Transcription of Stochastic Difierential Equations
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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 . See Chapters 5,10, 11 and figure is a computer simulation for the casex=r= 1, = mean value ofXt,E[Xt] = exp(t), is also drawn.
the stochastic calculus. Problem 4 is the Dirichlet problem. Although this is purely deterministic we outline in Chapters VII and VIII how the introduc-tion of an associated Ito difiusion (i.e. solution of a stochastic difierential equation) leads to a simple, intuitive and useful stochastic solution, which is
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