Oksendal B. - Stochastic Differential Equations (5th ed.).pdf

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BerntÂksendal
StochasticDi®erentialEquations
AnIntroductionwithApplications
FifthEdition,CorrectedPrinting
Springer-VerlagHeidelbergNewYork
Springer-Verlag
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ToMyFamily
Eva,Elise,AndersandKarina
2
Thefrontcovershowsfoursamplepaths X t ( ! 1 ) ;X t ( ! 2 ) ;X t ( ! 3 )and X t ( ! 4 )
ofageometricBrownianmotion X t ( ! ),i.e.ofthesolutionofa(1-dimensional)
stochasticdi®erentialequationoftheform
dX t
dt =( r + ®¢W t ) X t 0; X 0 = x
where x;r and ® areconstantsand W t = W t ( ! )iswhitenoise.Thisprocessis
oftenusedtomodel\exponentialgrowthunderuncertainty".SeeChapters5,
10,11and12.
The¯gureisacomputersimulationforthecase x = r =1, ® =0 : 6.
Themeanvalueof X t , E [ X t ]=exp( t ),isalsodrawn.CourtesyofJanUb¿e,
Stord/HaugesundCollege.
Wehavenotsucceededinansweringallourproblems.
Theanswerswehavefoundonlyservetoraiseawholeset
ofnewquestions.Insomewayswefeelweareasconfused
asever,butwebelieveweareconfusedonahigherlevel
andaboutmoreimportantthings.
Postedoutsidethemathematicsreadingroom,
Troms¿University
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