文档介绍：A BRIEF INTR ODUCTION TO NUMERICAL ANAL YSIS OF
ORDINAR Y
STOCHASTIC DIFFERENTIAL EQUA TIONS WITHOUT TEARS
By
Henri Sc h urz
IMA Preprin t Series
Decemb er
INSTITUTE F OR MA THEMA TICS AND ITS APPLICA TIONS
UNIVERSITY OF MINNESOT A
Vincent Hall
Church Street S
E
Minneap olis
Minnesota
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Dedicated to the T riumph of
Academic Honest y and Sincerit y
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than to mend a he art those wor ds have br oken
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but a harsh wor d stirs up anger
Pro v erbs
NIV
i
Preface
Abstract
Keyw ords
This surv ey rep ort sk etc hes some basic asp ects on n umerical analysis of systems
of
ordinary
sto c hastic di
eren tial equations
SDEs
at a fairly in tro ductory lev el
mean t to b e written for b oth engineers and mathematicians on the edge to w ards the
st cen tury
SDEs are often met in Natural and En vironmen tal Sciences
Financial
Mark ets and Mark eting when
uncertain t y
in mo deling or
erratic beha vior
of
the en vironmen t pla ys an essen tial role
Mostly
the analytical solutions are not
kno wn
and one has to resort to n umerical tec hniques
Starting from the w ell
kno wn Ito F orm ula
as sto c hastic coun terpart of deter
ministic c hain rule and as the link b et w een con tin uous and discrete time sto c hastic
a ylor expansions of functionals of SDEs
dynamical systems
w e arriv eatstoc hastic T
W agner
Platen Expansion
As in deterministic analysis
the latter form ulas are
essen tial for the systematic construction of sto c hastic
n umerical metho ds and the
in v estigation of the lo cal beha vior of their appro ximating tra jectories
prehensiv e to olb o x for n umerical sc hemes to build a zo o of sto c hastic n umerical
metho ds is presen ted thereafter
Starting from these main to ols
w e presen t a fairly general