随机过程--鞅

II1II1011011101210131014102102110221023103-10311032-103310410411042104310510511052-10531054———————-——-—--—2030Ville(1939)(Levy1934(Doob)19532070Pliska&KrepsII2-(Doob-Meyerdecomposition)(stoppingtime)——(equivalentmartingaletransformation)--(Cameron-Martin-Girsanovtheorem)101martingale(doublestrategy)fairgamenn+1nXnn11)|(−−=nnnXXXE01X()nEX20801011(trend)(submartingale)(supermartingale)41)},,{PFΩ-FP-F∈A0)(=APAN⊂F∈N22)(filtration)+∈ZnnS)(3+∈Znn)(Fonm123+ZII31011……⊆⊆onmFFFnFn--+∈=ZFnn)(F(filter)4(10-1)},,,{FPFΩ(filtered)(stochasticbasis)10-15(informationstructure)(spreadprocess)(recombining)uu[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}{ud4+∈=ZFnn)(F(FamaE.)5Dothan(1990)Rebonato(1998)II4]1[u0aFcFbFbFcFaF1+N31)}},...,,{{},{210nωωω=∅Ω=F2)}}{},...,{},{{21nNωωω=F3)tstFsF0Ω()0N33)0≥nnSnFnSnF(Meyer)nSnF(nF–adapted)610-2R→Ω:X'x0001110-1'x2})({'=uux0})({'=udx0})({'=dux2})({'−=ddx}}{},{},{},{{ddduuduua=F}}{},,{},{{ddduuduuf=F}}{},{},,{{duuddduug=F'xfF})({'udx})({'dux0fFgF},{dduuaF'xaF()})({'udx})({'dux0aFfFaF'xxxxFfxFF='''x610-2332II5bf'xuu(2)dd(-2)uddu(0)'xfF'''x000ii(2,1=i)321})({'''=+=uux121})({'''−=−=udx121})({'''=+−=dux321})({'''−=−−=ddx'''x(path-dependent)'''xaxFF='''1)(),...,1(),0(Nxxx)(nxnFt)(nF)(nF2))(nx)1(−nF(predictableorprevisible)(tradingstrategy))}(),...,1(),0({Nθθθθ=)1(−nF())(niθ)(nF74)nS1012NnSESEnNN=),|()(FPPnNnFP1011+∈ZnnS)({}F,,,PFΩnF-1)+∈∞ZnSEn,)(2)1013++∈=ZnSSEnnnn,)|(1F+∈ZnnS)(F8+∈ZnnS)(nS710.1.28FFSFnaturalfiltrationII6[])|()|(|)(11nnnnnnnnnnSESESSEFFF−=−++nS)(1+nnSEnSnSnF)(nnSEnnS10140)|(=∆nnnSEFnS0(martingaleproperty)nS∆(martingaledifference)(partialsummation)0)|(1=∆∑=knkknSEF101110122')++∈ZnSSEnnnn,)|(1F+∈ZnnS)(2)++∈ZnSSEnnnn,)|(1F+∈ZnnS)(1+∈ZnnS)(+∈−ZnnS)(2+∈ZnnS)(+∈ZnnX)()(nnXS+3+∈ZnnS)((.)f)(nnSfX=1,)(≥+∈λλZnnS1012t),0[∞],0[T(regularize)1R→∞),0[:fregular2)(tX],0[Ts∈)(lim)(tXsXst→=II7],0[.),,(lim),(TtsasXtXts∈∀=↓ωω3),0[)(∞∈ttX(regularrightcontinuous)tF-Ω∈ω),0[∞∈ta)tutXuXtu=→),,(),(limωωb)tssXtXts=−→),,(lim),(ωωXttXttab10-2RCLLright-continuoswithleftlimitscàdlàg9),0[)(∞∈ttX),0[)(∞∈ttYtωΩ∈∈∀=ωωω];,0[),()(TtYXttP2(almostthesame)10≥t1}{==ttYXP(modification)(version