In mathematics, the integral representation theorem for classical Wiener space is a result in the fields of measure theory and stochastic analysis. Essentially, it shows how to decompose a function on classical Wiener space into the sum of its expected value and an Itō integral.
Let
C
0
(
[
0
,
T
]
;
R
)
(or simply
C
0
for short) be classical Wiener space with classical Wiener measure
γ
. If
F
∈
L
2
(
C
0
;
R
)
, then there exists a unique Itō integrable process
α
F
:
[
0
,
T
]
×
C
0
→
R
(i.e. in
L
2
(
B
)
, where
B
is canonical Brownian motion) such that
F
(
σ
)
=
∫
C
0
F
(
p
)
d
γ
(
p
)
+
∫
0
T
α
F
(
σ
)
t
d
σ
t
for
γ
-almost all
σ
∈
C
0
.
In the above,
∫
C
0
F
(
p
)
d
γ
(
p
)
=
E
[
F
]
is the expected value of
F
; and
the integral
∫
0
T
⋯
d
σ
t
is an Itō integral.
The proof of the integral representation theorem requires the Clark-Ocone theorem from the Malliavin calculus.
Let
(
Ω
,
F
,
P
)
be a probability space. Let
B
:
[
0
,
T
]
×
Ω
→
R
be a Brownian motion (i.e. a stochastic process whose law is Wiener measure). Let
{
F
t
|
0
≤
t
≤
T
}
be the natural filtration of
F
by the Brownian motion
B
:
Suppose that
f
∈
L
2
(
Ω
;
R
)
is
F
T
-measurable. Then there is a unique Itō integrable process
a
f
∈
L
2
(
B
)
such that
