An algorithmic introduction to numerical simulatio

AnIntroductiontotheNumericalSimulationofStochasticDierentialEquationsDesmondJ.HighamDepartmentofMathematicsUniversityofStrathclydeGlasgow,G11XHScotland,U.K.Telephone:[phone]Fax:[phone]PeterE.KloedenFachbereichMathematikJohanWolfgangGoethe-UniversitatD-60054FrankfurtamMainGermanyLecturenotesforaCompactCourseforStudentsoftheBavarianGraduateSchoolinComputationalEngineering.cDesmondJ.HighamandPeterE.KloedenNOTFORFURTHERDISTRIBUTION.March15,20062PrefaceEXPLANATION:Thesenotesarebasedonabookthatiscurrentlyinpreparation.Pleasedonotdistributethemwithoutrstcontactingtheauthors.Thebookwillcontainmorematerial,butthesenotesareintendedtobefairlypolished.Pleasefeelfreetoemailcomments/listsoftyposto:djh\atmaths.strath.ac.uk.Ourintentioninthisbookistoprovideapunchy,accessibleintroductiontothenumericalsolutionofstochasticdierentialequations(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.Inaneorttoavoidthis,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