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1、CHAPTER4elementarystochasticcalculusInthisChapter...•allthestochasticcalculusyouneedtoknow,andnomore•themeaningofMarkovandmartingale•Brownianmotion•stochasticintegration•stochasticdifferentialequations•Ito’slemmainoneandmoredimensionsˆ4.1INTRODUCTIONStocha
2、sticcalculusisveryimportantinthemathematicalmodelingoffinancialprocesses.Thisisbecauseoftheassumedunderlyingrandomnatureoffinancialmarkets.BecausestochasticcalculusissuchanimportanttoolIwanttoensurethatitcanbeusedbyeveryone.Tothatend,Iamgoingtotrytomakethisc
3、hapterasaccessibleandintuitiveaspossible.Bytheend,Ihopethatthereaderwillknowwhatvarioustechnicaltermsmean(andrarelyaretheyverycomplicated),but,moreimportantly,willalsoknowhowtousethetechniqueswiththeminimumoffuss.Mostacademicarticlesinfinancehavea‘pure’math
4、ematicaltheme.Themathematicalrigorintheseworksisoccasionallyjustified,butmoreoftenthannotitonlysucceedsinobscuringthecontent.Whenasubjectisyoung,asismathematicalfinance(youngish),thereisatendencyfortechnicalrigortofeatureveryprominentlyinresearch.Thisisdueto
5、lackofconfidenceinthemethodsandresults.Asthesubjectages,researcherswillbecomemorecavalierintheirattitudesandwewillseemuchmorerapidprogress.4.2AMOTIVATINGEXAMPLETossacoin.EverytimeyouthrowaheadIgiveyou$1,everytimeyouthrowatailyougiveme$1.Figure4.1showshowmuc
6、hmoneyyouhaveaftersixtosses.InthisexperimentthesequencewasTHHTHT,andwefinishedeven.72PartOnemathematicalandfinancialfoundations2100123456WinningsNumberofcointosses−1−2Figure4.1Theoutcomeofacointossingexperiment.IfIuseRitomeantherandomamount,either$1or−$1,you
7、makeontheithtossthenwehaveE[R]=0,E[R2]=1andE[RR]=0.iiijInthisexampleitdoesn’tmatterwhetherornottheseexpectationsareconditionalonthepast.Inotherwords,ifIthrewfiveheadsinarowitdoesnotaffecttheoutcomeofthesixthtoss.Tothegamblersoutthere,thispropertyisalsoshare
8、dbyafairdie,abalancedroulettewheel,butnotbythedeckofcardsinBlackjack.InBlackjackthesamedeckisusedforgameaftergame,theoddsduringonegamedependonwhatcardsweredealtoutfromthesamedeckinpreviousgames.Thatiswhyyouca
