# Wiener-Itô  Chaos Expansion of Hilbert Space Valued Random Variables

Authors: M. A. Alshanskiy
Journal: Journal of Probability
 (2014)

---

## Abstract

The notion ofn-fold iterated Itô integral with respect to a cylindrical Hilbert space valued Wiener process is introduced and the Wiener-Itô chaos expansion is obtained for a square Bochner integrable Hilbert space valued random variable. The expansion can serve a basis for developing the Hilbert space valued analog of Malliavin calculus of variations which can then be applied to the study of stochastic differential equations in Hilbert spaces and their solutions.


---

## Body

## 1. Introduction

The Wiener-Itô chaos expansion of a square integrable random variable which was first proved in [1] plays fundamental role in Malliavin calculus of variations [2, 3] which appeared to be a powerful instrument in the analysis of functionals of Brownian motion. The Malliavin calculus has found extensive applications to stochastic differential equations arising as models of various random phenomena. One of the important sources of such equations is markets modeling in financial mathematics [4, 5].In the last decades, many researchers’ interest has been drawn to stochastic differential equations in infinite dimensional Hilbert spaces driven by a cylindrical Wiener process or, equivalently, by countable set of Brownian motions [6, 7]. For example, in financial mathematics, such equations are used in modeling interest rates term structure or zero-coupon bond market [8, 9]. The present work was motivated by the need to make the Malliavin calculus applicable to Hilbert space valued stochastic processes. The first step in this direction is to obtain the generalization of the Wiener-Itô chaos expansion for Hilbert space valued random variables.In order to achieve this, we first prove the Hilbert space valued version of the Itô representation theorem in Section2. This generalization is established in Theorem 9 and Corollary 11.In Section3, we introduce iterated Itô integrals and multiple Itô integrals with respect to a cylindrical Wiener process. In the Hilbert space valued case, the integrand of an n-fold iterated Itô integral is a function defined on a certain subset of ℝ
n and taking values in a certain space of Hilbert space valued continuous n-linear forms defined on the nth Cartesian power of the Hilbert space where the Wiener process takes values.Section4 contains main results of the paper which are stated in Theorem 9 and Corollary 11. The proof of the theorem follows the scheme of the proof of the Wiener-Itô chaos expansion in the ℝ-valued case in [5].


