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In
for random variables that are given as Itô integrals.
Let
W
:
[
0
,
T
]
×
Ω
→
R
{\displaystyle W:[0,T]\times \Omega \to \mathbb {R} }
denote the canonical real-valued Wiener process defined up to time
T
>
0
{\displaystyle T>0}
, and let
X
:
[
0
,
T
]
×
Ω
→
R
{\displaystyle X:[0,T]\times \Omega \to \mathbb {R} }
be a stochastic process that is adapted to the natural filtration
F
∗
W
{\displaystyle {\mathcal {F}}_{*}^{W}}
of the Wiener process.[clarification needed ] Then
E
[
(
∫
0
T
X
t
d
W
t
)
2
]
=
E
[
∫
0
T
X
t
2
d
t
]
,
{\displaystyle \operatorname {E} \left[\left(\int _{0}^{T}X_{t}\,\mathrm {d} W_{t}\right)^{2}\right]=\operatorname {E} \left[\int _{0}^{T}X_{t}^{2}\,\mathrm {d} t\right],}
where
E
{\displaystyle \operatorname {E} }
denotes
classical Wiener measure
.
In other words, the Itô integral, as a function from the space
L
a
d
2
(
[
0
,
T
]
×
Ω
)
{\displaystyle L_{\mathrm {ad} }^{2}([0,T]\times \Omega )}
of square-integrable adapted processes to the space
L
2
(
Ω
)
{\displaystyle L^{2}(\Omega )}
of square-integrable random variables, is an
inner products
(
X
,
Y
)
L
a
d
2
(
[
0
,
T
]
×
Ω
)
:=
E
(
∫
0
T
X
t
Y
t
d
t
)
{\displaystyle {\begin{aligned}(X,Y)_{L_{\mathrm {ad} }^{2}([0,T]\times \Omega )}&:=\operatorname {E} \left(\int _{0}^{T}X_{t}\,Y_{t}\,\mathrm {d} t\right)\end{aligned}}}
and
(
A
,
B
)
L
2
(
Ω
)
:=
E
(
A
B
)
.
{\displaystyle (A,B)_{L^{2}(\Omega )}:=\operatorname {E} (AB).}
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
]
{\displaystyle \operatorname {E} \left[\left(\int _{0}^{T}X_{t}\,\mathrm {d} W_{t}\right)\left(\int _{0}^{T}Y_{t}\,\mathrm {d} W_{t}\right)\right]=\operatorname {E} \left[\int _{0}^{T}X_{t}Y_{t}\,\mathrm {d} t\right]}
for
X
,
Y
∈
L
a
d
2
(
[
0
,
T
]
×
Ω
)
{\displaystyle X,Y\in L_{\mathrm {ad} }^{2}([0,T]\times \Omega )}
.
References