4
DAVID DOUGLAS ENGEL
T
J f(t)dX(t) = X(I).
0
By linearity we can extend this definition to all step functions, that
is if f(t) = a. for tel. where I ,...,1 are disjoint intervals contained
in [0,T], then
T
f
f(t)dX(t) = £ a±X(I).
L n=l
0
Clearly each of these integrals is a Gauss function of o with mean zero.
Computing the variance we get
T k
f(t)dX(t)||2 = ||S a. X(I±)||2
n=l
0
k k
*i *—% n m n m
n=l m=l
k
n=l
= V a E(X(I )2) since X(I ) I I X(I ) n * m
*"* n n n ---
LJ
m
k
£ anm(In
^-r, n n
n= l
| f ( t ) l l
2
2
o
So the operation of taking the stochastic integral preserves L -mean.
2
Therefore, if f is any function in L ([0,T]), by taking a sequence of
2
step functions f such that f - f in L ([0,T] ), we can define
T n
T
n
f(t)dX(t) = L.I.M.
n - °°
fn(t)dX(t),
The important properties of X(t) used in defining this stochastic
integral are:
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