## 2. Itô Representation Theorem for Hilbert Space Valued Random Variables

Let(
Ω
,
ℱ
,
P
) be a probability space. For any separable Hilbert space ℍ, we denote by L
2
(
Ω
,
ℱ
,
P
;
ℍ
), or L
2
(
Ω
;
ℍ
) for short, the space of all Bochner square integrable ℍ-valued random variables on (
Ω
,
ℱ
,
P
). Let {
e
j
} be an orthonormal basis in ℍ and let {
β
j
(
t
)
∣
t
∈
ℝ
}
j
=
1
∞ be a sequence of independent identically distributed Brownian motions on the probability space. Consider the corresponding cylinder Wiener process, defined by
(1)
W
(
t
)
=
∑
j
=
1
∞
‍
β
j
(
t
)
e
j
,
t
∈
ℝ
.
It is easy to see that the series is not convergent in L
2
(
Ω
;
ℍ
); however, for any x
∈
ℍ, we have that (
W
(
t
)
,
x
)
:
=
∑
j
=
1
∞
β
j
(
e
j
,
x
)
ℍ is a random variable belonging to L
2
(
Ω
;
ℝ
). Denote by ℬ
t
a
,
t
≥
a the σ-algebra, generated by the Wiener process at [
a
;
t
], that is, the σ-algebra, generated by the set of random variables (
W
(
s
)
,
x
), where a
≤
s
≤
t, x
∈
ℍ. (Note that the series (1) is not convergent in L
2
(
Ω
;
ℍ
) at t
≠
0 although (
W
(
s
)
,
x
)
∈
L
2
(
Ω
;
ℝ
) for any s
∈
ℝ, x
∈
ℍ.) The family {
ℬ
t
a
} is called the filtration generated by the Wiener process W
(
t
)
,
t
≥
a. Note that the Brownian motions β
j
(
t
)
,
j
∈
ℕ
,
t
≥
a are martingales with respect to the filtration ℬ
t
a.LetH be another separable Hilbert space and let {
g
i
}
i
=
1
∞ be an orthonormal basis in H. Then, the family of operators {
g
i
⊗
e
j
}
i
,
j
=
1
∞, defined by the equality
(2)
(
g
i
⊗
e
j
)
x
:
=
g
i
(
e
j
,
x
)
ℍ
,
x
∈
ℍ
,
forms an orthonormal basis in the space ℒ
2
(
ℍ
,
H
) of all Hilbert-Schmidt operators acting from ℍ to H. Any A
∈
ℒ
2
(
ℍ
,
H
) has the decomposition
(3)
A
=
∑
i
,
j
=
1
∞
‍
a
i
j
(
g
i
⊗
e
j
)
,
where a
i
j
=
(
g
i
,
A
e
j
)
H and
(4)
∥
A
∥
ℒ
2
(
ℍ
,
H
)
2
=
∑
i
,
j
=
1
∞
‍
|
a
i
j
|
2
<
∞
.
For any ℒ
2
(
ℍ
;
H
)-valued random process φ
(
t
) adapted to the filtration ℬ
t
a, where a
≤
t
≤
b, satisfying the property
(5)
E
[
∫
a
b
‍
∥
φ
(
t
)
∥
ℒ
2
(
ℍ
,
H
)
2
d
t
]
<
∞
,
the stochastic Itô integral with respect to the cylindrical Wiener process
(6)
∫
a
b
‍
φ
(
t
,
ω
)
d
W
(
t
)
is well-defined and is an element of the space L
2
(
Ω
;
H
) (see the definition and the properties in [6]). Note that if the function φ
(
t
)
=
φ
(
t
,
ω
) is ℬ
(
ℝ
)
×
ℱ-measurable, the equality (5) implies
(7)
E
[
∫
a
b
‍
∥
φ
(
t
)
∥
ℒ
2
(
ℍ
,
H
)
2
d
t
]
=
∫
a
b
‍
E
[
∥
φ
(
t
)
∥
ℒ
2
(
ℍ
,
H
)
2
]
d
t
.Theorem 1.
For anyℬ
b
a-measurable random variable F
∈
L
2
(
Ω
;
H
), there exists a unique ℬ
t
a-adapted ℒ
2
(
ℍ
,
H
)-valued random process φ
(
t
) satisfying (5) such that
(8)
F
=
E
[
F
]
+
∫
a
b
‍
φ
(
t
,
ω
)
d
W
(
t
)
,
∥
F
∥
L
2
(
Ω
;
H
)
2
=
∥
E
[
F
]
∥
H
2
+
E
[
∫
a
b
‍
∥
φ
(
t
)
∥
ℒ
2
(
ℍ
,
H
)
2
d
t
]
.Proof.
For anyi
∈
ℕ, we have F
i
∶
=
(
F
,
g
i
)
H
∈
L
2
(
Ω
;
ℝ
). Therefore, we have the following decomposition for F:
(9)
F
=
∑
i
=
1
‍
F
i
g
i
.
Denote F
i
n
=
E
[
F
i
∣
ℬ
b
a
,
(
n
)
], where ℬ
b
a
,
(
n
) is the filtration generated by the n-dimensional Wiener process
(10)
W
n
(
t
)
=
∑
j
=
1
n
‍
β
j
(
t
)
e
j
,
a
≤
t
≤
b
.
By the finite dimensional Itô representation theorem ([10], Theorem 4.3
.
3.), there exists a unique ℝ
n-valued random process
(11)
φ
i
n
(
t
)
=
[
ν
i
