Oksendal B. - Stochastic Differential Equations (5th ed.).pdf
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1291 KB
)
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134563925 UNPDF
BerntÂksendal
StochasticDi®erentialEquations
AnIntroductionwithApplications
FifthEdition,CorrectedPrinting
Springer-VerlagHeidelbergNewYork
Springer-Verlag
BerlinHeidelbergNewYork
LondonParisTokyo
HongKongBarcelona
Budapest
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
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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