Lecture 12
Stochastic Integration
12.1
The Itˆo integral of simple processes
We will define the Itˆo stochastic integral in this lecture through an approximation procedure, same
as in the definition of the Riemann integral. Let
W
be a standard Brownian motion and
{F
t
}
t
≥
0
be the Brownian filtration. An adapted process
h
=
{
h
t
}
is said to be
simple
if it is a random step
function:
h
t
=
k
X
i
=1
ξ
i
I
(
t
i
, t
i
+1
]
(
t
)
,
(12.1)
for some positive integer
k
, a finite sequence 0
< t
1
<
· · ·
< t
k
+1
≤
T
, and random variables
ξ
1
, ..., ξ
k
such that
ξ
i
is
F
t
i
measurable.
Definition 12.1
For a simple process
h
, define the Itˆo integral
I
(
h
) =
Z
T
0
h
t
dW
t
=
k
X
i
=1
ξ
i
(
W
t
i
+1

W
t
i
)
.
(12.2)
Proposition 12.1
I
(
h
)
satisfies the following properties:
EI
(
h
) = 0
(12.3)
E
[
I
(
h
)]
2
=
Z
T
0
Eh
2
t
dt
(12.4)
Z
T
0
(
ah
t
+
bh
0
t
)
dW
t
=
a
Z
T
0
h
t
dW
t
+
b
Z
T
0
h
0
t
dW
t
(12.5)
for any constants
a, b
and any simple processes
h
and
h
0
.
I
(
h
)
defined in
(
12.2
)
is
F
t
k
+1
measurable.
Note:
It is a good exercise to show (12.3) and (12.4). In particular, you will see why is crucial to
have independence between
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 Spring '08
 Staff
 Fundamental Theorem Of Calculus, Stochastic process, Wiener process

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