

Double Lusin Condition for the It o-Henstock Integrable Operator-Valued Stochastic Process


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 11, No. 4, 2018, 1003-1013
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Double Lusin Condition for the Itô-Henstock Integrable
Operator-Valued Stochastic Process

Mhelmar A. Labendia1,∗, Jayrold P. Arcede2

1 Department of Mathematics and Statistics, College of Science and Mathematics, Mindanao
State University-Iligan Institute of Technology, 9200 Iligan City, Philippines
2 Department of Mathematics, College of Arts and Sciences, Caraga State University, 8600
Butuan City, Philippines

Abstract. In this paper, using double Lusin condition, we give an equivalent definition of the Itô-
Henstock integral of an operator-valued stochastic process with respect to a Hilbert space-valued
Wiener process.

2010 Mathematics Subject Classifications: 60H30, 60H05
Key Words and Phrases: Itô-Henstock integral, Q-Wiener process, double Lusin condition

1. Introduction

The Henstock integral, which was studied independently by Henstock and Kurzweil
in the 1950s and later known as the Henstock-Kurzweil integral, is one of the notable
integrals that was introduced which in some sense is more general than the Lebesgue
integral. To avoid an extensive study of measure theory, Henstock-Kurzweil integration
had been deeply studied and investigated by numerous authors, see [3–5, 8–10]. The
Henstock-Kurzweil integral is a Riemann-type definition of an integral which is more
explicit and minimizes the technicalities in the classical approach of the Lebesgue integral.
This approach to integration is known as the generalized Riemann approach or Henstock
approach.

In the classical approach to stochastic integration, the Itô integral of a real-valued
stochastic process, which is adapted to a filtration, is attained from a limit of Itô integrals
of simple processes. To give a more explicit definition and reduce the technicalities in the
classical way of defining the Itô integral in the real-valued case, Henstock approach to
stochastic integration had already been studied in several papers, see [12, 13, 17–19].

In infinite dimensional spaces, the Itô integral of an operator-valued stochastic process,
adapted to a normal filtration, is obtained by extending an isometry from the space of

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v11i4.3310

Email addresses: mhelmar.labendia@g.msuiit.edu.ph (M. Labendia),
jparcede@carsu.edu.ph (J. Arcede)

http://www.ejpam.com 1003 c© 2018 EJPAM All rights reserved.


M. Labendia, J. Arcede / Eur. J. Pure Appl. Math, 11 (4) (2018), 1003-1013 1004

elementary processes to the space of continuous square-integrable martingales. In this
case, the value of the integrand is a Hilbert-Schmidt operator and the integrator is a Q-
Wiener process, a Hilbert space-valued Wiener process which is dependent on a symmetric
nonnegative definite trace-class operator Q. In [7], the authors defined the Itô-Henstock
integral of an operator-valued stochastic process with respect to a Q-Wiener process and
formulated a version of Itô’s formula, the stochastic counterpart of the classical chain rule
of differentiation.

In this paper, we revisit the concept of Itô-Henstock integral for the operator-valued
stochastic process with respect to a Q-Wiener process and characterize Itô-Henstock inte-
grability by using the concept of double Lusin condition and AC2[0, T ]-property, a version
of absolute continuity.

2. Preliminaries

Throughout this paper, (Ω,F , {Ft},P) be a filtered probability space, B(H) be the Borel
σ-field of a separable Banach space H, and L(h) be the probability distribution or the law
of a random variable h : Ω→ H.

A stochastic process f : [0, T ] × Ω → H, or simply a process {ft}0≤t≤T , is said to be
adapted to a filtration {Ft} if ft is Ft-measurable for all t ∈ [0, T ]. When no confusion
arises, we may refer to a process adapted to {Ft} as simply an adapted process.

Let U and V be separable Hilbert spaces. Denote by L(U, V ) the space of all bounded
linear operators from U to V , L(U) := L(U,U), Qu := Q(u) if Q ∈ L(U, V ), and L2(Ω, V )
the space of all square-integrable random variables from Ω to V . An operator Q ∈ L(U)
is said to be self-adjoint or symmetric if for all u, u′ ∈ U , 〈Qu, u′〉U = 〈u,Qu′〉U and is
said to be nonnegative definite if for every u ∈ U , 〈Qu, u〉U ≥ 0. Using the Square-root
Lemma [16, p.196], if Q ∈ L(U) is nonnegative definite, then there exists a unique operator

Q
1
2 ∈ L(U) such that Q

1
2 is nonnegative definite and (Q

1
2 )2 = Q.

