)lim()(limnnnnXEXE∞→∞→=822FatoulemmaYZ}{nX1n1YX∫∞→PXnndlim∫∞→PXnndlim2n1XnZ∫∞→PXnndlim∫∞→PXnndlim3YXnX↑n1YXnZXXn→∫∫→PXPXndd823Lebesgue’sdominatedconvergencetheoremY||nXYnX1XnXX∫∫ΩΩ→ddPXPXn8.3ABBA)|(BAP∞→nlim∞→nlim711408conditionalprobability8.3.1A)|(BAPBABB)(BPABAB)(ABPAB)|(BAP0)(,)()()|(=BPBPABPBAP8-130)(=BP0)(=ABP)|(BAP)|(BAPA1P(O|B)=12)|(BAPF∈A,03jiAABAPBAPjiiiii≠∀∅==⎟⎟⎠⎞⎜⎜⎝⎛∑∑∞=∞=,,)|(|11)|(.BP{O,F}{OBPF,}1multiplicationruleF∈BA,0)(AP0)(BP8-13)|()()|()()(ABPAPBAPBPABP==nniAi∈∈,F131211|()|()()(AAPAAPAPAPnii==IA2)I11)|(-=niinAAP2totalprobabilityF∈A0)(AP}{iBF∑⊂iiBA∑==1)|(}{)(iiiBAPBPAP84093Bayes’sformulaF∈A0)(AP}{iBF∑=⊂1iiBAiB∑==njjjiiiAPBAPBPBAPABP1)()|()()|()|(8.3.2{O,BPF,}XBPBXconditionalexpectation∫∫ΩΩ==)|d()(d)|(BPXPXBXEBww8-14)|(.BP},{F}}∈AABC0∫BBPX)|d()(wwBA⊂F∈A)()()()()|(BPAPBPABPBAP==8-14∫=BPXBPBXE)d()()(1)|(ww8-15BX∫=BwXBXEBP)()|()(P(dw)=E(X1B)8-16⎩⎨⎧∉∈=BBB)(10)(BP)|(BXE0)(=BP)|(BXEAX1=F∈A)()|1()(ABPBEBPA=8-17410)|1()|(BEBAPA=)|(