In this paper, we study the stochastic integral of processes taking values of generalized operators based on a triple
E
⊂
H
⊂
E
^{∗}
, where
H
is a Hilbert space,
E
is a countable Hilbert space and
E
^{∗}
is the strong dual space of
E
. For our purpose, we study
E
valued Wiener processes and then introduce the stochastic integral of
L
(
E
,
F
^{∗}
)valued process with respect to an
E
valued Wiener process, where
F
^{∗}
is the strong dual space of another countable Hilbert space
F
.
AMS Mathematics Subject Classification : 60H05.
1. Introduction
Since the stochastic integral for a standard Brownian motion initiated by K. Itô in
[7]
, so called the Itô integral, the stochastic calculus has been successfully and extensively developed with wide applications to various fields with randomness. The stochastic integral is one of the main topics in the study of stochastic calculus. In Itô integral, the noise term to explain the diffusion part of a stochastic dynamics is corresponding to the white noise which is formally understood as the time derivative of a standard Brownian motion. Recently, to understand various random phenomena, the stochastic calculus associated with various noise processes has been studied by several authors.
In another direction to study of stochastic calculus, the stochastic integrals of operatorvalued (stochastic) processes with respect to vectorvalued noise processes have been studied by several authors
[1
,
2
,
3
,
5
,
8
,
10
,
4]
, and the references cited therein. In particular, Applebaum
[2]
, Curtain & Falb
[3]
, Kunita
[8]
and Da Prato & Zabczyk
[4]
studied the stochastic integral of operatorvalued processes in a Hilbert space, and Albeverio & R¨udiger
[1]
, Mamporia
[10]
and van Neerven & Weis
[12]
studied the stochastic integral of operatorvalued processes with respect to a Banach space valued Wiener process. Also, Yadava
[13]
studied the stochastic integrals of operatorvalued processes with respect to a Wiener process taking values in a locally convex space.
The main purpose of this paper is to develop the stochastic integral of generalized operatorvalued stochastic process with respect to a Wiener process taking values in a countable Hilbert space. For our purpose, we focus on the following twofold:
(1) Wiener process values in a countable Hilbert space;
(2) the stochastic integral of processes taking values of generalized operators.
By taking
E
=
H
, from our Wiener process we construct a Wiener process taking values in a Hilbert space (see Remark 3.2). Therefore, our results are generalizations of the results for the case of Hilbert space. Also, our approach can be applied for the study of the Wiener process taking values in a locally convex space and the stochastic integral of processes taking values of generalized operators between locally convex spaces (see Remark 3.2).
This paper is organized as follows. In Section 2, we briefly review a standard construction of a countable Hilbert space
E
from a Hilbert space
H
and a triple:
E
⊂
H
⊂
E
^{∗}
, which is necessary for our study. In Section 3, we construct an
E
valued Wiener process. In Section 4, we introduce a stochastic integral of processes taking values of generalized operators from a countable Hilbert space into the strong dual space of another countable Hilbert space.
2. Countable Hilbert Spaces
Let
H
be a real separable Hilbert space with the norm ║·║
_{0}
induced by the inner product ⟨·, ·⟩. Let
A
be a positive selfadjont operator with the dense domain Dom(
A
) ⊂
H
such that inf Spec (
A
) > 0. For each
p
≥ 0, the dense subspace Dom(
A^{p}
) ⊂
H
equipped with the norm
becomes a Hilbert space, which is denoted by
E_{p}
. Note that since inf Spec (
A
) > 0,
A
^{−1}
is a bounded operator on
H
of which the operator norm is given by
For each
p
≥ 0, we define the norm ║·║
_{−p}
on
H
by
Let
E_{−p}
be the completion of
H
with respect to the norm ║·║
_{−p}
. Then we have a chain of Hilbert spaces:
where each inclusion is continuous and has a dense image, and
From a chain (1), we obtain the projective limit space:
equipped with the locally convex topology generated by the family {║·║
_{p}
}
_{p}
_{≥0}
of Hilbertian norms. Note that
E
is sequentially complete. Since the norms are linearly ordered as given in (2), we may choose a countable set of defining norms and so
E
becomes a countable Hilbert space. The strong dual space
E
^{∗}
of
E
is identified with the inductive limit space of {
E_{−p}
}
_{p≥0}
, i.e.,
Thus, we have a triple:
Note that if there exists
p
> 0 such that
A^{−p}
is a HilbertSchmidt operator, then
E
becomes a countable Hilbert nuclear space, and in this case, the triple given as in (3) is called a Gelfand triple. The canonical real bilinear form in
E
^{∗}
×
E
is denoted by ⟨·, ·⟩. For further study of the triple, we refer to
[6
,
11]
.
3.Evalued Wiener Processes
Let (Ω,
F, P
) be a complete probability space. A function
X
: Ω →
E
is said to be Bochner measurable if there exists a sequence
of simple functions from Ω to
E
such that
X_{n}
(
ω
) converges to
X
(
ω
) in
E
for almost all
ω
∈ Ω. A function
X
: Ω →
E
is said to be Bochner integrable if there exists a sequence
of simple functions from Ω to
E
such that
X_{n}
(
ω
) converges to
X
(
ω
) in
E
for almost all
ω
∈ Ω and
for any
p
≥ 0, and in this case, we define
A Bochner integrable function
X
: Ω →
E
is called a (
E
valued) random variable and for which, we denote
which is called the expectation of
X
with respect to
P
.
For each 1 ≤ ℓ < ∞, we denote
L
^{ℓ}
(Ω,
Ƒ, P
;
E
) the space of all Bochner integrable functions
X
: Ω →
E
such that
for all
p
≥ 0.
For two locally convex spaces
E
and
F
, we denote
L
(
E, F
) the space of all continuous linear operators from
E
into
F
. If
E
is a nuclear Fréchet space and
F
is a Fréchet space, then
by the kernel theorem (see
[11]
), where
E
^{∗}
⊗
_{π}
F
^{∗}
is the
π
tensor product of
E
^{∗}
and
F
^{∗}
. Therefore, an operator in
L
(
E, F
^{∗}
) is sometimes called a generalized operator. It is wellknown that for each Φ ∈
L
(
E, F
^{∗}
), there exist
p, q
≥ 0 such that Φ ∈
L
(
E_{q}
,
F_{−p}
). In this case, the operator norm of Φ is denoted by ║Φ║
_{q,−p}
.
From now on, let
Q
∈
L
(
E
^{∗}
,
E
) be given by
for some
S
∈
L
(
E
^{∗}
,
E
), and let
be a sequence of mutually independent realvalued Brownian motions.
Proposition 3.1.
Suppose that for any p
≥ 0,
for a complete orthonormal basis
⊂
E for H. Then there exists an Evalued stochastic process
{
W_{t}
}
_{t}
_{≥0}
⊂
L
^{2}
(Ω,
Ƒ, P
;
E
)
such that