Let {ej}∞j=1, or simply {ej}, be an orthonormal basis (abbrev. as ONB) in U . If Q ∈
L(U) is nonnegative definite, then the trace of Q is defined by tr Q =

∑∞
j=1 〈Qej , ej〉U . It

is shown in [16, p.206] that tr Q is well-defined and may be defined in terms of an arbitrary

ONB. An operator Q : U → U is said to be trace-class if tr [Q] := tr (QQ∗)
1
2 <∞. Denote

by L1(U) the space of all trace-class operators on U , which is known [16, p.209] to be a
Banach space with norm ‖Q‖1 = tr [Q]. If Q ∈ L(U) is a symmetric nonnegative definite
trace-class operator, then there exists an ONB {ej} ⊂ U and a sequence of nonnegative
real numbers {λj} such that Qej = λjej for all j ∈ N, {λj} ∈ `1, and λj → 0 as j → ∞
[16, p.203]. We shall call the sequence of pairs {λj , ej} an eigensequence defined by Q.

Let Q : U → U be a symmetric nonnegative definite trace-class operator. Let {λj , ej}
be an eigensequence defined by Q. Then the subspace UQ := Q

1
2U of U equipped with the

inner product 〈u, v〉UQ
=
〈
Q−1/2u,Q−1/2v

〉
U

, where Q1/2 is being restricted to [KerQ1/2]⊥

is a separable Hilbert space with
{√

λjej
}

as its ONB, see [15, p.90], [2, p.23].
Let {fj} be an ONB in UQ. An operator S ∈ L(UQ, V ) is said to be Hilbert-Schmidt

if
∑∞

j=1 ‖Sfj‖
2
V =

∑∞
j=1 〈Sfj , Sfj〉V < ∞. Denote by L2(UQ, V ) the space of all Hilbert-


M. Labendia, J. Arcede / Eur. J. Pure Appl. Math, 11 (4) (2018), 1003-1013 1005

Schmidt operators from UQ to V , which is known [14, p.112] to be a separable Hilbert space

with norm ‖S‖L2(UQ,V ) =
√∑∞

j=1 ‖Sfj‖
2
V . The Hilbert-Schmidt operator S ∈ L2(UQ, V )

and the norm ‖S‖L2(UQ,V ) may be defined in terms of an arbitrary ONB, see [15, p.418],

[14, p.111]. It is shown in [2, p.25] that L(U, V ) is properly contained in L2(UQ, V ). We
also note that L2(UQ, V ) contains genuinely unbounded linear operators from U to V .

Let Q : U → U be a symmetric nonnegative definite trace-class operator, {λj , ej} be
an eigensequence defined by Q, and {Bj} be a sequence of independent Brownian motions
(abbrev. as BM) defined on (Ω,F , {Ft},P). The process

W̃t :=

∞∑
j=1

√
λjBj(t)ej (1)

is called a Q-Wiener process in U . The series in (1) converges in L2(Ω, U). For each

u ∈ U , denote W̃t(u) :=
∞∑
j=1

√
λjBj(t) 〈ej , u〉U , with the series converging in L2(Ω,R).

Since the operator Q is assumed to be symmetric nonnegative definite trace-class, there
exists a U -valued process W such that

W̃t(u)(ω) = 〈Wt(ω), u〉U P-almost surely (abbrev. as P-a.s.). (2)

We call the process W a U -valued Q-Wiener process. This process is a multidimentional

BM . It should be noted that if we assume that λj > 0 for all j,
Wt(ej)√

λj
, j = 1, 2, . . . , is a

sequence of real-valued BM defined on (Ω,F , {Ft},P), see [15, p.87].
A filtration {Ft} on a probability space (Ω,F ,P) is called normal if (i) F0 contains

all elements A ∈ F such that P(A) = 0, and (ii) Ft = Ft+ :=
⋂
s>t

Fs for all t ∈ [0, T ]. A

Q-Wiener process Wt, t ∈ [0, T ] is called a Q-Wiener process with respect to a filtration
{Ft} if (i) Wt is adapted to {Ft}, t ∈ [0, T ] and (ii) Wt −Ws is independent of Fs for all
0 ≤ s ≤ t ≤ T . It is shown in [14, p.16] that a U -valued Q-Wiener process W (t), t ∈ [0, T ],
is a Q-Wiener process with respect to a normal filtration. From now onwards, a filtered
probability space (Ω,F , {Ft},P) shall mean a probability space equipped with a normal
filtration.