1
(
n
)
(
t
)
,
…
,
ν
i
n
(
n
)
(
t
)
]
,
a
≤
t
≤
b
,
such that the following conditions hold:(i)
all the mappingsν
i
k
(
n
)
:
(
t
,
ω
)
↦
ν
i
k
(
n
)
(
t
,
ω
) are ℬ
(
ℝ
)
×
ℱ-measurable,
(ii)
E
[
∫
a
b
‍
ν
i
k
(
n
)
(
t
)
2
d
t
]
<
∞
,
k
=
1
,
…
,
n,
(iii)
the processesν
i
k
(
n
)
(
t
)
,
k
=
1
,
…
,
n are adapted to the filtration ℬ
t
a.
Moreover, we have(12)
F
i
n
=
E
[
F
i
n
]
+
∫
a
b
‍
φ
i
n
(
t
)
d
B
n
(
t
)
=
E
[
F
i
]
+
∑
k
=
1
n
‍
∫
a
b
‍
ν
i
k
(
n
)
(
t
)
d
β
k
(
t
)
.
Here, B
n
(
t
) denotes the n-dimensional Brownian motion
(13)
B
n
(
t
)
=
[
β
1
(
t
)
⋮
β
n
(
t
)
]
.
Since ℬ
b
a
,
(
n
)
⊆
ℬ
b
a
,
(
m
) for m
>
n, we have
(14)
F
j
n
=
E
[
F
j
∣
ℬ
b
a
,
(
n
)
]
=
E
[
E
[
F
j
∣
ℬ
b
a
,
(
m
)
]
∣
ℬ
b
a
,
(
n
)
]
=
E
[
F
j
m
∣
ℬ
b
a
,
(
n
)
]
=
E
[
E
[
F
i
m
]
+
∑
k
=
1
m
‍
∫
a
b
‍
ν
i
k
(
m
)
(
t
)
d
β
k
(
t
)
∣
ℬ
b
a
,
(
n
)
]
=
E
[
F
i
]
+
∑
k
=
1
n
‍
∫
a
b
‍
ν
i
k
(
m
)
(
t
)
d
β
k
(
t
)
.
It follows from the uniqueness of the representation (12) that ν
i
k
(
m
)
≡
ν
i
k
(
n
) for k
≤
n
<
m. Thus, the equality (12) can be rewritten as
(15)
F
i
n
=
E
[
F
i
n
]
+
∫
a
b
‍
φ
i
n
(
t
)
d
B
n
(
t
)
=
E
[
F
i
]
+
∑
k
=
1
n
‍
∫
a
b
‍
ν
i
k
(
t
)
d
β
k
(
t
)
.
By the Jensen inequality, we have
(16)
F
i
n
2
=
E
[
F
i
∣
ℬ
b
a
,
(
n
)
]
2
≤
E
[
F
i
2
∣
ℬ
b
a
,
(
n
)
]
.
Therefore,
(17)
E
[
F
i
n
2
]
≤
E
[
E
[
F
i
2
∣
ℬ
b
a
,
(
n
)
]
]
=
E
[
F
i
2
]
<
∞
.
Thus,
(18)
E
[
F
i
n
2
]
=
E
[
(
E
[
F
i
]
+
∑
k
=
1
n
‍
∫
a
b
‍
ν
i
k
(
t
)
d
β
k
(
t
)
)
2
]
=
E
[
F
i
]
2
+
∑
k
=
1
n
‍
E
[
∫
a
b
‍
ν
i
k
2
(
t
)
d
t
]
≤
E
[
F
i
2
]
<
∞
.
Consequently,
(19)
∑
k
=
1
∞
‍
E
[
∫
a
b
‍
ν
i
k
2
(
t
)
d
t
]
<
∞
.
Since ℬ
b
a
,
(
n
)
⊆
ℬ
b
a
,
(
n
+
1
) and ℬ
b
a
=
⋃
n
∈
ℕ
‍
ℬ
b
a
,
(
n
), we have
(20)
F
i
=
lim
⁡
n
→
∞
E
[
F
i
∣
ℬ
b
a
,
(
n
)
]
=
E
[
F
i
]
+
∑
k
=
1
∞
‍
∫
a
b
‍
ν
i
k
(
t
)
d
β
k
(
t
)
,
E
[
F
i
2
]
=
lim
⁡
n
→
∞
E
[
F
i
n
2
]
=
E
[
F
i
]
2
+
∑
k
=
1
∞
‍
E
[
∫
a
b
‍
ν
i
k
2
(
t
)
d
t
]
.
For any N
∈
ℕ, it holds that
(21)
E
[
∑
i
=
1
N
‍
F
i
2
]
=
∑
i
=
1
N
‍
E
[
F
i
]
2
+
∑
i
=
1
N
‍
∑
k
=
1
∞
‍
E
[
∫
a
b
‍
ν
i
k
2
(
t
)
d
t
]
≤
E
[
∥
F
∥
H
2
]
.
It follows from (20) that
(22)
∑
i
=
1
∞
‍
F
i
g
i
=
∑
i
=
1
∞
‍
E
[
F
i
]
g
i
+
∑
i
=
1
∞
‍
g
i
∑
k
=
1
∞
‍
∫
a
b
‍
ν
i
k
(
t
)
d
β
k
(
t
)
,
since the series in the right-hand side of the equality are convergent in L
2
(
Ω
;
H
) by the estimate (21). Thus, the equality (8) holds true. It also follows from this estimate by the Levy theorem that we can pass to the limits at N
→
∞ in the equality. As a result, we obtain
(23)
E
[
∥
F
∥
H
2
]
=
∑
i
=
1
∞
‍
E
[
F
i
]
2
+
∑
i
=
1
∞
‍
∑
k
=
1
∞
‍
E
[
∫
a
b
‍
ν
i
k
2
(
t
)
d
t
]
=
∥
E
[
F
]
∥
H
2
+
E
[
∫
a
b
‍
∑
i
,
k
=
1
∞
‍
ν
i
k
2
(
t
)
d
t
]
=
∥
E
[
F
]
∥
H
2
+
E
[
∫
a
b
‍
∥
φ
(
t
)
∥
ℒ
2
(
ℍ
,
H
)
2
d
t
]
,
where
(24)
φ
(
t
)
∶
=
∑
i
,
k
=
1
∞
‍
ν
i
k
(
t
)
(
g
i
⊗
e
k
)
.
By the polarization identity, we obtain the following assertion.Corollary 2.
Letξ
(
t
) and η
(
t
) be ℒ
2
(
ℍ
,
H
)-valued random processes, adapted to the filtration ℬ
t
a
,
a
≤
t
≤
b, such that
(25)
∫
a
b
‍
E
[
∥
ξ
(
t
)
∥
ℒ
2
(
ℍ
,
H
)
2
]
<
∞
,
∫
a
b
‍
E
[
∥
η
(
t
)
∥
ℒ
2
(
ℍ
,
H
)
2
]
<
∞
.
Then,
(26)
E
[
(
∫
a
b
‍
ξ
(
t
)
d
W
(
t
)
,
∫
a
b
‍
η
(
t
)
d
W
(
t
)
)
H
]
=
∫
a
b
‍
E
[
(
ξ
(
t
)
,
η
(
t
)
)
ℒ
2
(
ℍ
,
H
)
]
d
t
.