(i)E[Wt] = 0for all t≥ 0;

(ii)for any p≥ 0, it holds that

for any s, t≥ 0,where C(p)is given as in(5);

(iii)for any ϕ, ψ∈E∗,E[⟨ϕ, Ws⟩ ⟨ψ, Wt⟩] = min{s, t} ⟨ϕ, Qψ⟩;

(iv) {Wt}t≥0has independent increments.
Proof
. The proof is a modification of the proof of Proposition 4.2 in
[4]
. Let
⊂
E
be a complete orthonormal basis for
H
such that (5) holds. For each
t
≥ 0, we can easily see that the series of
E
valued random variables:
is summable in
E
almost surely on Ω. In fact, for each
p
≥ 0 and any
m, n
∈ ℕ, we obtain that
which implies that the sequence
converges in
L
^{2}
(Ω,
Ƒ, P
;
E_{p}
) for any
p
≥ 0, and so converges in
L
^{2}
(Ω,
Ƒ, P
;
E
). Therefore, for each
t
≥ 0, we define
W_{t}
by
Then {
W_{t}
}
_{t}
_{≥0}
is an
E
valued stochastic process and it is obvious that
E
[
W_{t}
] = 0 for all
t
≥ 0. On the other hand, by applying the dominated convergence theorem, we obtain that
which proves (6). For any
ϕ, ψ
∈
E
^{∗}
, we obtain that
which gives the proof of (iii). Let 0 ≤
s
<
t
≤
u
<
v
and
ϕ, ψ
∈
E
^{∗}
. Then we obtain that
which means that {
W_{t}
}
_{t}
_{≥0}
has independent increments. □
Remark 3.2.
Since every absolutely summable sequence is square summable, the condition that
is stronger than the condition given as in (5). For a special case of the operator
S
and a locally convex space
E
, a condition similar to the condition given as in (9) can be found in
[13]
. In the case of Hilbert space valued Wiener processes studied in
[4]
, the condition given as in (9) implies that
Q
is a HilbertSchmidt operator. But for the study of Hilbert space valued Wiener processes, it is enough to assume that
Q
is a traceclass operator (see
[4]
). In this sense, the condition given as in (5) is natural.
Remark 3.3.
In Proposition 3.1, the identity given as in (7) is essential. If
E
is a locally convex space, then as a counterpart of (7), by applying the CauchySchwarz inequality, we obtain that
for which we used only the triangle inequality and the CauchySchwarz inequality, and so it is a sharp iequality. Therefore, for the study of
Q
Wiener processes in a locally convex space, the condition given as in (9) is natural.
Definition 3.4.
Let
Q
=
S
^{∗}
S
∈
L
(
E
^{∗}
,
E
) be a nonnegative symmetric operator for an operator
S
∈
L
(
E
^{∗}
,
E
) satisfying (5). An
E
valued stochastic process {
W_{t}
}
_{t}
_{≥0}
⊂
L
^{2}
(Ω,
F, P
;
E
) is called a
Q
Wiener process if