3. Itô-Hentock Integral and Double Lusin Condition

In [19], Chew et al. introduced the Itô-Henstock integral of a real-valued process with
respect to a Brownian motion. We shall use the same definition of belated partial division
employed by the authors in [19] to define the Itô-Henstock integral of an L(U, V )-valued
stochastic process with respect to a U -valued Q-Wiener process. We note that the given
closed and bounded interval [0, T ] is nondegenerate, i.e. 0 < T , which can be replaced

with any interval [a, b]. If no confusion arises, we may write (D)
∑

instead of

n∑
i=1

for the

given finite collection D.


M. Labendia, J. Arcede / Eur. J. Pure Appl. Math, 11 (4) (2018), 1003-1013 1006

Definition 1. Let δ be a positive function on [0, T ]. A finite collection D of interval-point
pairs {((ξi, vi], ξi)}ni=1 is a δ-fine belated partial division of [0, T ] if

(i) (ξi, vi], i = 1, 2, . . . , n, are disjoint subintervals of [0, T ]; and

(ii) each (ξi, vi] is δ-fine belated, that is, (ξi, vi] ⊂ [ξi, ξi + δ(ξi)).

The term partial is used in Definition 1 since the finite collection of disjoint left-open
subintervals of [0, T ] may not cover the entire interval [0, T ]. Using the Vitali covering
lemma, the following concept can be defined.

Definition 2. Given η > 0, a given δ-fine belated partial division D = {((ξ, v], ξ)} is said
to be a (δ, η)-fine belated partial division of [0, T ] if it fails to cover [0, T ] by at most length
η, that is, ∣∣∣T − (D)

∑
(v − ξ)

∣∣∣ ≤ η.
This type of partial division is the basis to which we define the Itô-Henstock integral.
Throughout the succeeding discussions, assume that U and V are separable Hilbert

spaces, Q : U → U is a symmetric nonnegative definite trace-class operator, {λj , ej} is
an eigensequence defined by Q, and W is a U -valued Q-Wiener process. A stochastic
process f : [0, T ]× Ω → L(U, V ) means a process measurable as mappings from ([0, T ]×
Ω,B([0, T ])⊗F) to (L2(UQ, V ),B(L2(UQ, V ))).

Definition 3. Let f : [0, T ] × Ω → L(U, V ) be an adapted process. Then f is said to
be Itô-Henstock integrable, or IH-integrable, on [0, T ] with respect to W if there exists
A ∈ L2(Ω, V ) such that for every ε > 0, there is a positive function δ on [0, T ] and a positive
number η > 0 such that for any (δ, η)-fine belated partial division D = {((ξi, vi], ξi)}ni=1 of
[0, T ], we have

E
[
‖S(f,D, δ, η)−A‖2V

]
< ε,

where

S(f,D, δ, η) := (D)
∑

fξ(Wv −Wξ) :=
n∑
i=1

fξi(Wvi −Wξi).

In this case, f is IH-integrable to A on [0, T ] and A is called the IH-integral of f which

will be denoted by (IH)

∫ T

0
ft dWt or (IH)

∫ T

0
f dW . For convenience, we shall denote

(IH)

∫ 0

0
ft dWt by the zero random variable 0 ∈ L2(Ω, V ).

Example 1. f : [0, T ]×Ω→ L(U, V ) be an adapted process such that E
[
‖ft‖2L2(UQ,V )

]
= 0

for all t ∈ [0, T ] except on a set of Lebesgue measure zero. Then f is IH-integrable to 0
on [0, T ].