## 3. Multiple Itô Integrals with respect to the Cylindrical Wiener Process

Letξ
(
t
1
,
t
2
,
ω
) be a function defined for a
≤
t
1
≤
t
2
≤
b, ω
∈
Ω taking values in the space ℒ
2
(
ℍ
;
ℒ
2
(
ℍ
;
H
)
) and ℬ
(
ℝ
2
)
×
ℱ-measurable. If for any t
2
∈
[
a
;
b
] the random process ξ
(
t
,
t
2
)
,
a
≤
t
≤
t
2 is adapted to the filtration ℬ
t
a
,
0
≤
t
≤
t
2 and the condition
(27)
E
[
∫
a
t
2
‍
∥
ξ
(
t
,
t
2
)
∥
ℒ
2
(
ℍ
;
ℒ
2
(
ℍ
;
H
)
)
2
d
t
]
<
∞
holds true, then the Itô integral
(28)
∫
a
t
2
‍
ξ
(
t
1
,
t
2
)
d
W
(
t
1
)
is well defined and is an element of the space L
2
(
Ω
;
ℒ
2
(
ℍ
;
H
)
) for all t
2
∈
[
a
;
b
]. As a function of t
2, it is an ℒ
2
(
ℍ
;
H
)-valued random process adapted to the filtration ℬ
t
a. We also have
(29)
E
[
∥
∫
a
t
2
‍
ξ
(
t
1
,
t
2
)
d
W
(
t
1
)
∥
ℒ
2
(
ℍ
;
H
)
2
]
=
E
[
∫
a
t
2
‍
∥
ξ
(
t
1
,
t
2
)
∥
ℒ
2
(
ℍ
;
ℒ
2
(
ℍ
;
H
)
)
2
d
t
1
]
<
∞
.
If moreover we have
(30)
E
[
∫
a
b
‍
∫
a
t
2
‍
∥
ξ
(
t
1
,
t
2
)
∥
ℒ
2
(
ℍ
;
ℒ
2
(
ℍ
;
H
)
)
2
d
t
1
d
t
2
]
<
∞
,
then the following iterated Itô integral is well defined:
(31)
∫
a
b
‍
∫
a
t
2
‍
ξ
(
t
1
,
t
2
)
d
W
(
t
1
)
d
W
(
t
2
)
and it satisfies the equality
(32)
E
[
∥
∫
a
b
‍
∫
a
t
2
‍
ξ
(
t
1
,
t
2
)
d
W
(
t
1
)
d
W
(
t
2
)
∥
H
2
]
=
E
[
∫
a
b
‍
∫
a
t
2
‍
∥
ξ
(
t
1
,
t
2
)
∥
ℒ
2
(
ℍ
;
H
)
2
d
t
1
d
t
2
]
.
Note that the operators g
i
⊗
e
j
⊗
e
k defined by the equality
(33)
(
g
i
⊗
e
j
⊗
e
k
)
x
∶
=
(
g
i
⊗
e
j
)
(
e
k
,
x
)
ℍ
,
i
,
j
,
k
∈
ℕ
form an orthonormal basis in the space ℒ
2
(
ℍ
;
ℒ
2
(
ℍ
;
H
)
) and any operator A
∈
ℒ
2
(
ℍ
;
ℒ
2
(
ℍ
;
H
)
) has the following decomposition:
(34)
A
=
∑
i
,
j
,
k
‍
a
i
j
k
(
g
i
⊗
e
j
⊗
e
k
)
,
∥
A
∥
ℒ
2
(
ℍ
;
ℒ
2
(
ℍ
;
H
)
)
2
=
∑
i
,
j
,
k
‍
a
i
j
k
2
<
∞
.
We can identify the decomposition (34) with the H-valued bilinear form on ℍ
×
ℍ defined by the equality (
g
i
⊗
e
j
⊗
e
k
)
(
x
,
y
)
∶
=
g
i
(
e
j
,
x
)
ℍ
(
e
k
,
y
)
ℍ. Thus, the space ℒ
2
(
ℍ
;
ℒ
2
(
ℍ
;
H
)
) can be identified with the Hilbert space ℒ
2
2
(
ℍ
×
ℍ
;
H
) of all continuous bilinear forms A of the form (34) with the norm generated by the scalar product
(35)
(
A
,
B
)
ℒ
2
2
(
ℍ
×
ℍ
;
H
)
2
=
∑
i
,
j
,
k
‍
a
i
j
k
b
i
j
k
,
where A
=
∑
i
,
j
,
k
‍
a
i
j
k
(
g
i
⊗
e
j
⊗
e
k
), B
=
∑
i
,
j
,
k
‍
b
i
j
k
(
g
i
⊗
e
j
⊗
e
k
). So, the iterated integral (31) with respect to the cylindrical Wiener process W
(
t
) is well defined for any ℒ
2
2
(
ℍ
×
ℍ
;
H
)-valued ℬ
(
ℝ
2
)
×
ℱ-measurable function ξ
(
t
1
,
t
2
,
ω
), defined on S
2
a
,
b
×
Ω, where
(36)
S
2
a
,
b
=
{
(
t
1
,
t
2
)
∣
a
≤
t
1
≤
t
2
≤
b
}
,
if it satisfies the condition (30).One can easily extend the above definition to the case of arbitraryn
>
2, defining the n times iterated integral
(37)
J
n
(
ξ
)
=
∫
a
b
‍
∫
a
t
2
‍
…
∫
a
t
n
-
1
‍
ξ
(
t
1
,
t
2
,
…
,
t
n
,
ω
)
d
W
(
t
1
)
,
…
,
d
W
(
t
n
)
inductively for any function ξ
:
S
n
×
Ω
→
ℒ
2
n
(
ℍ
×
n
;
H
), where
(38)
S
n
a
,
b
=
{
(
t
1
,
…
,
t
n
)
∣
a
≤
t
1
≤
,
…
,
t
n
≤
b
}
,
ℒ
2
n
(
ℍ
×
n
;
H
) is the space of all continuous H-valued n-linear forms on ℍ
×
⋯
×
ℍ having form
(39)
A
=
∑
i
,
k
1
,
…
,
k
n
‍
a
i
,
k
1
,
…
,
k
n
(
g
i
⊗
e
k
1
⊗
,
…
,
⊗
e
k
n
)
such that
(40)
∥
A
∥
ℒ
2
n
(
ℍ
×
n
;
H
)
2
=
∑
i
,
k
1
,
…
,
k
n
‍
a
i
,
k
1
,
…
,
k
n
2
<
∞
,
if it satisfies the following conditions: (n-i)
ξ is ℬ
(
ℝ
n
)
×
ℱ-measurable;
(n-ii)
ξ
(
t
1
,
t
2
,
…
,
t
n
) is ℬ
t
1
a-measurable for any (
t
1
,
t
2
,
…
,
t
n
)
∈
S
n
a
,
b;
(n-iii)
E
[
∫
a
b
‍
∫
a
t
2
‍
⋯
∫
a
t
n
-
1
‍
∥
ξ
(
t
1
,
…
,
t
n
)
∥
L
2
n
(
H
×
n
;
H
)
2
d
t
1
,
…
,
d
t
n
]
<
∞.For the defined iterated integral, we have(41)
E
[
∥
J
n
(
ξ
)
∥
H
2
]
=
∫
a
b
‍
∫
a
t
2
‍
⋯
∫
a
t
n
-
1
‍
E
[
∥
ξ
(
t
1
,
…
,
t
n
)
∥
ℒ
2
n
(
ℍ
×
n
;
H
)
2
]
d
t
1
,
…
,
d
t
n
.
We state without proof the next two lemmas which are straightforward generalizations of the corresponding properties of the ℝ-valued iterated Itô integrals.Lemma 3.
Letξ
:
S
n
1
a
,
b
×
Ω
→
ℒ
2
n
1
(
ℍ
×
n
1
;
H
) and η
:
S
n
2
a
,
b
×
Ω
→
ℒ
2
n
2
(
ℍ
×
n
2
;
H
) satisfy the conditions   (n-i),(n-ii) and (n-iii) with n
=
n
1 and n
=
n
2 correspondingly. Then,
(42)
E
[
(
J
n
1
(
ξ
)
,
J
n
2
(
η
)
)
H
]
=
{
0
,
n
1
≠
n
2
,
(
ξ
,
η
)
L
2
(
S
n
a
,
b
;
ℒ
2
n
(
ℍ
×
n
;
H
)
)
,
n
1
=
n
2
=
n
.Lemma 4.
Letξ
:
S
n
a
,
b
×
Ω
→
ℒ
2
n
(
ℍ
×
n
;
H
) satisfy the conditions (n-i),(n-ii) and (n-iii). Then,
(43)
E
[
J
n
(
ξ
)
|
ℬ
t
a
,
b
]
=
∫
a
t
‍
∫
a
t
2
‍
⋯
∫
a
t
n
-
1
‍
ξ
(
t
1
,
t
2
,
…
,
t
n
,
ω
)
d
W
(
t
1
)
,
…
,
d
W
(
t
n
)
.
Denote by L
^
2
(
[
a
;
b
]
n
;
ℒ
2
n
(
ℍ
×
n
;
H
)
) the space of all symmetric functions f
∈
:
[
a
;
b
]
n
→
ℒ
2
n
(
ℍ
×
n
;
H
) satisfying the condition
(44)
∥
f
∥
L
2
(
[
a
;
b
]
n
;
ℒ
2
n
(
ℍ
×
n
;
H
)
)
2
=
∫
‍
⋯
[
a
;
b
]
n
∫
‍
∥
f
n
(
t
1
,
…
,
t
n
)
∥
ℒ
2
n
(
ℍ
×
n
;
H
)
2
d
t
1
,
…
,
d
t
n
<
∞
.Definition 5.
For anyf
∈
L
^
2
(
[
a
;
b
]
n
;
ℒ
2
n
(
ℍ
×
n
;
H
)
), define the multiple n-fold Itô integral by the equality
(45)
I
n
(
f
)
∶
=
n
!
J
n
(
f
)
,
if the right-hand side iterated integral exists.


