微观经济学 数学基础 第10章 随机过程II

II1II77891011111011011101210131014102102110221023103-10311032-103310410411042104310510511052-10531054()----2030Ville(1939)(Levy1934)(Doob)1953()(HarrisonJM)(KrepsDM)1979(PliskaSR)1981(freelunch)II21()()2(stoppingtime)-(Doob-Meyerdecomposition)104104110421043——(equivalentmartingaletransformation)--(Cameron-Martin-Girsanovtheorem)101martingale(doublestrategy)5241+2+4+8=15516161512667003(fairgame)nn+1nXnn11)|(−−=nnnXXXE012()1143II34X()nEX20801011(trend)(submartingale)(supermartingale)41)},,{PFΩ-FP-F∈A0)(=APAN⊂F∈N52)(filtration)+∈ZnnS)(6+∈Znn)(Fonm1011……⊆⊆onmFFFnFn--+∈=ZFnn)(F(filter)7(10-1)},,,{FPFΩ(filtered)(stochasticbasis)10-18(informationstructure)(spreadprocess)(recombining)456+Z7+∈=ZFnn)(F(FamaE)(generated)8Dothan(1990)Rebonato(1998)II4uu[2]u[1]0du,ud[0]d[-1]dd[-2]t0t1t210-1ud4}}{},{},{},{{ddduuduu=Ω}}{},{},{},{{ddduuduua=F},,,{ddduuduub=F}}{},{},,{{ddduuduuc=F}}{},{},,{},{{ddduuduuuud=F}}{},{},{{duuduue=FaFbFcFΩdFeFdF}{uuef}{ddbF00},{0Ω∅=FaF2cF1},{uduu]1[d}{dd}{ud]1[u0aFcFbFacbFFF1+N31)}},...,,{{},{210nωωω=∅Ω=F2)}}{},...,{},{{21nNωωω=F3)tstFsF0Ω()0N3II53)0≥nnSnFnSnF(Meyer)nSnF(nFadapted)910-2R→Ω:X'x0001110-1'x2})({'=uux0})({'=udx0})({'=dux2})({'−=ddx}}{},{},{},{{ddduuduua=F}}{},,{},{{ddduuduuf=F}}{},{},,{{duuddduug=F'xfF})({'udx})({'dux0fFgF},{dduuaF'xaF()})({'udx})({'dux0aFfFaF'xxx(generating)xFfxFF='''xbf'xuu(2)dd(-2)uddu(0)'xfF'''x000ii(2,1=i)321})({'''=+=uux121})({'''−=−=udx121})({'''=+−=dux321})({'''−=−−=ddx'''x(path-dependent)'''xaxFF='''910-2332II61))(),...,1(),0(Nxxx)(nxnFt)(nF)(nF2))(nx)1(−nF(predictableprevisible)(tradingstrategy))}(),...,1(),0({Nθθθθ=)1(−nF())(niθ)(nF104)nS1012NnSESEnNN=),|()(FPPnNnFP()1011+∈ZnnS)({}F,,,PFΩnF-1)+∈∞ZnSEn,)(2)1013++∈=ZnSSEnnnn,)|(1F+∈ZnnS)((F)11+∈ZnnS)(nS[])|()|(|)(11nnnnnnnnnnSESESSEFFF−=−++nS)(1+nnSEnSnSnF)(nnSEnnS10140)|(=∆nnnSEFnS0(martingaleproperty)nS∆(martingaledifference)(partialsummation)0)|(1=∆∑=knkknSEF10101211FFSF(naturalfiltration)II7101110122')++∈ZnSSEnnnn,)|(1F+∈ZnnS)(2)++∈ZnSSEnnnn,)|(1F+∈ZnnS)(1)+∈ZnnS)(+∈−ZnnS)(2)+∈ZnnS)(+∈ZnnX)(())(nnXS+()3)+∈ZnnS)((.)f)(nnSfX=1,)(≥+∈λλZnnS1012t),0[∞],0[T()(regularize)1)R→∞),0[:f(regular)2))(tX],0[Ts∈)(lim)(tXsXst→=],0[.),,(lim),(TtsasXtXts∈∀=↓ωω3)),0[)(∞∈ttX(regularrightcontinuous)tF-Ω∈ω),0[∞∈ta)tutXuXtu=→),,(),(limωωb)tssXtXts=−→),,(lim),(ωωII8XttXttab10-2RCLL(right-continuoswithleftlimitscàdlàg12)(equivalent)),0[)(∞∈ttX),0[)(∞∈ttYtωΩ∈∈∀=ωωω];,0[),()(TtYXttP2(almostthesame)1)0≥t1}{==ttYXP(modification