订单流驱动的限价订单簿动态演化

285Vol.28No.520119JOURNALOFSHENZHENUNIVERSITYSCIENCEANDENGINEERINGSep．20111000-2618201105-0454-06【】2011-03-092011-06-01110011791969-、[e-mail]11215180552518060．．Kolmogorov．．O211.9A、、、1．、．2-7．8-10．．1111．E=12…n．p∈EXpttXpt＞0XptXpt＜0－XptXptt≥0Z=…－3－2－10123…．Xt=X1tX2t…XntXtt≥0http//journal．szu．edu．cn5455∑nk=1Xktt∑nk=1Xktt．n+10．tpAt=infpp∈EXpt＞0∧n+1tpBt=suppp∈EXpt＜0∨0．pAt＞pBt．pMtpMt=pAt+pBt/2．pdtpdt=pAt－pBt．2．、．1p．Xpt=iαii．2p．Xpt=iβi．3μ．4．3T0=inftt＞0pMt≠pM0T0t=0．T0．T1T2…Tk=inftt＞Tk－1pMt≠pMTk－1k=12…．0≤t＜T0pAt=pA0pBt=pB0Tk－1≤t＜Tkk=123…pAt=pATk－1pBt=pBTk－1．Tk－1．0≤t＜T0t=T0．．3.10≤t＜T0p=pA0、pA0＜p≤n、p=pB01≤p＜pB0pB0＜p＜pA0pd0=1．3.1.1pA0＜p≤nXpt012…．0≤t＜T0．Xpt0≤t＜T0p．12Xpt=ipαiβii．ppA0＜p≤nXpt0≤t＜T0http//journal．szu．edu．cn45628012…．Kolmogorov．1Xpt0≤t＜T0αii≥0βii＞0．【】hPijhPXpt+h=jXpt=i．i≥0j=i+1Pii+1h=PXpt+h=i+1Xpt=i=PXpt+h－Xpt=1Xpt=i=Pptt+hXpt=i112pXt=iαi．h＞0tt+hαihβihPptt+hXpt=i=αih11e－αih2Pptt+hXpt=i=βih00e－βih3231Pii+1h=PXpt+h=i+1Xpt=i=αih+oh4i＞0j=i－1Pii+1hPii－1h=PXpt+h=i－1Xpt=i=βih+oh5i≥0j－i≥2hPijh=PXpt+h=jXpt=i=oh6i≥0j=iPiih=1－αi+βih+oh74～7Xpt0≤t＜T0Pijh=αih+ohi≥0j=i+1βih+ohi＞0j=i－11－αi+βih+ohi≥0j=iohi≥0j－i≥288Xpt0≤t＜T0αii≥0βi＞0i≥0．qij=limh↓0Pijh－δij/h．qijqij=αii≥0j=i+1βii＞0j=i－1－αi+βii≥0j=i0i≥0j－i≥2Xpt0≤t＜T0Kolmogorov．Pijt=PXpt=jXp0=i．2PijtKolmogorovPijt=αiPi+1jt+βiPi－1jt－αi+βiPijti＞0P0jt=α0P1jt－α0P0jti=0PijtKolmogorovPijt=Pij－1tαj－1+Pij+1tβj+1－Pijtαi+βij＞0Pi0t=Pi1tβ1－Pi0tα0j=03.1.2p=pA0XPAt123…．XPAt0≤t＜T0．1XPAt0≤t＜T0αii≥1βi+μi≥1．hPXPAt+h=jXPAt=i=αih+ohj=i+1βi+μh+ohj=i－11－αi+βi－μh+ohj=iohj－i≥2http//journal．szu．edu．cn5457PAijt=PXPAt=jXPA0=iXPAt0≤t＜T03．3PAijtKolmogorovPAijt=αiPAi+1jt+βi+μPAi－1jt－αi+βi+μPAijti＞0PA0jt=α0PA1jt－α0PA0jti=0．PAijtKolmogorovPAijt=PAij－1tαj－1+PAij+1tβj+1+μ－PAijtαj+βj+μj＞0PAi0t=PAi1tβ1+μ－PAi0tα0j=0．3.1.3p=pB0XPBt…－3－2－1．XPBt0≤t＜T0．XPBtpB0XPBt0≤t＜T0．1XPBt0≤t＜T0αii≥1βi+μi≥1．PXPBt+h=j|XPBt=i=αih+ohj=i+1βi+μh+ohj=i－11－αi+βi－μh+ohj=iohj－i≥2PBijt=PXPBt=j|XPB0=iXPBt0≤t＜T04．4PBijtKolmogorovPBijt=αiPBi+1jt+βi+μPBi－1jt－αi+βi+μPBijti＞0PB0jt=α0PB1jt－α0PB0jti=0PBijtKolmogorovPBijt=PBij－1tαj－1+PBij+1tβj+1+μ－PBijtαj+βj+μj＞0PBi0t=PBi1tβ1+μ－PBi0tα0j=0．3.1.41≤p＜pB0Xpt…－3－2－10．0≤t＜T0．Xpt0≤t＜T0p．XptpXpt0≤t＜T0．1Xpt0≤t＜T0αii≥0βii＞0．hPXPBt+h=j|XPBt=i=αih+ohj=i+1βih+ohj=i－11－αi+βih+ohj=iohj－i≥2Ppijt=PXpt=j|Xp0=iXpt0≤t＜T0pA0＜p≤nXpt0≤t＜T0Kolmogorov2．3.1.5pB0＜p＜pA00≤t＜T0Xpt≡0．．3.2t=T0．．①pd0=1pA0pB0．②pd0＞1pA0pB0http//journal．szu．edu．cn45828．．、．t=T0．①．t=T0．②1T01t=T0．t=T0．③pB0＜p＜pA01．t=T0．．Kolmogorov．．、．1CaoCHanschOWangX．