## 4. The Decomposition Theorem

To establish the main result, we need a few lemmas.Lemma 6.
The set of random variables(46)
{
φ
(
β
j
1
(
t
i
1
)
,
…
,
β
j
n
(
t
i
n
)
)
|
j
k
∈
ℕ
,
t
k
∈
[
a
;
b
]
,
s
s
φ
∈
C
0
∞
(
ℝ
n
)
,
n
∈
ℕ
t
i
n
t
i
n
}
is dense in L
2
(
ℬ
b
a
,
P
).Proof.
Let{
t
i
}
i
=
1
∞ be a dense subset in [
a
;
b
]. Let {
ζ
k
}
k
=
1
∞ be a fixed ordering of the countable set of random variables {
β
j
(
t
i
)
}
j
,
i
=
1
∞. Denote by ℋ
p be the σ-algebra generated by {
ζ
k
}
k
=
1
p. We have ℋ
p
⊂
ℋ
p
+
1 for all p
∈
ℕ and ℬ
b
a is the smallest σ-algebra containing all ℋ
p.
For anyg
∈
L
2
(
ℬ
b
a
,
P
), we have
(47)
g
=
E
(
g
∣
ℬ
b
a
)
=
lim
⁡
p
→
∞
E
(
g
∣
ℋ
p
)
,
where the limit is pointwise a.e. with respect to P and in L
2
(
ℬ
b
a
,
P
). By the Doob-Dynkin lemma for any p
∈
ℕ, there exists a Borel function g
p
:
ℝ
p
→
ℝ such that
(48)
E
[
g
∣
ℋ
p
]
=
g
p
(
ζ
1
,
…
,
ζ
p
)
.
Let P
ζ
1
,
…
,
ζ
p be the probability measure on ℝ
p generated by ζ
1
,
…
,
ζ
p. Since g
p can be approximated in L
2
(
ℝ
p
,
P
ζ
1
,
…
,
ζ
p
) by functions φ
∈
C
0
∞
(
ℝ
p
), the assertion follows.Consider the following exponential functionals of Brownian motionsβ
j:
(49)
ℰ
θ
j
,
a
,
b
∶
=
exp
⁡
[
∫
a
b
‍
θ
(
t
)
d
β
j
(
t
)
-
|
θ
|
0
2
2
]
,
s
s
s
s
s
s
s
s
s
s
s
s
s
j
∈
ℕ
,
θ
∈
L
2
[
a
;
b
]
.Lemma 7.
The linear span of the set(50)
{
∏
k
=
1
p
‍
ℰ
θ
k
j
k
,
a
,
b
|
j
k
∈
ℕ
,
θ
k
∈
L
2
[
a
;
b
]
,
k
=
1
,
…
,
p
,
p
∈
ℕ
}
is dense in L
2
(
ℬ
b
a
,
P
).Proof.
Letg
∈
L
2
(
ℬ
b
a
,
P
) be orthogonal to all functions of the family (50). Take θ
k
(
t
)
=
λ
k
1
[
a
;
t
k
], where {
t
k
}
k
=
1
p
⊂
[
a
;
b
], λ
¯
=
(
λ
1
,
…
,
λ
p
)
∈
ℝ
p. Then, we have
(51)
∏
k
=
1
p
‍
ℰ
θ
k
j
k
,
a
,
b
=
∏
k
=
1
p
‍
exp
⁡
[
λ
k
∫
a
t
k
‍
d
β
j
k
(
t
)
-
t
k
-
a
2
]
=
exp
⁡
[
-
1
2
∑
k
=
1
p
‍
(
t
k
-
a
)
]
·
exp
⁡
[
∑
k
=
1
p
‍
λ
k
β
j
k
(
t
k
)
]
.
It follows that
(52)
G
(
λ
¯
)
∶
=
∫
Ω
‍
exp
⁡
[
∑
k
=
1
p
‍
λ
k
β
j
k
(
t
k
)
]
g
(
ω
)
d
P
(
ω
)
=
0
for all p
∈
ℕ, λ
¯
∈
ℝ
p, j
1
,
…
,
j
p
∈
ℕ, t
1
,
…
,
t
p
∈
[
a
;
b
].
Fixp
,
j
1
,
…
,
j
p
,
t
1
,
…
,
t
p. The corresponding function G
(
λ
¯
) is real analytic. Consider its analytic extension onto ℂ
p. It is also equal to zero. Taking λ
¯
=
(
i
y
1
,
…
,
i
y
p
), we obtain
(53)
∫
Ω
‍
exp
⁡
[
i
∑
k
=
1
p
‍
y
k
β
j
k
(
t
k
)
]
g
(
ω
)
d
P
(
ω
)
=
0
.
For any φ
∈
C
0
∞
(
ℝ
p
) denoting by φ
^ its Fourier transform, we obtain
(54)
∫
Ω
‍
φ
(
β
j
1
(
t
1
)
,
…
,
β
j
p
(
t
p
)
)
g
(
ω
)
d
P
(
ω
)
=
∫
Ω
‍
1
(
2
π
)
p
/
2
ssssssss
×
∫
ℝ
p
‍
φ
^
(
y
¯
)
exp
⁡
[
i
∑
k
=
1
p
‍
y
k
β
j
k
(
t
k
)
]
d
y
¯
g
(
ω
)
d
P
(
ω
)
=
1
(
2
π
)
p
/
2
∫
ℝ
p
‍
φ
^
(
y
¯
)
×
∫
Ω
‍
exp
⁡
[
i
∑
k
=
1
p
‍
y
k
β
j
k
(
t
k
)
]
g
(
ω
)
d
P
(
ω
)
d
y
¯
=
0
.
Thus, g is orthogonal to a dense subset in L
2
(
ℬ
b
a
,
P
).Lemma 8.
For anyα
1
,
…
,
α
n
∈
ℕ
∪
{
0
}, θ
1
,
…
,
θ
n
∈
𝒮, the product
(55)
∏
j
=
1
n
‍
h
α
j
(
∫
a
b
‍
θ
j
(
t
)
d
β
j
(
t
)
|
θ
j
·
1
[
a
;
b
]
|
0
)
is a linear combination of the iterated Itô integrals of the form
(56)
∫
a
b
‍
∫
a
t
p
‍
⋯
∫
a
t
2
‍
θ
j
1
(
t
1
)
,
…
,
θ
j
p
(
t
p
)
d
β
j
1
(
t
1
)
,
…
,
d
β
j
p
(
t
p
)
,
where p
=
α
1
+
⋯
+
α
n and j
1
,
…
,
j
p
∈
{
1
,
…
,
n
}.Proof.
We use induction with respect top. The assertion is evident for p
=
1. Suppose it is true for some p
∈
ℕ. Let α
1
+
⋯
+
α
n
=
p
+
1. We introduce the following notation:
(57)
X
j
(
t
)
∶
=
|
θ
j
·
1
[
a
;
