An algorithmic introduction to numerical simulatio

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AnIntroductiontotheNumericalSimulationofStochasticDi erentialEquationsDesmondJ.HighamDepartmentofMathematicsUniversityofStrathclydeGlasgow,G11XHScotland,U.K.Telephone:+441415483716Fax:+441415528657PeterE.KloedenFachbereichMathematikJohanWolfgangGoethe-UniversitatD-60054FrankfurtamMainGermanyLecturenotesforaCompactCourseforStudentsoftheBavarianGraduateSchoolinComputationalEngineering.cDesmondJ.HighamandPeterE.KloedenNOTFORFURTHERDISTRIBUTION.March15,20062PrefaceEXPLANATION:Thesenotesarebasedonabookthatiscurrentlyinpreparation.Pleasedonotdistributethemwithout rstcontactingtheauthors.Thebookwillcontainmorematerial,butthesenotesareintendedtobefairlypolished.Pleasefeelfreetoemailcomments/listsoftyposto:djh\atmaths.strath.ac.uk.Ourintentioninthisbookistoprovideapunchy,accessibleintroductiontothenumericalsolutionofstochasticdi erentialequations(SDEs).Withtheaimofmakingthistopicavailabletothewidestpossiblereadership,wehavekepttheprerequisitestoaminimum.Weassumeonlyacompetenceinalgebraandcalcu-lusatthelevelreachedbyatypical rstyearundergraduatemathematicsclass.Somefamiliaritywithbasicconceptsfromnumericalanalysisandprobabilityisalsodesirable,butnotabsolutelynecessary.Ourintendedreadershipincludesundergraduateandbeginninggraduatestudentsinmathematics,statistics,physics,economics, nance,business,computerscience,engineeringandthelifesciences,who,perhapshavingbeenexposedtoSDEmodels,wishtolearnmoreabouthowtosimulatethem,researchersintheabovedisciplineswhoroutinelyperformSDEsimulationsandwouldliketounderstandsomeoftheunderlyingtheory,andresearchersfromdeterministicappliedmathematicsornumericalanalysisbackgroundswhoareseekingtobroadentheirinterests.Abigmotivationforthisbookhasbeentheamountofpositivefeedbackthatoneofushasreceivedfromthearticle[16]thatappearedintheEducationsectionofSIAMReview.Basedonthatfeedback,andondiscussionswithscientists,webelievethatthereisade nitenicheforaself-contained,low-leveltextthatputsacrossthefundamentalsassuccinctlyaspossible.Followingthestyleof[16]weiiiPREFACEhavemadeheavyuseofcomputationalexamplesandillustrative gures.Thereis,ofcourse,muchmorematerialherethanin[16].Ourguidingprinciplesweretoaddenoughbackgroundmaterialtoallowanoutlineproofofthekeypropertiesofweakconvergence(Chapter8)andstrongconvergence(Chapter9)oftheEuler{Maruyamamethod,andtoexplaintherelevanceofIto'slemmainthederivationofhigherordermethods.BecausecomprehensiveandrigorousnumericalSDEbooks,suchas[27,35,36],havealreadybeenpublished,wefeeljusti edinfocusingonaccessibilityandbrevity.Thisisnotarigoroustext.LikeThomasMikosch,authoroftheexcellentintroductorySDEtext[34],wearenotashamedofthislackofrigour1.However,wearesomewhatwarythatreadersmaybefooledintothinkingthatwearepresentingthewholestory.Inane orttoavoidthis,wehaveincludedmanypointerstothewealthofhigh-level,technicalliteraturethatcanbeusedto llinthemanygaps.MentionthatwearedoingSDEsimulationratherthane,g.solvingFokker-Planck,refertoK&Pforjusti cation.YoucangetareasonablefeelforthecontentofthisbookbyskimmingthroughtheOutlinebulletpointsthatbegineachchapter.Becausethisisarelativelynewandrapidlyexpanding eld,eveninanintroductorytextlikethiswewereabletoincludematerialontopicsthat,toourknowledge,donotappearelsewhereoutsidethedomainofresearcharticles.ThereareanumberoftopicsrelatedtonumericalSDEsthatcanbecon-fusing,andcanraisequestionsthatareasmuchphilosophicalasmathematical.Withinthelimitationsofaccessibility,wehavetriedtoexplainclearlytheissuessurroundingstrongversusweaksolutions,ItoversusStratonovichcalculus,strongversusweakconvergence,mean-squareversusasymptoticstability.ThepopularityandimportanceofSDEsisdrivenbytheirrelevanceasmodelsinmanyapplicationareas,mostnotablymathematical nance,physicsandthelifesciences.Toacknowledgethis,andalsotogivetheopportunitytopresentsomefairlyrealisticcomputationalscenarios,wehaveincludedCaseStudieschap-tersthatdescribenot-so-small-scaleSDEsimulations.Eachchapterincludesalistofexercisesthataredesignedtoreinforcethetextand llinsomeofthedetails.Athreestarlabellingsystemhasbeenused;onestarforquestionswiththeshortest/easiestanswersandthreestarsforquestionswiththelongest/hardestanswers.Workedsolutionstotheodd-numberedexercises1Seethequoteattheendofthispreface.iiiareobtainablefromthewebsitementionedinthenextparagraph.Thisleavestheeven-numberedquestionsasanexaminingresourceforteachers.Wehavealsosprinkledafewquotesattheendofeachchapter.Youarenotallowedtoreadtheseuntilyouhavedonealltheexercises.Awebsiteforthebookhasbeencreatedat??????Itincludesworkedsolutionstotheodd-numberedexercises,listingsofeachProgramoftheChapter,linkstoallwebsitesmentionedinthebook,bonusquotes,andwillnodoubtalsohousealistofcorrections.Anumberofpeopledeserveacknowledgement.DJHwishestothankXuerongMaoandAndrewStuart,whohavegenerouslysharedtheirSDEexpertiseinresearchcollaborations.BothauthorsaregratefultoXYZwhohaveprovidedcriticalfeedbackononeormoreportionsofdraftsofthisbook.ivPREFACEMATLABProgramsInourexperience,thebestwaytounderstandacomputationalalgorithmistoexperimentw

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