)1021]},0[,{=∈∀=TtYXPtt(indistinguishable)11},,,{FPFΩ(usualconditions)1)-FP-F∈A0)(=APAN⊂F∈N9càdlàgcontinuàdroite,limitesàgauche10(stochasticallyequivalence)XY11Elliott&Kopp(1999)p102II82)-0FFP-F∈A0)(=AP0F∈A3)),0[}{∞∈=ttFF0tItuut=FFItuuFtuuF--jointmeasurability],0[TR),0(TB],0[TBorel-)],0[(TB[[[F⊗-12],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)13F14F--PM(progressive-field)-4-Op(optional-field)——F-Op5-Pr(predictable-field)——F-Prtt***⇒RCLL⇒⇒-],0[TBFPMOpPr⊗⊂⊂⊂**nielsen175431013),0[)(∞∈ttS{}F,,,PFΩ1)),0[,)(∞∈∞tSEt2),)|(ttTtSSE=FtT∀tS()tF12],0[T×ΩBA×F∈A)],0[(TBB[[[∈-)],0[(TB[[[F⊗×Pλ-0-13Chung&Doob(1965)14Meyer1966p68II915RCLL161013nnS1+n)/()1(dudp−−=nuSp−1ndS1015−−−−=+duudSduduSSnnn111ud01016nnnnnnSduudduSduudSduduSSSE=−−+−=−−+−−=+)1()1(11)|(1nF-10170,01=+=+SSSnnnεnε10181,0,,1,01=++≥−=qrpqrpqrpnε)]([qpnSn−−1019)())(1()())(1()(]|))(1([]|))(1([1qpnSqpnqpSqpnESqpnSEqpnSEnnnnnnnnn−−=−+−−+=−+−+=−+−+=−+−+εεFF)(2qpnSn−−)]()[()]([222qpqpnqpnSn+−−+−−pqaanS/,=nF-),0[)(∞∈ttW[15Karatzas&Shreve199116Hunt&Kennedy2000p49II101)00=W2))(ωt[Wt→3)t≤sstWW−0st−WW0tFF}0),({tss≤≤W-wtF0tF0-),0[}{∞∈ttWF[W)(WFWσ=t17W0)()|(=−=−∆+∆+tttttttEEWWFWWtWtFtt−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∆tttttXtataXXE=∆∆−=∆+)21exp()21exp(]|[22Ftaate221−W(Wald’smartingale)10-3(zerocouponbond)BNnBBEnNn,)(NnSSEnnN≥−,0)|(F1011()QNnSSeEnnNnNrn=−−,)|()(FQ18Elliot&Kopp1999p125-26II12101431L-boundeduniformlyintegrablesquareintegrableuniformlyintegrabilityHunt+Dothan8421,≥pLp∞)|(|pXEXpLtX∞≥)|(|sup0pttXEpL-pLtX2L-1L-Mt∞|)(|tMEJensents|)(||)|(|)]||(|[|)(|sststtMEMEEMEEME=≥=FF|)(|lim|)(|sup0ttttMEME∞→≥=1L-∞→t**1L-martingaleconvergencetheorem1014DoobM1L-RCLL)(lim)(ωωttMM∞→∞=Revuz&Yor(1991)Fatou∞≤∞|)(|inflim|)(|tMEME11L∞→MMt∞→t0|)(|→−∞MMEt2)|(ttMEMF∞=tM——10151L-C∞→εC∈X∫≥}|{|||εXdPX0tM1L-1,pLp-II13tailbehaviorM1911L∞→MMn21LM∈∞nnMME=∞)|(FM∞Mclosed[kopp105]——∞)(2nME20DoobDoob’smaximalinequality21******()1),(1sup−≤pXEppXEpppt102(stoppingtime)22tamethecontinuumoftimeChung1982231021ttFt)(ωT})(,{t≤ωωTtt100n=Tn100nFnTnn{}F,,,PFΩ}{),0[:∞∪∞→ΩT+∈Rt245431021tttFTT∈≤=≤})(,{}{ωω19kopp83-85martingaleconvergenceWilliamsRogers&Williams202