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1、BerntÂksendalStochasticDi®erentialEquationsAnIntroductionwithApplicationsFifthEdition,CorrectedPrintingSpringer-VerlagHeidelbergNewYorkSpringer-VerlagBerlinHeidelbergNewYorkLondonParisTokyoHongKongBarcelonaBudapestToMyFamilyEva,Elise,AndersandKarina2ThefrontcovershowsfoursamplepathsXt(!
2、1);Xt(!2);Xt(!3)andXt(!4)ofageometricBrownianmotionXt(!),i.e.ofthesolutionofa(1-dimensional)stochasticdi®erentialequationoftheformdXt=(r+®¢Wt)Xtt¸0;X0=xdtwherex;rand®areconstantsandWt=Wt(!)iswhitenoise.Thisprocessisoftenusedtomodelexponentialgrowthunderuncertainty".SeeChapters5,10,11an
3、d12.The¯gureisacomputersimulationforthecasex=r=1,®=0:6.ThemeanvalueofXt,E[Xt]=exp(t),isalsodrawn.CourtesyofJanUb¿e,Stord/HaugesundCollege.Wehavenotsucceededinansweringallourproblems.Theanswerswehavefoundonlyservetoraiseawholesetofnewquestions.Insomewayswefeelweareasconfusedasever,butweb
4、elieveweareconfusedonahigherlevelandaboutmoreimportantthings.Postedoutsidethemathematicsreadingroom,Troms¿UniversityPrefacetoCorrectedPrinting,FifthEditionThemaincorrectionsandimprovementsinthiscorrectedprintingarefromChaper12.Ihavebene¯ttedfromusefulcommentsfromanumberofpeo-ple,includi
5、ng(inalphabeticalorder)FredrikDahl,SimoneDeparis,UlrichHaussmann,YaozhongHu,MarianneHuebner,CarlPeterKirkeb¿,Niko-layKolev,TakashiKumagai,ShlomoLevental,GeirMagnussen,AndersÂksendal,JÄurgenPottho®,ColinRowat,StigSandnes,LonesSmith,Set-suoTaniguchiandBj¿rnThunestvedt.Iwanttothankthemallf
6、orhelpingmemakingthebookbetter.IalsowanttothankDinaHaraldssonforpro¯cienttyping.Blindern,May2000BerntÂksendalVIPrefacetotheFifthEditionThemainnewfeatureofthe¯ftheditionistheadditionofanewchapter,Chapter12,onapplicationstomathematical¯nance.Ifounditnaturaltoincludethismaterialasanotherma
7、jorapplicationofstochasticanalysis,inviewoftheamazingdevelopmentinthis¯eldduringthelast10{20years.Moreover,theclosecontactbetweenthetheoreticalachievementsandtheapplicationsinthisareaisstriking.Forexample,todayveryfew¯rms(ifany)tradewithoptionswithoutconsultingtheBlack&Scholesf