J．200929116-41．2DanielsMGFarmerJDIoriG．J．20039010108102-1-108102-4．3LuckockH．J．200335385-404．4SmithEFarmerJDGillemotL．J．200336481-514．5FarmerJDPatelliPZovkoI．J．200510262254-2259．6MikeSFarmerJD．J．2008321200-234．7ContRStoikovSTalrejaR．J．2010583549-563．8BiaisBHillionPSpattC．J．19955051655-1689．9LoISappS．J．、2010203213-237．10．、J．20101266-73．11AsmussenS．M．2003340-352．Abstract1000-2618201105-0458-EA【AppliedMathematics】ThedynamicevolutionoflimitorderbookdrivenbyorderflowsWANGSu-sheng1JIANGGuo-chao1andDONGHai-ling21ShenzhenGraduateSchoolHarbinInstituteofTechnologyShenzhen518055P．R．China2CollegeofMathematicsandComputationalScienceShenzhenUniversityShenzhen518060P．R．ChinaAbstractThelimitorderbookincontinuousauctionstockmarketcouldbeviewedasamulti-serverqueueingsys-http//journal．szu．edu．cn5459temwhereeachpricelevelwasseenasaserver．Thenthelimitorderbookcouldbemodeledbyamulti-dimension-alrandomprocess．Asordersubmissionstrategiesofmarketparticipantsoftendependedonthedepthoflimitorderbookstate-dependentPoissonprocesseswereintroducedtomodelthearrivalandcancellationprocessesoflimitor-derflowsnamelytheparametersofthePoissonprocessesdependedonthestateoflimitorderbook．Thedynamice-volutionoflimitorderbookdrivenbyorderflowswasstudiedpiecewiseaccordingtothemomentsofmid-pricemoves．Thedevelopmentoforderquantityateachpricelevelwasprovedtobeabirth-deathprocessbeforemid-pricemove．ThetransitionprobabilitiesandtheirKolmogorovbackwardandforwarddifferentialequationsofthesebirth-deathprocessesweregiven．Atthemomentofmid-pricemovethechangesofbid/askpriceandorderquantityateachpricelevelcausedbydifferenttypesofordersubmissionswereanalyzed．KeywordsstockmarketslimitorderbookstochasticprocessesMarkovprocessesstochasticmodelsPoissonprocessesbirth-deathprocessesThisworkwassupportedbytheNationalNaturalScienceFoundationofChina11001179．References1CaoCHanschOWangX．TheinformationalcontentofanopenlimitorderbookJ．TheJournalofFuturesMar-kets200929116-41．2DanielsMGFarmerJDIoriGetal．QuantitativemodelofpricediffusionandmarketfrictionbasedontradingasamechanisticrandomprocessJ．PhysicalReviewLetters20039010108102-1-108102-4．3LuckockH．Asteady-statemodelofthecontinuousdoubleauctionJ．QuantitativeFinance200335385-404．4SmithEFarmerJDGillemotLetal．Statisticaltheo-ryofthecontinuousdoubleauctionJ．QuantitativeFi-nance200336481-514．5FarmerJDPatelliPZovkoI．ThepredictivepowerofzerointelligenceinfinancialmarketsJ．ProceedingsoftheNationalAcademyofSciencesoftheUnitedStatesofA-meric