t
]
|
0
α
j
α
j
!
h
α
j
(
∫
a
t
‍
θ
j
(
τ
)
d
β
j
(
τ
)
|
θ
j
·
1
[
a
;
t
]
|
0
)
=
∫
a
t
‍
∫
a
t
α
j
‍
⋯
∫
a
t
2
‍
θ
j
(
t
1
)
,
…
,
θ
j
(
t
α
j
)
d
β
j
(
t
1
)
,
…
,
d
β
j
(
t
α
j
)
=
:
∫
a
t
‍
Y
j
(
t
α
j
,
ω
)
d
β
j
(
t
α
j
)
,
j
=
1
,
…
,
n
.
This means that X
j satisfy the following stochastic differential equations:
(58)
d
X
j
(
t
)
=
Y
j
(
t
,
ω
)
d
β
j
(
t
α
j
)
,
j
=
1
,
…
,
n
,
and we have
(59)
Y
j
(
t
)
=
|
θ
j
·
1
[
a
;
t
]
|
0
α
j
-
1
(
α
j
-
1
)
!
h
α
j
-
1
(
∫
a
t
‍
θ
j
(
τ
)
d
β
j
(
τ
)
|
θ
j
·
1
[
a
;
t
]
|
0
)
.
By the Itô formula, we obtain
(60)
d
(
∏
j
=
1
n
‍
X
j
(
t
)
)
=
∑
j
=
1
n
‍
[
∏
i
≠
j
‍
X
i
(
t
)
]
d
X
j
(
t
)
+
∑
i
≠
j
‍
d
X
i
(
t
)
·
d
X
j
(
t
)
.
Since d
X
i
(
t
)
·
d
X
j
(
t
)
=
Y
i
(
t
,
ω
)
Y
j
(
t
,
ω
)
d
β
i
(
t
)
d
β
j
(
t
)
=
0, we come to the equality
(61)
∏
j
=
1
n
‍
X
j
(
b
)
=
∑
j
=
1
n
‍
∫
a
b
‍
[
∏
i
≠
j
‍
X
i
(
t
)
]
Y
j
(
t
,
ω
)
d
β
j
(
t
)
=
∑
j
=
1
n
‍
∫
a
b
‍
[
∏
i
≠
j
‍
|
θ
j
·
1
[
a
;
t
]
|
0
α
j
α
j
h
α
j
(
∫
a
t
‍
θ
j
(
τ
)
d
β
j
(
τ
)
)
s
s
s
s
s
s
s
s
s
s
·
|
θ
j
·
1
[
a
;
t
]
|
0
α
j
-
1
(
α
j
-
1
)
!
h
α
j
-
1
(
∫
a
t
‍
θ
j
(
τ
)
d
β
j
(
τ
)
|
θ
j
·
1
[
a
;
t
]
|
0
)
]
×
θ
j
(
t
)
d
β
j
(
t
)
.
Since ∑
i
≠
j
‍
α
i
+
α
j
-
1
=
p, we can apply the assumption to the products in the brackets, thus completing the proof.Theorem 9.
Anyℬ
b
a-measurable random variable F
∈
L
2
(
Ω
;
H
) has the unique decomposition
(62)
F
=
∑
n
=
0
∞
‍
I
n
(
f
n
)
,
where f
n
∈
L
^
2
(
[
a
;
b
]
n
;
ℒ
2
n
(
ℍ
×
n
;
H
)
)
,
n
∈
ℕ, I
0
(
f
0
)
:
=
f
0
∈
H. It holds that
(63)
∥
F
∥
L
2
(
Ω
;
H
)
2
=
∑
n
=
0
∞
‍
n
!
∥
f
n
∥
L
2
(
[
a
;
b
]
n
;
ℒ
2
n
(
ℍ
×
n
;
H
)
)
2
,
where ∥
f
0
∥
L
2
(
[
a
;
b
]
0
;
ℒ
2
0
(
ℍ
×
0
;
H
)
)
2
∶
=
∥
f
0
∥
H
2.Proof.
By Theorem1, we have
(64)
F
=
E
[
F
]
+
∫
a
b
‍
φ
1
(
t
1
,
ω
)
d
W
(
t
1
)
,
where φ
1
(
t
) is a ℒ
2
1
(
ℍ
;
H
)-valued random variable, ℬ
t
a-measurable for any t. For its norm, we have
(65)
∥
F
∥
L
2
(
Ω
;
H
)
2
=
∥
E
[
F
]
∥
H
2
+
E
[
∫
a
b
‍
∥
φ
1
(
t
)
∥
ℒ
2
(
ℍ
;
H
)
2
d
t
1
]
.
Let ψ
0
∶
=
E
[
F
]. For any t
1
∈
[
a
,
b
] by Theorem 1, taking t
1 for b and ℒ
2
(
ℍ
;
H
) for H, we obtain the following representation:
(66)
φ
1
(
t
1
)
=
E
[
φ
1
(
t
1
)
]
+
∫
a
t
1
‍
φ
2
(
t
1
,
t
2
,
ω
)
d
W
(
t
2
)
,
where φ
2
:
S
2
a
,
b
×
Ω
→
ℒ
2
(
ℍ
;
ℒ
2
(
ℍ
;
H
)
)
≅
ℒ
2
2
(
ℍ
×
2
;
H
), with the following equality for the norm:
(67)
∥
φ
1
(
t
1
)
∥
ℒ
2
(
Ω
;
ℒ
2
(
ℍ
;
H
)
)
2
=
∥
E
[
φ
1
(
t
1
)
]
∥
ℒ
2
(
ℍ
;
H
)
2
+
E
[
∫
a
t
1
‍
∥
φ
2
(
t
1
,
t
2
)
∥
ℒ
2
2
(
ℍ
×
2
;
H
)
2
d
t
2
]
.
Set ψ
1
(
t
1
)
∶
=
E
[
φ
1
(
t
1
)
]. Substituting the representation (66) into (64), we come to the equality
(68)
F
=
J
0
(
ψ
0
)
+
∫
a
b
‍
ψ
1
(
t
1
)
d
W
(
t
1
)
+
∫
a
b
‍
∫
a
t
1
‍
φ
2
(
t
1
,
t
2
,
ω
)
d
W
(
t
2
)
d
W
(
t
1
)
.
From (65) and (67), it follows that
(69)
∥
F
∥
L
2
(
Ω
;
H
)
2
=
∥
ψ
0
∥
H
2
+
∫
a
b
‍
∥
ψ
1
(
t
1
)
∥
ℒ
2
1
(
ℍ
;
H
)
2
d
t
1
+
E
[
∫
a
b
‍
∫
a
t
1
‍
∥
φ
2
(
t
1
,
t
2
)
∥
ℒ
2
2
(
ℍ
×
2
;
H
)
2
d
t
2
d
t
1
]
=
∥
ψ
0
∥
H
2
+
∫
a
b
‍
∥
ψ
1
(
t
1
)
∥
ℒ
2
1
(
ℍ
;
H
)
2
d
t
1
+
∫
a
b
‍
∫
a
t
1
‍
E
[
∥
φ
2
(
t
1
,
t
2
)
∥
ℒ
2
2
(
ℍ
×
2
;
H
)
2
]
d
t
2
d
t
1
.
Applying further in similar manner Theorem 1 and setting
(70)
ψ
n
(
t
1
,
…
,
t
n
)
∶
=
E
[
φ
n
(
t
1
,
…
,
t
n
)
]
∈
ℒ
2
n
(
ℍ
×
n
;
H
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
(
t
1
,
…
,
t
n
)
∈
S
n
a
,
b
at the nth step of this process, we obtain
