膨胀的Poisson过程及其在金融中的应用.doc

膨胀的Poisson过程及其在金融中的应用应用数学MATHEMATICAAPPLICATA2006.19(4):793798DilatedPoissonProcessesandtheirApplicationsinFinanceHUANGGuanghui(黄光辉),WANJian-ping(万建平)(DeP口rtme;ofMathematics,HuazhongUniversityofScience&Technology,Wuhan430074,China)Abstract:Akindofpurejumpprocesseswithdependentincrements,calleddilatedpoissonprocesses,wasdefined.Ageneralized疗-jumppriceprocesswasusedtodescribethemovementofstockprice.Usingthiskindofstochasticprocesses,wedecomposedtheprofitofthemonetarymarketportfolio,generatingthesamecashflowatthepointedtimeasitsstockmarketcounterpart,intoadeterministictermandadilatedPoissonterm.ItwasshowedthatinvestmentinstockmarketiSmoreriskythaninthemonetarymarket,consistingwithourobservationoftherealworld.Keywords:Stochasticintegral;CompoundPoissonprocess;InvestmentriskCLCNumber:AMS(2000)SubjectClassification:60H05Documentcode:AArticleID:1001-9847(2006)040793061.IntrOductiOnAndersonandMOiler引.demonstratethepresenceoflong-termdependenceandheavytaileddistributioninhighfrequencyfinancialdata.BecausetheincrementsoffractionalBrownianmotionarelongtermdependenceandheavytaileddistribution,manyauthorsusefractionalBrownianmotiontoreplacetheBrownianmotioninthestandardfinancialmodels.Alongthisline,wewanttousethedilatedpoissonprocesstodescribethebehavioroftheprofitprocessinthemonetarymarket.Thereareseveralkindsofprocesseswhoseincrementshavelongtermdependenceandheavytaileddistribution.Sait6LusesthefractionalpowersofL6vyLaplacianactingonwhitenoisedistributionstogeneratestableprocesses.Wang嘲replacetheBrownianmotionintheintegralrepresentationoffractionalBrownianmotionbyPoissonmartingale,generatingthePoissonfractionalprocesses.LaskinusestheRiemannLiouvilletypefractionalderivativeoperatoractingontheKolmogorovFellequationofPoissonprocesstogeneratethefractionalReceiveddate:Mar15,2006Foundationitem:SupportedbyNaturalScienceFoundationofChinaGrantBiography:HUANGGuanghui,male,Han,Sichuan.PhDcandidate,majorinStochasticAnalysisandMathematicalFinance.794MATHEMATICAAPPLICATA2006Po.is1sonprocIeIss.PipirasandTaqt.qugUe3neursaetethtehekderilnaetledwfitrhacati.dnilaaltisotnabflae.mto.rtdi.nins.tBheecaiunsteegtrhaelwithrespectt.astablemotion.gneahd.d:u.u:二二+ion,wemaymethodsgeneratingt+hoseprocessesaretsimilartothatoIractlonalDrOW|lllltil:lllxuttavlkernelt.callthetmfrarctioantaeldtPyp.ep.rnocpers.secess.sIenstahnisdpaapppelry,twheemwillu,.sesomell,lateoJl_mteg,.tolmancemarkettneuxygene甜扑ynthedefn0nofthednatedPoiThepapersorganizedasfollows.InSectionZwegVetnedenunuuuuus.npr.ce:ss,usingtheintegralw.ithrie.spect.toaTchommpo.mundnP.o.isso.nhpo.c.ewssannddthhe,depiennd.-fienceoncrementsalsobegiveninthssection.1nen()【Ieuu.2.ThedilatedpoisssonprocessesSupposethatN一N,t0)isaPoissonprocesswithintensity,>0,andZi(1,2,N)isanindependentandidenticallydistributedsequenceofrandomvariablesindependentofN.Thenx=lzt(o)isacompoundPoionp?WedenotebyK,withfl>0,theintegralkernel加三withthisintegralkernelwedefinetheintegral.perat.rJ.nLf.reVeryLasJ垒K*.Lemma1iscontinuousonL.ProofNoticethatJ车(z)Idzl:.er.cdIdzI8(t川池:+_dflldJ一.J一一rI车()IdtfdtII车()IJ一.o0一-bOOl()ldtI1一l()lI1I1J一.L1.(1)(2)whichpcl1onmarpleteth.p.j,finea?-t?f.reveryRk1FromthecontinuityofonLwecandestochasticmtegratemar卫.,tneauccLwithrespectt.anyc.mp.undP.iss.npr.cesspathbypath?Thatisthef.ll.wingintegraliswelldefined广十a.x2()=I卫()dF(5,),Ja.(3)wherReF(s.,.we)thiserae.mp.ndP.eralkernel,weisalsoadilationfactorinourgnamethest.chasti.BecausetherenteralLIaucLllcI上cessesgeneratedbythiskernelthedilatedPoisonpO.e.仁仁.I1一NO.4HuANGGuang-huieta1.:DilatedPoiss.nPr.cessesandtheirAppli.ati.795Definition1Forevery卢>0,thedilatedPoiss.nPr.cess()isdefinedasr+oo()II1【0.f(s)dx(s),Joowherex()isac.mp.undPoiss.nprocess,and1.istheindicat.r.fEo,.Actingtheintegraloperatorontheindicator1.,wgtI1c.?(z):卢er1o司()d一l-er,一r,X0;0<zSRemark2IfourcompoundPoissonprocessX()isdefinedgralin(4)canbewrittenas垒lIt0,f()dx(z)一re_(epc一1)dX(z)+z(一1)dX(x).