continuous_time_review

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1、JohnH.Cochrane1.RevisedSeptember22,20132ContinuousTimeSummary/ReviewWeneedastatisticalmodelthatcanapplytostockprices,returns,etc.Thepointofthesenotesistoquicklypresentthestandarddiffusionmodelthatweuseinassetpricing.Weusecontinuoustimewhenit’ssimpler.Inpointoffactallourdataarediscretelys

2、ampled.Whethertimeis“really”continuousordiscreteisanissuewe’llleavetophysicistsandphilosophers.MostoftheapproximationsthattookalotofeffortinChapter1ofAssetpricingworkmuchmoreeasilyincontinuoustime.2.1Preview/summary:2.1.1Continuoustime1.Brownianmotion+∆−˜(0∆)2.Differential=lim(

3、+∆−)∆&03.dzanddt√˜()2=()=0()=(2)=2=4.Diffusions=()+()()=()2()=2()5.Examples=+=−(−)+1Pleasereporttypostojohn.cochrane@chicagobooth.edu.Makesureyouhavethelatestversionofthedocument.Ifyoucan

4、reportthecontextofthetypoandnotjustthepagenumberthatwillhelpmetofinditmorequickly.96.Ito’slemma=+;=()=⇒122=++22∙¸∙¸122=+++227.Stochasticcalculus:“dosecond-orderexpansions,keeptheandterms.”2.1.2Solvingstochasticdifferent

5、ialequations1.Stochasticintegral“addupthechanges”ZX−0=↔−0==0=1theresultisjustawayofsayingisanormallydistributedrandomvariableZ=−0˜(0)=02.Example1:diffusion=+ZZZ=+=0=0=0−0=+(−0)3.Example2:lognormaldiffusion=+µ¶12ln

6、=−+2µ¶Z1ln−ln=−2+02012=(−2)+004.Example3:AR(1)=−(−)+Z−=−(−)+−(−)0−=0105.Continuous-timeMAprocessesZ=(−)=06.Withsolutions,youcanfindmoments.Examples(a)Lognormaldiffusion12=(−2)+00(−12)+12()=

7、022=0(b)AR(1)Z(−)=−(−)+−(−)0=0(−)=−(−)00Z−22−2(−)21−2(−)===02(c)Continuous-timeMA.Z=(−)0µZ¶Z0()=0(−)=(−)()=000µZ¶2Z(2)=(−)=(−)200002.1.3FindingMomentsandDensities1.Backwardequation.W