(71)
F
=
J
0
(
ψ
0
)
+
J
1
(
ψ
1
)
+
⋯
+
J
n
(
ψ
n
)
+
J
n
+
1
(
φ
n
+
1
)
with
(72)
∥
F
∥
L
2
(
Ω
;
H
)
2
=
∥
ψ
0
∥
H
2
+
∫
a
b
‍
∥
ψ
1
(
t
1
)
∥
ℒ
2
1
(
ℍ
;
H
)
2
d
t
1
+
∫
a
b
‍
∫
a
t
1
‍
∥
ψ
2
(
t
1
,
t
2
)
∥
ℒ
2
2
(
ℍ
×
2
;
H
)
2
d
t
2
d
t
1
+
⋯
+
∫
a
b
‍
∫
a
t
1
‍
…
∫
a
t
n
-
1
‍
∥
ψ
n
(
t
1
,
…
,
t
n
)
∥
ℒ
2
n
(
ℍ
×
n
;
H
)
2
d
t
n
,
…
,
d
t
1
+
∫
a
b
‍
∫
a
t
1
‍
…
∫
a
t
n
‍
E
[
∥
φ
n
+
1
(
t
1
,
…
,
t
n
+
1
)
∥
ℒ
2
n
+
1
(
ℍ
×
(
n
+
1
)
;
H
)
2
]
d
t
n
+
1
,
…
,
d
t
1
.
It follows from here that
(73)
∥
ψ
0
∥
H
2
+
∑
k
=
1
n
‍
∫
a
b
‍
∫
a
t
1
‍
⋯
∫
a
t
k
-
1
‍
∥
ψ
k
(
t
1
,
…
,
t
k
)
∥
ℒ
2
k
(
ℍ
×
k
;
H
)
2
d
t
k
,
…
,
d
t
1
≤
∥
F
∥
L
2
(
Ω
;
H
)
2
and consequently
(74)
∥
ψ
0
∥
H
2
+
∑
k
=
1
∞
‍
∫
a
b
‍
∫
a
t
1
‍
⋯
∫
a
t
k
-
1
‍
∥
ψ
k
(
t
1
,
…
,
t
k
)
∥
ℒ
2
k
(
ℍ
×
k
;
H
)
2
d
t
k
,
…
,
d
t
1
<
∞
.
This means that the series
(75)
∑
n
=
0
∞
‍
J
n
(
ψ
n
)
is convergent in the space L
2
(
Ω
;
H
) and
(76)
Φ
∶
=
lim
⁡
n
→
∞
J
n
+
1
(
φ
n
+
1
)
is an element of this space. By Lemma 3, the integral J
n
+
1
(
φ
n
+
1
) is orthogonal in L
2
(
Ω
;
H
) to the integrals J
0
(
ψ
0
)
,
…
,
J
n
(
ψ
n
) for all ψ
0
∈
H, ψ
1
∈
L
2
(
S
1
a
,
b
;
ℒ
2
1
(
ℍ
;
H
)
)
,
…
,
ψ
n
∈
L
2
(
S
n
a
,
b
;
ℒ
2
n
(
ℍ
×
n
;
H
)
). It follows that Φ is orthogonal to J
n
(
ψ
n
) for all n
∈
ℕ
∪
{
0
}, ψ
n
∈
L
2
(
S
n
a
,
b
;
ℒ
2
n
(
ℍ
×
n
;
H
)
).
Note that by the well-known connection between iterated Itô integrals with respect to a Brownian motion and the Hermite polynomials, we have(77)
J
n
(
g
i
⊗
e
j
⊗
n
θ
⊗
n
)
=
g
i
∫
a
b
‍
∫
a
t
1
‍
⋯
∫
a
t
n
-
1
‍
θ
⊗
n
(
t
1
,
…
,
t
n
)
d
β
j
(
t
n
)
…
d
β
j
(
t
1
)
=
g
i
|
θ
·
1
[
a
;
b
]
|
0
n
h
n
(
∫
a
b
‍
θ
(
t
)
d
β
j
(
t
)
|
θ
·
1
[
a
;
b
]
|
0
)
for any n
,
i
,
j
∈
ℕ, θ
∈
𝒮.
LetΦ
=
∑
i
=
1
∞
Φ
i
g
i, where Φ
i
∈
L
2
(
Ω
;
ℝ
). For any n
,
i
,
j
1
,
…
,
j
n
∈
ℕ and any θ
j
1
,
…
,
θ
j
n
∈
𝒮, we have
(78)
E
[
Φ
i
·
∫
a
b
‍
∫
a
t
n
‍
⋯
∫
a
t
2
‍
θ
j
1
(
t
1
)
,
…
,
θ
j
n
(
t
n
)
d
β
j
1
(
t
1
)
,
…
,
d
β
j
n
(
t
n
)
∫
a
b
‍
]
=
E
[
(
Φ
,
g
i
∫
a
b
‍
∫
a
t
n
‍
⋯
∫
a
t
2
‍
θ
j
1
(
t
1
)
,
…
,
θ
j
n
(
t
n
)
d
β
j
1
(
t
1
)
,
…
,
d
β
j
n
(
t
n
)
∫
a
b
‍
∫
a
t
n
‍
)
H
]
=
(
Φ
,
J
n
(
g
i
⊗
e
j
1
⊗
⋯
⊗
e
j
n
⊗
θ
j
1
⊗
⋯
⊗
θ
j
n
)
)
=
0
,
from where, by Lemma 8 and the equality (77), it follows that, for any i
∈
ℕ, Φ
i is orthogonal in L
2
(
Ω
;
ℝ
) to the products of the form
(79)
∏
j
=
1
n
‍
h
α
j
(
∫
a
b
‍
θ
j
(
t
)
d
β
j
(
t
)
|
θ
j
·
1
[
a
;
b
]
|
0
)
.
It follows from here and the connection between the Hermite polynomials and general powers that
(80)
E
[
Φ
i
∏
j
=
1
n
‍
(
∫
a
b
‍
θ
j
(
t
)
d
β
j
(
t
)
)
α
j
]
=
0
.
It follows from the expansion
(81)
ℰ
θ
j
,
a
,
b
=
exp
⁡
[
∫
a
b
‍
θ
(
t
)
d
β
j
(
t
)
-
|
θ
|
0
2
2
]
=
e
-
(
|
θ
|
0
2
/
2
)
∑
k
=
0
∞
‍
(
∫
a
b
‍
θ
(
t
)
d
β
j
(
t
)
)
k
k
!
that a product of the form
(82)
∏
k
=
1
n
‍
ℰ
θ
k
j
k
,
a
,
b
has a decomposition into the series of the products of the form
(83)
∏
j
=
1
n
‍
(
∫
a
b
‍
θ
j
(
t
)
d
β
j
(
t
)
)
α
j
,
α
j
∈
ℕ
∪
{
0
}
,
j
=
1
,
…
,
n
,
convergent in the space L
2
(
Ω
;
ℝ
). Consequently, Φ
i is orthogonal to any product of the form (82). By Lemma 7, this means that Φ
i
=
0 for any i
∈
ℕ; that is, Φ
=
0. Passing to the limit in the equality (71), we obtain
(84)
F
=
∑
n
=
0
∞
‍
J
n
(
ψ
n
)
.
Setting the functionsψ
n to be equal to zero on [
a
;
b
]
n
∖
S
n
a
,
b and defining f
n
∶
=
ψ
^
n, where ψ
^
