Introduction to Stochastic Calculus 4
Example 2 We want to compute
R
T
0
W
t
dW
t
. Towards this end, let 0 = t
n
0
< t
n
1
< t
n
2
< . . . < t
n
n
= T be a
partition of [0, T ] and define
X
n
t
:=
n−1
X
i=0
W
t
n
i
I
[t
n
i
,t
n
i+1
)
(t)
where I
[t
n
i
,t
n
i+1
)
(t) = 1 if t ∈ [t
n
i
, t
n
i+1
) and is 0 otherwise. Then X
n
t
is an adapted elementary process and, by
continuity of Brownian motion, satisfies lim
n→∞
X
n
t
= W
t
almost surely as max
i
|t
n
i+1
− t
n
i
| → 0. The Itˆo
integral of X
n
t
is given by
Z
T
0
X
n
t
dW
t
=
n−1
X
i=0
W
t
n
i
(W
t
n
i+1
− W
t
n
i
)
=
1
2
n−1
X
i=0
³
W
2
t
n
i+1
− W
2
t
n
i
− (W
t
n
i+1
− W
t
n
i
)
2
´
=
1
2
W
2
T
−
1
2
W
2
0
−
1
2
n−1
X
i=0
(W
t
n
i+1
− W
t
n
i
)
2
. (4)
By the definition of quadratic variation the sum on the right-hand-side of (4) converges in probability to T . And
since W
0
= 0 we obtain
Z
T
0
W
t
dW
t
= lim
n→∞
Z
T
0
X
n
t
dW
t
=
1
2
W
2
T
−
1
2
T.
Note that we will generally evaluate stochastic integrals using Itˆo’s Lemma (to be discussed later) without
having to take limits of elementary processes as we did in Example 2.
Definition 8 We define the space L
2
[0, T] to be the space of processes, X
t
(ω), such that
E
"
Z
T
0
X
t
(ω)
2
dt
#
< ∞.
Theorem 4 (Itˆo’s Isometry) For any X
t
(ω) ∈ L
2
[0, T ] we have
E
Ã
Z
T
0
X
t
(ω) dW
t
(ω)
!
2
= E
"
Z
T
0
X
t
(ω)
2
dt
#
.
Proof: (For the case where X
t
is an elementary process)
Let X
t
=
P
i
e
i
(ω)I
[t
i
,t
i+1
)
(t) be an elementary process where the e
i
(ω)’s and t
i
’s are as defined in Definition 6.
We therefore have
R
T
0
X
t
(ω) dW
t
(ω) :=
P
n−1
i=0
e
i
(ω)
¡
W
t
i+1
(ω) − W
t
i
(ω)
¢
We then have
E
Ã
Z
T
0
X
t
(ω) dW
t
(ω)
!
2
= E
Ã
n−1
X
i=0
e
i
(ω)
¡
W
t
i+1
(ω) − W
t
i
(ω)
¢
!
2
=
n−1
X
i=0
E
h
e
2
i
(ω)
¡
W
t
i+1
(ω) − W
t
i
(ω)
¢
2
i
+ 2
n−1
X
0≤i<j≤n−1
E
£
e
i
e
j
(ω)
¡
W
t
i+1
(ω) − W
t
i
(ω)
¢¡
W
t
j+1
(ω) − W
t
j
(ω)
¢¤