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1、Chapter12MonteCarloandQuasi-MonteCarloMethodsInthischapter,wediscussMonteCarloandQuasi-MonteCarlomethodsandshowhowtheycanbeusedtocomputefunctionalsofmultidimensionaldiffusions.12.1MonteCarloMethodsMonteCarlo(MC)methodscanbeemployedtocomputefunctionalsofmultidime
2、n-sionaldiffusions,e.g.byusingtheinversetransformmethod,seee.g.Sect.6.1andPlatenandBruti-Liberati(2010),ifthetransitiondensityisknownexplicitly.How-ever,whenthetransitiondensityisnotknownexplicitly,onecanemploydiscretiza-tionschemes,suchastheEulerScheme,seeKloed
3、enandPlaten(1999),toapprox-imatelysamplefromthetransitiondensity.Thediscretizationschemeintroducesanerror,whichcanbestudiedusingthetechniquesinKloedenandPlaten(1999).Theaimofthissectionistointroducetwoalternativestodiscretizationschemes,whichallowustoeliminateth
4、ediscretizationerrorintroducedbydiscretizationschemesandtorecovertheMonteCarloconvergencerate.Thesealternativesaretheexactsimulationmethods,duetoRobertsandcollaborators,seeBeskosetal.(2006,2008,2009),BeskosandRoberts(2005),andalsoChenandHuang(2012b),andmulti-lev
5、elmethodsduetoGilesandcoauthors,seeGiles(2008a,2008b).WeTrstlyprovideaverybriefintroductiontoMonteCarlomethodsandthenbrie?yillustratetheEulerdiscretizationscheme,whichmotivatestheexactsim-ulationandmultilevelmethods.FordetailedreferencesonMonteCarlomethodsapplie
6、dtoTnance,wereferthereadertoKloedenandPlaten(1999),Glasser-man(2004),Jckel(2002),PlatenandBruti-Liberati(2010),andKornetal.(2010).J.Baldeaux,E.Platen,FunctionalsofMultidimensionalDiffusionswithApplications299toFinance,Bocconi&SpringerSeries5,DOI10.1007/978-3-319
7、-00747-2_12,?SpringerInternationalPublishingSwitzerland201330012MonteCarloandQuasi-MonteCarloMethods12.1.1MonteCarloMethodsMonteCarlomethodsareeasilyillustratedbyconsideringtheproblemofestimatingtheintegral1a=f(x)dx.0Thisintegralcanbeinterpretedastheexpectedval
8、ueEf(U),whereUisuniformlydistributedovertheinterval[0,1],assumingthatfisinte-grable.Nowconsiderthei.i.d.randomvariablesU1,U2,...,UN,uniformlydis-tributedover[0,1],t
