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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.

ter V we use this to solve some stochastic difierential equations, including the flrst two problems in the introduction. In Chapter VI we present a solution of the linear flltering problem (of which problem 3 is an example), using the stochastic calculus. Problem 4 is the Dirichlet problem. Although this is

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