In mathematics, the Itô isometry, named after Kiyoshi Itô, is a crucial fact about Itô stochastic integrals. One of its main applications is to enable the computation of variances for random variables that are given as Itô integrals.
Let
W
:
[
0
,
T
]
×
Ω
→
R
denote the canonical real-valued Wiener process defined up to time
T
>
0
, and let
X
:
[
0
,
T
]
×
Ω
→
R
be a stochastic process that is adapted to the natural filtration
F
∗
W
of the Wiener process. Then
E
[
(
∫
0
T
X
t
d
W
t
)
2
]
=
E
[
∫
0
T
X
t
2
d
t
]
,
where
E
denotes expectation with respect to classical Wiener measure.
In other words, the Itô integral, as a function from the space
L
a
d
2
(
[
0
,
T
]
×
Ω
)
of square-integrable adapted processes to the space
L
2
(
Ω
)
of square-integrable random variables, is an isometry of normed vector spaces with respect to the norms induced by the inner products
(
X
,
Y
)
L
a
d
2
(
[
0
,
T
]
×
Ω
)
:=
E
(
∫
0
T
X
t
Y
t
d
t
)
and
(
A
,
B
)
L
2
(
Ω
)
:=
E
(
A
B
)
.
As a consequence, the Itô integral respects these inner products as well, i.e. we can write
E
[
(
∫
0
T
X
t
d
W
t
)
(
∫
0
T
Y
t
d
W
t
)
]
=
E
[
∫
0
T
X
t
Y
t
d
t
]
for
X
,
Y
∈
L
a
d
2
(
[
0
,
T
]
×
Ω
)
.