The following statements show that the Itô-Henstock integral possesses the standard
properties of an integral. Refer to [8] for the proofs.


M. Labendia, J. Arcede / Eur. J. Pure Appl. Math, 11 (4) (2018), 1003-1013 1007

(1) The Itô-Henstock integral is uniquely determined, in the sense that if A1 and A2 are
two Itô-Henstock integrals of f in Definition 3, then ‖A1 −A2‖L2(Ω,V ) = 0.

(2) Let α ∈ R. If f and g are IH-integrable on [0, T ], then

(i) f + g is IH-integrable on [0, T ], and

(IH)

∫ T

0
(f + g) dW = (IH)

∫ T

0
f dW + (IH)

∫ T

0
g dW ;

(ii) αf is IH-integrable on [0, T ], and

(IH)

∫ T

0
(αf) dW = α · (IH)

∫ T

0
f dW.

(3) If f : [0, T ]×Ω→ L(U, V ) is IH-integrable on [0, c] and [c, T ] where c ∈ (0, T ), then
f is IH-integrable on [0, T ] and

(IH)

∫ T

0
f dW = (IH)

∫ c

0
f dW + (IH)

∫ T

c
f dW.

(4) If f : [0, T ] × Ω → L(U, V ) is IH-integrable on [0, T ], then f is also IH-integrable
on every subinteval [c, d] of [0, T ].

(5) A process f : [0, T ] × Ω → L(U, V ) is IH-integrable on [0, T ] if and only if there
exist A ∈ L2(Ω, V ), a decreasing sequence {δn} of positive functions defined on [0, T ],
and a decreasing sequence of positive numbers {ηn} such that for any (δn, ηn)-fine
belated partial division Dn of [0, T ], we have

lim
n→∞

E
[
‖S(f,Dn, δn, ηn)−A‖2V

]
= 0.

In this case,

A = (IH)

∫ T

0
ft dWt.

(6) (Cauchy Criterion). A process f : [0, T ]× Ω→ L(U, V ) is IH-integrable on [0, T ] if
and only if for every ε > 0, there exist a positive function δ on [0, T ] and a positive
number η such that for any two (δ, η)-fine belated partial divisions D and D′ of
[0, T ], we have

E
[∥∥S(f,D, δ, η)− S(f,D′, δ, η)

∥∥2

V

]
< ε.

(7) (Weak Version of Saks-Henstock Lemma). Let f be IH-integrable on [0, T ] and

F (u, v] := (IH)

∫ v

u
ft dWt for any (u, v] ⊂ [0, T ]. Then for every ε > 0, there

exist a positive function δ on [0, T ] such that for any δ-fine belated partial division
D = {((ξ, v], ξ)} of [0, T ], we have

E
[∥∥∥(D)

∑
{fξ(Wv −Wξ)− F (ξ, v]}

∥∥∥2

V

]
< ε.


M. Labendia, J. Arcede / Eur. J. Pure Appl. Math, 11 (4) (2018), 1003-1013 1008

(8) (Itô Isometry). Let f be IH-integrable on [0, T ]. Then E
[
‖ft‖2L2(UQ,V )

]
is Lebesgue

integrable on [0, T ] and

E

[∥∥∥∥(IH)

∫ T

0
ft dWt

∥∥∥∥2

V

]
= (L)

∫ T

0
E
[
‖ft‖2L2(UQ,V )

]
dt <∞.

In [6], the Itô-Henstock integral has been characterized using AC2[0, T ]-property, a
version of absolute continuity. Throughout the following, denote by J , the collection of
all closed intervals (u, v] ⊂ [0, T ]. In the following definition, when no confusion arises, we
may refer to F ((u, v], ·) or F ((u, v], ω) as simply F (u, v].

Definition 4. A function F : J ×Ω→ V is said to be AC2[0, T ] if for every ε > 0, there
exists η > 0 such that for any finite collection D = {(ξ, v]} of non-overlapping subintervals

of [0, T ] with (D)
∑

(v − ξ) < η, we have E
[∥∥∥(D)

∑
F (ξ, v]

∥∥∥2

V

]
< ε.