n is the symmetrization of ψ
n, we obtain (62). By the equality
(85)
∥
f
n
∥
L
2
(
[
a
;
b
]
n
;
ℒ
2
n
(
ℍ
×
n
;
H
)
)
2
=
∫
‍
⋯
[
a
;
b
]
n
∫
‍
∥
f
n
(
t
1
,
…
,
t
n
)
∥
ℒ
2
n
(
ℍ
×
n
;
H
)
2
d
t
1
,
…
,
d
t
n
=
n
!
∫
‍
⋯
S
n
a
,
b
∫
‍
∥
f
n
(
t
1
,
…
,
t
n
)
∥
ℒ
2
n
(
ℍ
×
n
;
H
)
2
d
t
1
,
…
,
d
t
n
=
n
!
∫
‍
⋯
S
n
a
,
b
∫
‍
∥
ψ
n
(
t
1
,
…
,
t
n
)
n
!
∥
ℒ
2
n
(
ℍ
×
n
;
H
)
2
d
t
1
,
…
,
d
t
n
=
1
n
!
∫
‍
⋯
S
n
a
,
b
∫
‍
∥
ψ
n
(
t
1
,
…
,
t
n
)
∥
ℒ
2
n
(
ℍ
×
n
;
H
)
2
d
t
1
,
…
,
d
t
n
,
passing to the limit in (72), we obtain (63).Definition 10.
For anyf
∈
L
^
2
(
ℝ
n
;
ℒ
2
n
(
ℍ
×
n
;
H
)
), define the n-fold Itô integral I
n
(
f
) by the equality (45), where the iterated integral
(86)
J
n
(
f
)
=
∫
-
∞
+
∞
‍
∫
-
∞
t
2
‍
⋯
∫
-
∞
t
n
-
1
‍
ξ
(
t
1
,
t
2
,
…
,
t
n
,
ω
)
d
W
(
t
1
)
,
…
,
d
W
(
t
n
)
is understood as the limit of the integrals J
n
(
f
·
1
[
-
A
;
A
]
n
) at A
→
+
∞ in the space L
2
(
Ω
;
H
) if the limit exists.Corollary 11.
AnyF
∈
L
2
(
Ω
;
H
) can be uniquely decomposed into the series
(87)
F
=
∑
n
=
0
∞
‍
I
n
(
f
n
)
,
where f
n
∈
L
^
2
(
ℝ
n
;
ℒ
2
n
(
ℍ
×
n
;
H
)
)
,
n
∈
ℕ, I
0
(
f
0
)
∶
=
f
0
∈
H. The equality
(88)
∥
F
∥
L
2
(
Ω
;
H
)
2
=
∑
n
=
0
∞
‍
n
!
∥
f
n
∥
L
2
(
ℝ
n
;
ℒ
2
n
(
ℍ
×
n
;
H
)
)
2
holds true, where ∥
f
0
∥
L
2
(
ℝ
0
;
ℒ
2
0
(
ℍ
×
0
;
H
)
)
2
∶
=
∥
f
0
∥
H
2.Proof.
LetF
N
=
E
[
F
∣
ℬ
N
-
N
]
,
N
∈
ℕ. For any N
∈
ℕ, it can be uniquely represented in the form
(89)
F
N
=
∑
n
=
0
∞
‍
I
n
(
f
N
,
n
)
=
∑
n
=
0
∞
‍
J
n
(
ψ
N
,
n
)
,
where ψ
N
,
n
∈
L
2
(
S
n
-
N
,
N
;
ℒ
2
n
(
ℍ
×
n
;
H
)
) and f
N
,
n is the symmetrization of the function ψ
N
,
n set to be equal to zero at the set [
-
N
;
N
]
n
∖
S
n
-
N
,
N. For its norm, we have
(90)
∥
F
N
∥
L
2
(
Ω
;
H
)
2
=
∑
n
=
0
∞
‍
n
!
∥
f
N
,
n
∥
L
2
(
[
a
;
b
]
n
;
ℒ
2
n
(
ℍ
×
n
;
H
)
)
2
=
∑
n
=
0
∞
‍
∥
ψ
N
,
n
∥
L
2
(
S
n
-
N
,
N
;
ℒ
2
n
(
ℍ
×
n
;
H
)
)
2
.
Suppose that the functions f
N
,
n and ψ
N
,
n are defined everywhere in ℝ
n and are equal to zero on ℝ
n
∖
[
-
N
,
N
]
n and ℝ
n
∖
S
n
-
N
′
,
N
′
, correspondingly.
It follows from the equalityE
[
F
N
′
′
|
ℬ
N
′
-
N
′
]
=
F
N
′ where N
′
′
>
N
′, from the uniqueness of the representation (89), and from the assertion of Lemma 4 that the equality ψ
N
′
,
n
=
ψ
N
′
′
,
n
·
1
S
n
-
N
′
,
N
′ holds true and therefore f
N
′
,
n
=
f
N
′
′
,
n
·
1
[
-
N
′
,
N
′
]
n for N
′
′
>
N
′ and any n
∈
ℕ. It follows from here that for any n
∈
ℕ there exists a unique function ψ
n such that for any N
∈
ℕ the equality ψ
n
=
ψ
N
,
n
·
1
S
n
-
N
′
,
N
′ holds true and there exists a unique function f
n such that for any N
∈
ℕ it holds f
n
=
f
N
,
n
·
1
[
-
N
′
,
N
′
]
n. By the estimates
(91)
∥
ψ
N
,
n
∥
L
2
(
ℝ
n
;
ℒ
2
n
(
ℍ
×
n
;
H
)
)
2
=
∥
ψ
N
,
n
∥
L
2
(
S
n
-
N
,
N
;
ℒ
2
n
(
ℍ
×
n
;
H
)
)
2
≤
∥
F
N
∥
L
2
(
Ω
;
H
)
2
,
∥
f
N
,
n
∥
L
2
(
ℝ
n
;
ℒ
2
n
(
ℍ
×
n
;
H
)
)
2
=
∥
f
N
,
n
∥
L
2
(
[
-
N
,
N
]
n
;
ℒ
2
n
(
ℍ
×
n
;
H
)
)
2
≤
∥
F
N
∥
L
2
(
Ω
;
H
)
2
,
it follows that
(92)
ψ
n
:
=
lim
⁡
N
→
∞
ψ
N
,
n
,
f
n
∶
=
lim
⁡
N
→
∞
f
N
,
n
,
in the space L
2
(
ℝ
n
;
ℒ
2
n
(
ℍ
×
n
;
H
)
). Consequently,
(93)
J
n
(
ψ
n
)
∶
=
lim
⁡
N
→
∞
J
n
(
ψ
N
,
n
)
,
I
n
(
f
n
)
=
lim
⁡
N
→
∞
I
n
(
f
N
,
n
)
exist in the space L
2
(
Ω
;
H
). Passing to the limits in the equalities (89) and (90) we obtain (87) and (88).



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*Source: 786854-2014-04-07.xml*