(4)z>S.Onthewholeline,theinteInthispaper,wef0cusonthecompoundPoissonprocessesontherighthalfline,soourdiltedPoissonprocessesare尘Ie一(一1)dX(x).(5)WecanrewritethestochasticintegralrepresentationofthedilatedPoissonprocessesntothefollowingsummationform:-<Zi(1-e-)=.?O<IuiwherezisthetharrivaltimeofthecompoundPoissonpr0cess,andZlisthelevelofithjump.For0<l<2<3<4<+o.,weconsiderthefollowingtWOincrements:xxandXt一磁.Obvi0uslythetWOincrementsaredependent.Wecanalsofindthatthedisributionofincrementisdeterminednotonlybythelengthoftheintervall,tzb.utalsobythetimet1.SodilatedPoissonprocessesarenotL邑Vyprocesses.ToillustratethiswegivethecovarianceofthetWOincrementsinthefollowingproposition?Proposition1SupposeZissymmetricsuchthatEZ)=0,thenE(一磁)(磁一)一2fl(ef3一)(eple-2)(e2pt一1)+(2一p.)(.一)EZ2),andtheaboveformulaispositive.ProofFromthedefinitionofdilatedPoissonprocesse?,wehavet(xt磁)(x一xt)=(e-plep2)(e3ep4)796MATHEMATICAAPPLICATA2006Z(1一e一+lNtZz(1-ez),N1+1N2+Zi(1一e-),=N1+l4(1一e-).(6),Nt3+lTakenexpectationonthebothsidesof(6),thedesiredresultsfollows.3.TheMomentsofGrossReturnsWeadopttheRydbergandShephardframeworktodescribethepriceprocessSrSo+竺lZl,wheretheNrdenotesthetota1jumpsfromtimezerototime.wecangeneTatethesamecashflowattimetinthemonetarymarket,markingtothestockpricechanges.Attimezero,wedepositSoe-inthebank,withtheinterestrateWhentheithjumpofstockpricetakesplaceattimez,weaddourdepositwiththeamountofZe一.Notethatthenegativeamountofdepositmenasborrowing.ItisobviouslythatattimetthemarketvalueofourmonetaryportfolioisalsoSf.ItisobviouslythatthegrossreturnfrommonetarymarketcanbedecomposedintoN,s.一s.e+Zi(1一e-)i=1一S.(1一e一)+X,(7)thefirsttermin(7)isdeterministic,andthesecondtermisourdilatedPoissonprocess,arandoraterm.DenotehteprofitinstockmarketasYl一NtZ.Proposition2DenotetherandomjumplevelbyZ.ThenthefirstandsecondordermomentsofprofitfromstockmarketareEY=atEZ;E)一2tEZ2)+t(EZ);(8)VarY)一2tEZ).ProofFromRoss.inviewoftheindependenceofNandZi,wehaveZ,l/Z,l,Z,一e一一e,+,f,)z=,-,一e一+No.4HUANGGuanghuieta1.:DilatedPoissonProcessesandtheirApplications797EY):EZ)=EN)Ez);E)一2tE)+t(EZ),thedesiredresultsfollows.NotethatundertheconditionthatNl,thearrivaltimesX1,X2,Xhavethesamedistributionastheorderstatisticsofindependentrandomvariablesuniformlydistributedontheinterval(0,).Wehavethefollowingproposition.Proposition3DenotetherandomjumplevelbyZ.ThenthefirstandsecondordermomentsofXareExf):(卜1十1e一)E(Z);x一(t一+吾e一一1)EZ2)+(一1十1e一)(E(Z)ProofDenotebyUtherandomvariableuniformlydistributedontheinterval(0,).NoticethatEXf)一EN)EZ)E1一e-0(),E(Xf)一EN)EZ2)E(1一e一)+(EN)(EZ)(E1一e一),thedesiredresultsfollows.Proposition4ThefirstandsecondordermomentsofprofitfrommonetarymarketareE(:s.(1-e-a)+(一1十1e一)E(z;E)一S5(1一e-)+21So(1一e呻)(1十1e一)Ez)+(一翕+吾ee-2)EZz)+(一1十1e一)(Ez);Var)一t-+吾e呻一e_2)EZ2).ProofInviewof(7),wehaveE)一So(1一e一)+EXf),(9)andE)一S5(1一e-)+2S.(1一e-)EX)+E(Xf)z),(10)whichcompletetheproof.TheoremWhentheinvestmentinmonetarymarketmarkingtothestockpricechanges,investinginthestockmarketismoreriskythaninthemonetarymarket.ProofNoticethatthedifferentofthetWOvariancesis798MATHEMATICAAPPLICATA2006Var)-Vary):(一+舌e呻一e)Ez.)whichisobviouslynegative.4.ConclusionThepatternofstockpricemovementisdiscussedintheprevioussections,underthehypothesisthatitisajumpprocess.Andtheriskinstockmarketiscomparedwithrespecttothatofmonetarymarketmeasuredinvariance.Itisinterestingtoshowthatthepreviousoneismoreriskythanthelatteroneunderourassumption.Thisconclusionconsistswithourintuitionabouttherealworld.RefeFences:1233435367891OAndersonT,BollerslevT.Heterogeneousinformationarrivalsandreturnvolatilitydynamicsiuncoveringthelongruninhigh-frequencyreturnsJ.J.JFiance,1997,52:9751006.M011erWA,DacorognaMA,Picte0V.HeavytailsinhighfrequencyfinancialdataA.InapracticalguidetOheavytails:statisticaltechniquesandapplicationsC.AdlerRA,FeldmanRE,TaqquMS.Boston:Birkhuser,1998,5577.ElloittRJ,VanDJHoek.Ageneralfractionalwhitenoisetheoryan