Theorem 1. [6, Theorem 3.4] Let f : [0, T ]× Ω→ L(U, V ) be an adapted process. Then
f is IH-integrable on [0, T ] if and only if there exists a function F : J ×Ω→ V such that

(i) F is AC2[0, T ] and

(ii) for every ε > 0, there exist a positive function δ on [0, T ] such that whenever D =
{((ξ, v], ξ)} is a δ-fine belated partial division of [0, T ], we have

E
[∥∥∥(D)

∑
{fξ(Wv −Wξ)− F (ξ, v]}

∥∥∥2

V

]
< ε.

We remark that in Theorem 1 if f is IH-integrable on [0, T ], then the existing function

F that satisfies conditions (i) and (ii) is given by F (u, v] := (IH)

∫ v

u
ft dWt for each

(u, v] ∈ J , see [6, proof of Theorem 3.4].
Next, we present the double Lusin condition-property for a process f : [0, T ] × Ω →

L(U, V ) and a function F : J × Ω → V . This property is analogous to the double Lusin
condition used in [1, 11].

Definition 5. Let f : [0, T ] × Ω → L(U, V ) be an adapted process and F : J × Ω → V
be a function. For any given ε > 0, let Γε be the set of all interval-point pairs {((ξ, v], ξ)}
such that

E
[
‖fξ(Wv −Wξ)− F (ξ, v]‖2V

]
≥ εE

[
‖Wv −Wξ‖2U

]
= ε(v − ξ)tr Q.

Definition 6. A process f : [0, T ]×Ω→ L(U, V ) and a function F : J ×Ω→ V are said
to satisfy the double Lusin condition if for every ε > 0, there exists a positive function δ
on [0, T ] such that for any δ-fine belated partial division D = {((ξ, v], ξ)} ⊆ Γε of [0, T ],

E
[
‖(D)

∑
fξ(Wv −Wξ)‖2V

]
< ε and E

[
‖(D)

∑
F (ξ, v]‖2V

]
< ε.


M. Labendia, J. Arcede / Eur. J. Pure Appl. Math, 11 (4) (2018), 1003-1013 1009

Definition 7. A function F : J × Ω→ V is said to satisfy the double Lusin condition if
for every ε > 0, there exists a positive function δ on [0, T ] such that for any δ-fine belated
partial division D = {((ξ, v], ξ)} ⊆ Γε of [0, T ],

E
[
‖(D)

∑
(Wv −Wξ)‖2V

]
< ε and E

[
‖(D)

∑
F (ξ, v]‖2V

]
< ε.

Before giving an equivalent definition of IH-integrable operator-valued process, we
need to consider the following known results:

Lemma 1. [7, Lemma 3.6] Let f : [0, T ] × Ω → L(U, V ) be an adapted process and
{(ξi, vi]}ni=1 be a finite collection of disjoint subintervals of [0, T ]. Then

E

∥∥∥∥∥
n∑
i=1

fξi(Wvi −Wξi)

∥∥∥∥∥
2

V

 =
n∑
i=1

E
[
‖fξi(Wvi −Wξi)‖

2
V

]

=
n∑
i=1

(vi − ξi)E
[
‖fξi‖

2
L2(UQ,V )

]
.

Lemma 2. (Strong Version of Saks-Henstock Lemma). Let f be IH-integrable on [0, T ]

and F (u, v] := (IH)

∫ v

u
ft dWt for any (u, v] ⊂ [0, T ]. Then for every ε > 0, there exist a

positive function δ on [0, T ] such that for any δ-fine belated partial division D = {((ξ, v], ξ)}
of [0, T ], we have

(D)
∑

E
[
‖fξ(Wv −Wξ)− F (ξ, v]‖2V

]
< ε.

We shall now characterize the Itô-Henstock integral using the double Lusin condition.

Theorem 2. Let f : [0, T ]×Ω→ L(U, V ) be an adapted process. Then f is IH-integrable
on [0, T ] if and only if there exists an AC2[0, T ] function F : J × Ω→ V and that f and
F satisfy the double Lusin condition.

