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1、Chapter2FunctionalsofWienerProcessesInthischapter,wediscussscalar-andmultidimensionalprocesses,whicharebasedontheWienerprocess,andconsequentlyapplytheminthecontextofthebenchmarkapproach.2.1One-DimensionalFunctionalsofWienerProcessesWesummarizewell-knownSDEsandtransitiondensitiesformodelsandpr
2、ocessescloselyrelatedtotheWienerprocessorBrownianmotion,including:•theBacheliermodel;•theBlack-Scholesmodel;•theOrnstein-Uhlenbeck-process;•thegeometricOrnstein-Uhlenbeck-process.AlsowecollectresultsfromtheliteratureonfunctionalsofWienerprocessesandaddnewresultsandpresentations.Weremarkthatpa
3、rtsofthissectionarebasedonBorodinandSalminen(2002),Jeanblancetal.(2009),Chap.3,andPlatenandHeath(2010),Chap.4.2.1.1WienerProcessTheWienerprocessisacontinuousMarkovprocessandhasthefollowingtransitiondensity:1(y−x)2p(s,x;t,y)=√exp−,(2.1.1)2π(t−s)2(t−s)fort∈[0,∞),s∈[0,t]andx,y∈.Forthepurposeo
4、fillustration,wedisplaysometransitiondensitiesinFig.2.1.1asfunctionsoftimetandÞnalvaluey,wherewesettheinitialtimetos=0andtheinitialvaluetox=0.J.Baldeaux,E.Platen,FunctionalsofMultidimensionalDiffusionswithApplications23toFinance,Bocconi&SpringerSeries5,DOI10.1007/978-3-319-00747-2_2,©Springer
5、InternationalPublishingSwitzerland2013242FunctionalsofWienerProcessesFig.2.1.1ProbabilitydensitiesforthestandardWienerprocessTheWienerprocessenjoysthestrongMarkovproperty,whichallowsustofor-mulatethefollowinglemma:Lemma2.1.1Forafinitestoppingtimeτ,theprocessW˜={W˜t,t≥0},whereW˜t=Wτ+t−Wτ,(2.1.2
6、)isaWienerprocesswithrespecttoitsnaturalfiltration.WenowintroducethefollowingnotationTa=inf{t≥0:Wt=a}Mt=supWs0≤s≤tmt=infWs.0≤s≤tThefollowingproposition,commonlyreferredtoasreßectionprinciple,employsLemma2.1.1andthesymmetryoftheWienerprocess,seeLemma15.1.3.Proposition2.1.2Lety≥0,x≤y,thenonehasP
7、(Wt≤x,Mt≥y)=P(Wt≥2y−x).(2.1.3)Foraproof,seee.g.Jeanblancetal.(2009),Proposition3.1.1.1.Next,wediscussthejointdistributionof(Mt,Wt),seeTheorem3.1.1.2inJeanblancetal.(2009).Proposition2.1.3ForaBrownianmotionWtanditsrunningmaximumMt,thefollowing
