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Stochastic Difierential Equations

Bernt ksendalStochastic Differential EquationsAn Introduction with ApplicationsFifth Edition, Corrected PrintingSpringer-Verlag Heidelberg New YorkSpringer-VerlagBerlin Heidelberg NewYorkLondon Paris TokyoHong Kong BarcelonaBudapestTo My FamilyEva, Elise, Anders and Karina2 The front cover shows four sample pathsXt( 1), Xt( 2), Xt( 3) andXt( 4)of a geometric Brownian motionXt( ), of the solution of a (1-dimensional) Stochastic differential equation of the formdXtdt= (r+ Wt)Xtt 0 ;X0=xwherex, rand are constants andWt=Wt( ) is white noise. This process isoften used to model exponential growth under uncertainty.

stochastic difierential equation of the form dXt dt = (r +fi ¢Wt)Xt t ‚ 0 ; X0 = x where x;r and fi are constants and Wt = Wt(!) is white noise. This process is often used to model \exponential growth under uncertainty". See Chapters 5, 10, 11 and 12. The flgure is a computer simulation for the case x …

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