(i)W0= 0 a.s.;

(ii)Whas continuous sample paths;

(iii) for every 0 ≤s

for anyϕ, ψ∈E∗;

(iv)Whas independent increments.
Theorem 3.5.
Let Q
=
S
^{∗}
S
∈
L
(
E
^{∗}
,
E
)
be a nonnegative symmetric operator for an operator S
∈
L
(
E
^{∗}
,
E
)
satisfying
(5).
Then there exists an Evalued QWiener process
{
W_{t}
}
_{t}
_{≥0}
.
Proof
. The proof is immediate from Proposition 3.1. □
Proposition 3.6.
Let
{
W_{t}
}
_{t}
_{≥0}
be an Evalued QWiener process. Suppose that there exist a complete orthonormal basis
for H and a sequence
of nonnegative real numbers such that for any p
≥ 0,
Then there exists a sequence
of mutually independent realvalued Wiener processes such that
where the series given as in
(11)
converges in L
^{2}
(Ω,
Ƒ, P
;
E
).
Proof
. The proof is a modification of the proof of Proposition 4.1 in
[4]
. For each
n
∈ ℕ, define
and put Λ = {
n
∈ ℕ ⟨
Qe_{n}, e_{n}
⟩ > 0}. Then it is easy to see that {
ζ_{n}
}
_{∈Λ}
is a sequence of mutually independent Gaussian process such that for any
m, n
∈ Λ,
Therefore, we define
where {
η_{n}
}
_{n∈ℕ╲Λ}
is any sequence of mutually independent realvalued Brownian motions. Then it holds that
where the last identity is as in
L
^{2}
(Ω,
Ƒ, P
;
E
). In fact, it holds that
Therefore, to complete the proof, it is enough to see the identity:
and then, since the identity holds as in
L
^{2}
(Ω,
Ƒ, P
;
H
), it is enough to see that the series converges in
L
^{2}
(Ω,
Ƒ, P
;
E
), which follows from that
Hence, the proof is completed. □
4. Stochastic Integration ofL(E, F∗)valued Process
Let
H
and
K
be Hilbert spaces. Then we have chains such as (1): for
p, q
≥ 0,
Let {
W_{t}
}
_{t}
_{≥0}
be an
E
valued QWiener process. For each
t
≥ 0, we denote
Ƒ_{t}
the
σ
algebra generated by {
W_{s}
0 ≤
s
≤
t
}. Then
Ƒ
= {
Ƒ_{t}
}
_{t}
_{≥0}
is a filtration, i.e., an increasing family of
σ
algebras.
A
L
(
E, F
^{∗}
)valued stochastic process {Φ
_{t}
}
_{t}
_{≥0}
is said to be
Ƒ
adapted (or simply adapted) if for each
t
≥ 0, Φ
_{t}
is
Ƒ_{t}
measurable.
For a
L
(
E, F
^{∗}
)valued (adapted) simple process Φ = {Φ
_{t}
}
_{t}
_{≥0}
of the form
where 0 =
t
_{1}
<
t
_{2}
< · · · <
t_{N}
=
T
and Φ
_{ti}
(
i
= 1, · · · ,
N
−1) is
Ƒ_{ti}
measurable, the stochastic integral of Φ with respect to the
E
valued
Q
Wiener process {
W_{t}
}
_{t}
_{≥0}
is a
F
^{∗}
valued stochastic process defined by
Proposition 4.1.
For a L
(
E, F
^{∗}
)
valued simple process
Φ = {Φ
_{t}
}
_{t}
_{≥0}
of the form given as in (13), we have