Proof. Suppose that f is IH-integrable on [0, T ] and let F (u, v] = (IH)

∫ v

u
ft dWt for

each (u, v] ∈ J . By Theorem 1, F is AC2[0, T ]. Let ε > 0. By Theorem 1 and the strong
version of Saks-Henstock Lemma, for each k ∈ N, there exists a positive function δk on
[0, T ] such that for any δk-fine belated partial division Dk = {((ξ, v], ξ)} of [0, T ], we have

(Dk)
∑

E
[
‖fξ(Wv −Wξ)− F (ξ, v]‖2V

]
= E

[∥∥∥(Dk)
∑
{fξ(Wv −Wξ)− F (ξ, v]}

∥∥∥2

V

]
<

ε2(tr Q)

k · 2k+2
.

Moreover, there exists a positive function δ′ on [0, T ] such that for any δ′-fine belated
partial division D′ = {((ξ, v], ξ)} of [0, T ], we have

E
[∥∥∥(D′)

∑
{fξ(Wv −Wξ)− F (ξ, v]}

∥∥∥2

V

]
<
ε

4
.


M. Labendia, J. Arcede / Eur. J. Pure Appl. Math, 11 (4) (2018), 1003-1013 1010

For each k ∈ N, let Gk :=
{
t ∈ [0, T ] : k − 1 ≤ E

[
‖ft‖2L2(UQ,V )

]
< k

}
. Choose δ(ξ) =

min{δ′(ξ), δk(ξ)} if ξ ∈ Gk for some k ∈ N. Let D = {((ξ, v], ξ)} ⊆ Γε be a δ-fine belated
partial division of [0, T ]. For each k ∈ N, let Dk ⊆ D such that each tag in Dk is in Gk.
Then by Lemma 1,

E
[∥∥∥(D)

∑
fξ(Wv −Wξ)

∥∥∥2

V

]
= (D)

∑
(v − ξ)E

[
‖fξ‖2L2(UQ,V )

]
≤

∑
k∈N

(
(Dk)

∑
(v − ξ)E

[
‖fξ‖2L2(UQ,V )

])
≤

∑
k∈N

(
k · (Dk)

∑
(v − ξ)

)
≤

∑
k∈N

(
k

ε(tr Q)
E
[∥∥∥(Dk)

∑
{fξ(Wv −Wξ)− F (ξ, v]}

∥∥∥2

V

])
≤

∑
k∈N

k

ε(tr Q)
· ε

2(tr Q)

k · 2k+2
=
ε

4
.

Furthermore,

E
[∥∥∥(D)

∑
F (ξ, v]

∥∥∥2

V

]
≤ 2E

[∥∥∥(D)
∑
{fξ(Wv −Wξ)− F (ξ, v]}

∥∥∥2

V

]
+2E

[∥∥∥(D)
∑

fξ(Wv −Wξ)
∥∥∥2

V

]
<

ε

2
+
ε

2
= ε.

Conversely, suppose that there exists an AC2[0, T ] function F : J ×Ω→ V and that f
and F satisfy the double Lusin condition. Let ε > 0. Then there exists a positive function
δ on [0, T ] such that for any δ-fine belated partial division D′ = {((ξ, v], ξ)} ⊆ Γε of [0, T ],
we have

E
[
‖(D′)

∑
fξ(Wv −Wξ)‖2V

]
< ε and E

[
‖(D′)

∑
F (ξ, v]‖2V

]
< ε.

Let D = {((ξ, v], ξ)} be δ-fine belated partial division of [0, T ]. Then

E
[
‖(D)

∑
fξ(Wv −Wξ)− F (ξ, v]‖2V

]

≤ 2

(
(D \ Γε)

∑√
E
[
‖fξ(Wv −Wξ)− F (ξ, v]‖2V

])2

+4E
[∥∥∥(D ∩ Γε)

∑
fξ(Wv −Wξ)

∥∥∥2

V

]
+4E

[
‖(D ∩ Γε)F (ξ, v]‖2V

]
< 2

(
(D \ Γε)

∑√
ε(v − ξ)tr Q

)2
+ 4ε+ 4ε = ε(2T · tr Q+ 8).


M. Labendia, J. Arcede / Eur. J. Pure Appl. Math, 11 (4) (2018), 1003-1013 1011

By Theorem 1, f is IH-integrable on [0, T ]. �

Theorem 3. Let f : [0, T ]×Ω→ L(U, V ) be an adapted process. Then f is IH-integrable
on [0, T ] if and only if there exists an AC2[0, T ] function F : J ×Ω→ V that satisfies the
double Lusin condition.

