Simulating Markov chains

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1、Copyrightc2007byKarlSigman1SimulatingMarkovchainsManystochasticprocessesusedforthemodelingofnancialassetsandothersystemsinengi-neeringareMarkovian,andthismakesitrelativelyeasytosimulatefromthem.HerewepresentabriefintroductiontothesimulationofMarkovchains

2、.Ouremphasisisondiscrete-statechainsbothindiscreteandcontinuoustime,butsomeexampleswithageneralstatespacewillbediscussedtoo.1.1DenitionofaMarkovchainWeshallassumethatthestatespaceSofourMarkovchainisS=ZZ=f:::;2;1;0;1;2;:::g,theintegers,orapropersubsetof

3、theintegers.TypicalexamplesareS=IN=f0;1;2:::g,thenon-negativeintegers,orS=f0;1;2:::;ag,orS=fb;:::;0;1;2:::;agforsomeintegersa;b>0,inwhichcasethestatespaceisnite.Denition1.1AstochasticprocessfXn:n0giscalledaMarkovchainifforalltimesn0andallstatesi0;:::

4、;i;j2S,P(Xn+1=jjXn=i;Xn1=in1;:::;X0=i0)=P(Xn+1=jjXn=i)(1)=Pij:Pijdenotestheprobabilitythatthechain,wheneverinstatei,movesnext(oneunitoftimelater)intostatej,andisreferredtoasaone-steptransitionprobability.ThesquarematrixP=(Pij);i;j2S;iscalledtheone-stept

5、ransitionmatrix,andsincewhenleavingstateithechainmustmovetooneofthestatesj2S,eachrowsumstoone(e.g.,formsaprobabilitydistribution):ForeachiXPij=1:j2SWeareassumingthatthetransitionprobabilitiesdonotdependonthetimen,andso,inparticular,usingn=0in(1)yieldsPij=

6、P(X1=jjX0=i):(FormallyweareconsideringonlytimehomogenousMC'smeaningthattheirtransitionprob-abilitiesaretime-homogenous(timestationary).)Thedeningproperty(1)canbedescribedinwordsasthefutureisindependentofthepastgiventhepresentstate.Lettingnbethepresenttim

7、e,thefutureaftertimenisfXn+1;Xn+2;:::g,thepresentstateisXn,andthepastisfX0;:::;Xn1g.IfthevalueXn=iisknown,thenthefutureevolutionofthechainonlydepends(atmost)oni,inthatitisstochasticallyindependentofthepastvaluesXn1;:::;X0.MarkovProperty:Conditionalonthe

8、rvXn,thefuturesequenceofrvsfXn+1;Xn+2;:::gisindepen-dentofthepastsequenceofrvsfX0;:::;Xn1g.ThedeningMarkovpropertyabovedoesnotrequirethatthestatespacebediscrete,andingeneralsuchaprocesspossessingtheMarkovpropertyi