(i)

(ii)for some p≥ 0,


(iii)for some p, q≥ 0,where C(q)is given as in(5).
Proof
. (i) For any
ζ
∈
F
, we obtain that
Therefore, the proof of (i) is immediate since {
W_{t}
}
_{t}
_{≥0}
is an
E
valued
Q
Wiener process and
for any
ϕ
∈
E
^{∗}
.
(ii) For some
p, q
≥ 0, {Φ
_{t}
}
_{t}
_{≥0}
⊂
L
(
E_{q}
,
F_{−p}
) and so we obtain that
where ⟨·, ·⟩ is the bilinear form on
F
^{∗}
×
F
and
is a complete orthonormal basis for the given Hilbert space
K
. Therefore, we have
where
is a complete orthonormal basis for the given Hilbert space
K
. On the other hand, we obtain that
Hence we have
which gives the proof.
(iii) From (ii), we obtain that
which gives the proof. □
Let
L
^{2}
(
E, F
^{∗}
) be the space of all (adapted)
L
(
E, F
^{∗}
)valued process such that
for some
p, q
≥ 0, and let
the space of all (adapted) simple process Φ ∈
L
^{2}
(
E, F
^{∗}
) of the form given as in (13). We define a map
I
:
→
L
^{2}
(Ω,
Ƒ, P
;
F
^{∗}
) by
From (iii) in Proposition 4.1, we see that
I
is a continuous linear map from
into
L
^{2}
(Ω,
Ƒ, P
;
F
^{∗}
). Since
L
^{2}
(Ω,
Ƒ, P
;
F
^{∗}
) is complete and
is dense in
L
^{2}
(
E, F
^{∗}
), we extend
I
as a continuous linear map from
L
^{2}
(
E, F
^{∗}
) into
L
^{2}
(Ω,
Ƒ, P
;
F
^{∗}
). Therefore, for Φ ∈
L
^{2}
(
E, F
^{∗}
), we define
which is called the stochastic integral of Φ with respect to
Q
Wiener process {
W_{t}
}
_{t}
_{≥0}
.
Theorem 4.2.
For any L
(
E, F
^{∗}
)
valued process
Φ
in L
^{2}
(
E, F
^{∗}
)
, we have
Proof
. (i) It is obvious from (i) in Proposition 4.1.
(ii) Let
be a sequence of simple processes in
such that Φ
_{n}
converges to Φ in
L
^{2}
(
E, F
^{∗}
) and
I
(Φ
_{n}
) converges to
I
(Φ). Then for some
p, q
≥ 0 and any arbitrary
ϵ
> 0, by applying the CauchySchwarz inequality, we obtain that
from which, by taking limit as
n
→ ∞, we obtain that
Since
ϵ
> 0 is arbitrary, we prove the desired result. □
The study of stochastic differential equations and stochastic control systems (see
[9]
) is in progress.
BIO
Un Cig Ji received M.Sc. in 1991 and Ph.D in 1996 from Sogang University. He is currently a professor at Chungbuk National University since 2001. His research interests are Functional Analysis, Infinite Dimensional Analysis, White Noise Theory, Quantum Probability and Stochastic Analysis.
Department of Mathematics, Research institute of Mathematical Finance, Chungbuk National University, Cheongju 28644, Korea
uncigji@chungbuk.ac.kr
Byoung Jin Choi received M.Sc. and Ph.D. from Chunbuk National University. His research interests are Functional Analysis, Quantum Probability and Stochastic Analysis.
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea
choibj@skku.edu
Jin Pil Choi received M.Sc. from Chunbuk National University. His research interests are Stochastic Analysis and Applications.
Department of Mathematics, Chungbuk National University, Cheongju 28644, Korea
jinpilchoi@chungbuk.ac.kr
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