Proof. Suppose that f is IH-integrable on [0, T ] and let F (u, v] = (IH)

∫ v

u
ft dWt for

each (u, v] ∈ J . By Theorem 1, F is AC2[0, T ]. Let ε > 0. By Theorem 1 and the strong
version of Saks-Henstock Lemma, there exists a positive function δ on [0, T ] such that for
any δ-fine belated partial division D′ = {((ξ, v], ξ)} of [0, T ], we have

(D′)
∑

E
[
‖fξ(Wv −Wξ)− F (ξ, v]‖2V

]
= E

[∥∥∥(D′)
∑
{fξ(Wv −Wξ)− F (ξ, v]}

∥∥∥2

V

]
< ε2.

Let D = {((ξ, v], ξ)} ⊆ Γε be a δ-fine belated partial division of [0, T ]. Then by Lemma 1,

E
[∥∥∥(D)

∑
(Wv −Wξ)

∥∥∥2

V

]
= (D)

∑
(v − ξ)tr Q

≤ 1

ε
(D)

∑
E
[
‖fξ(Wv −Wξ)− F (ξ, v]‖2V

]
<

1

ε
· ε2 = ε.

Conversely, suppose that there exists an AC2[0, T ] function F : J × Ω → V that
satisfies the double Lusin condition. Let ε > 0. By Theorem 1 and the strong version of
Henstock Lemma, for each k ∈ N, there exists a positive function δk on [0, T ] such that
for any δk-fine belated partial division Dk = {((ξ, v], ξ)} of [0, T ], we have

(Dk)
∑

E
[
‖fξ(Wv −Wξ)− F (ξ, v]‖2V

]
= E

[∥∥∥(Dk)
∑
{fξ(Wv −Wξ)− F (ξ, v]}

∥∥∥2

V

]
<

ε2(tr Q)

k · 2k+1
.

For each k ∈ N, let Gk :=
{
t ∈ [0, T ] : k − 1 ≤ E

[
‖ft‖2L2(UQ,V )

]
< k

}
. Choose δ(ξ) ≤ δk(ξ)

if ξ ∈ Gk for some k ∈ N. Let D = {((ξ, v], ξ)} ⊆ Γε be a δ-fine belated partial division of
[0, T ]. For each k ∈ N, let Dk ⊆ D such that each tag in Dk is in Gk. Then by Lemma 1,

E
[∥∥∥(D)

∑
fξ(Wv −Wξ)

∥∥∥2

V

]
= (D)

∑
(v − ξ)E

[
‖fξ‖2L2(UQ,V )

]
≤

∑
k∈N

(
(Dk)

∑
(v − ξ)E

[
‖fξ‖2L2(UQ,V )

])
≤

∑
k∈N

(
k · (Dk)

∑
(v − ξ)

)


REFERENCES 1012

≤
∑
k∈N

(
k

ε(tr Q)
E
[∥∥∥(Dk)

∑
{fξ(Wv −Wξ)− F (ξ, v]}

∥∥∥2

V

])
≤

∑
k∈N

k

ε(tr Q)
· ε

2(tr Q)

k · 2k+1
< ε.

By Theorem 2, f is IH-integrable on [0, T ]. �

We remark that in Theorem 2, the double Lusin condition involving the process f and
the function F may be restated as the double Lusin condition involving the function F
only.

4. Conclusion and Recommendation

In this paper, we formulate an equivalent definition of the Itô-Henstock integral of
an operator-valued stochastic process with respect to a Hilbert space-valued Q-Wiener
process. To attain this objective, we use the concept of the double Lusin condition and
AC2[0, T ]-property, a version of absolute continuity. A worthwhile direction for further
investigation is to use Henstock-Kurzweil approach to define the stochastic integral with
respect to a cylindrical Wiener process.

Acknowledgements

The first author would like to express his sincerest gratitude to his wife for the inspi-
ration in completing this paper. Also, the authors would like to acknowledge the financial
support from the National Research Council of the Philippines (NRCP) and to thank the
unknown referee for the helpful comments for the improvement of this paper.

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