)(version)132)1]},0[,{=∈∀=TtYXPtt(indistinguishable)14},,,{FPFΩ(usualconditions)1)-FP-F∈A0)(=APAN⊂F∈N2)-0FFP-F∈A0)(=AP0F∈A3)),0[}{∞∈=ttFF0tItuut=FFItuuFtuuF--(jointmeasurability)],0[TR),0(TB],0[TBorel-)],0[(TB[[[F⊗-12càdlàgcontinuàdroitelimitesàgauche13(stochasticallyequivalence)XY14Elliott&Kopp(1999)p102II915],0[)(TttS∈R→×Ω],0[:TS1)],0[)(TttS∈-)],0[(TB[[[F⊗2)],0[)(TttS∈tF],0[)(TttS∈),0[}{∞∈=ttFF3)],0[)(TttS∈],0[Tt∈-]),0([ttB[[[F⊗(progressivelymeasurable)16F17F--PM(progressive-field)()-4)-Op(optional-field)F-Op5)-Pr(predictable-field)F-Pr18⇒RCLL⇒⇒-19],0[TBFPMOpPr⊗⊂⊂⊂1013),0[)(∞∈ttS{}F,,,PFΩ1)),0[,)(∞∈∞tSEt2),)|(ttTtSSE=FtT∀tS()tF20RCLL211013nnS1+n)/()1(dudP−−=15],0[T×ΩBA×F∈A)],0[(TBB[[[∈-)],0[(TB[[[F⊗×Pλ-0-16Chung&Doob(1965)17Meyer(1966)p681819Chung&Williams(1990)p6320Karatzas&Shreve(1991)21Hunt&Kennedy(2000)p49II10nuSP−1ndS1015−−−−=+duudSduduSSnnn111ud01016nnnnnnSduudduSduudSduduSSSE=−−+−=−−+−−=+)1()1(11)|(1nF-10170,01=+=+SSSnnnεnε10181,0,,1,01=++≥−=QRPQRPQRPnε)]([QPnSn−−1019)())(1()())(1()(]|))(1([]|))(1([1QPnSQPnQPSQPnESQPnSEQPnSEnnnnnnnnn−−=−+−−+=−+−+=−+−+=−+−+εεFF)(2QPnSn−−)]()[()]([222QPQPnQPnSn+−−+−−nSPQ)/(nF-),0[)(∞∈ttW[1)00=W2))(ωt[Wt→3)t≤sstWW−0st−WW0tFF}0),({tss≤≤W-wtF0tF0-),0[}{∞∈ttWF[W)(WFWσ=t22W0)()|(=−=−∆+∆+tttttttEEWWFWWII11tWtFtt−2W)()|()|(2]|)[()(]|2)[()()|(]|[2222ttEEEttEttEXEtttttttttttttttttttttttt∆+−−+−=∆+−−+−=∆+−=∆+∆+∆+∆+∆+∆+FWFWWFWWF−∆+tFt∆=∆2WtEEttttttt∆=−=−∆+∆+])[(]|)[(22WWFWW(833)2)|()()|(ttttttttEtEWFWWFWW==∆+∆+22)|(tttEWFW=tttttttXttttXE=−=∆−−−+∆=∆+2222]|[−2Wtt−2−Wa∆−−=∆+−=∆+∆+∆+ttttttttttttaaXEttaaEXEFWWFWF|]21)(exp[|)](21exp[]|[22{})](exp[)21exp(|]21)(exp[]|[22ttttttttttttaEtaXtaaEXXEWWFWWF−∆−=∆−−=∆+∆+∆+)(tttaWW−∆+0ta∆2)](exp[tttaWW−∆+)21exp(2ta∆22(2001)p1823Elliot&Kopp(1999)p125-26II12tttttXtataXXE=∆∆−=∆+)21exp()21exp(]|[22Ftaate221−W(Waldsmartingale)10-3(zerocouponbond)BNnBBEnNn,)(NnSSEnnN≥−,0)|(F1011()QNnSSeEnnNnNrn=−−,)|()(FQ101431L-(bounded)(uniformlyintegrable)(squareintegrable)8441,≥pLp∞)|(|pXEXpL()tX∞≥)|(|sup0pttXEpL-(pL)tX2L-1L-Mt∞|)(|tMEJensents|)(||)|(|)]||(|[|)(|sststtMEMEEMEEME=≥=FF|)(|lim|)(|sup0ttttMEME∞→≥=1L-∞→tII131L-(martingaleconvergencetheorem)1014(Doob)M1L-RCLL)(lim)(ωωttMM∞→∞=asRevuz&Yor(1991)Fatou∞≤∞|)(|inflim|)(|tMEME1)1L∞→MMt∞→t0|)(|→−∞MMEt2))|(ttMEMF∞=∞MMtM(uniformlyintegrability)10151L-C∞→εC∈X∫≥}|{|||εXdPX0tM1L-1,pLp-(tailbehavior)M241)1L∞→MMn2)1LM∈∞nnMME=∞)|(FM∞M(closed)——∞)(2nME25Doob(Doobsmaximalinequality)2624Rogers&Williams(1994)252L()10426DoobspLRogers&Will