A Descriptive Definition of the Ito-McShane Integral for the Hilbert-Schmidt-Valued Stochastic Process


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 1, 2019, 101-117
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

A Version of Fundamental Theorem for the
Itô-McShane Integral of an Operator-Valued Stochastic

Process

Jeffer Dave A. Cagubcob1, Mhelmar A. Labendia1,∗

1 Department of Mathematics and Statistics, College of Science and Mathematics,
Mindanao Sate University-Iligan Institute of Technology, 9200 Iligan City, Philippines

Abstract. In this paper, we formulate a descriptive definition or a version of fundamental theo-
rem for the Itô-McShane integral of an operator-valued stochastic process with respect to a Hilbert
space-valued Wiener process. For this reason, we introduce the concept of belated Mcshane differ-
entiability and a version of absolute continuity of a Hilbert space-valued stochastic process.

2010 Mathematics Subject Classifications: 60H30, 60H05

Key Words and Phrases: Itô-McShane integral, orthogonal increment property, Q-Wiener
process, AC2[0, T ]-property

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 in-
tegral. To avoid an extensive study of measure theory, Henstock-Kurzweil integration had
been deeply studied and investigated by numerous authors, see [2–4, 7–9]. 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 ap-
proach 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 [10, 11, 15–17].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i1.3331

Email addresses: jdacagubcob@gmail.com (J.D. Cagubcob), mhelmar.labendia@g.msuiit.edu.ph (M.
Labendia)

http://www.ejpam.com 101 c© 2019 EJPAM All rights reserved.



J.D. Cagubcob, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 101-117 102

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
elementary processes to the space of continuous square-integrable martingales. In this
case, the value of the integrand is an 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 this paper, we formulate a version of Fundamental Theorem for the Itô-McShane
integral, a Henstock approach integral, for the operator-valued stochastic process with
respect to a Q-Wiener process.

2. Preliminaries

Throughout this paper, let (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) for 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 [14, 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 ◦Q

1
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 [14, 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 [14, 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 → ∞
[14, 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 [13, p.90], [1, 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-

Schmidt operators from UQ to V , which is known [12, 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 )



J.D. Cagubcob, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 101-117 103

and the norm ‖S‖L2(UQ,V ) may be defined in terms of an arbitrary ONB, see [13, p.418],

[12, p.111]. It is shown in [1, 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 [13, 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 [12, 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-McShane Integral and Belated McShane Derivative

In this section, we introduce the Itô-McShane integral of a process f : [0, T ] × Ω →
L(U, V ) with respect to a U -valued Q-Wiener process W and the belated McShane deriva-
tive of a Hilbert space-valued function.

Throughout, assume that U and V are separable Hilberts spaces, Q : U → U is
a symmetric nonnegative definite trace-class operator, {λj , ej} is an eigensequence de-
fined 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 ))). Also, the given closed interval [0, T ] is nondegenerate, i.e.
0 < T and can be replaced with any closed interval [a, b]. If no confusion arises, we may

write (D)
∑

instead of
n∑
i=1

for the given finite collection D.



J.D. Cagubcob, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 101-117 104

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

(iii) δ-fine belated McShane division of [0, T ] if {[ui, vi]}ni=1 is a collection of non-overlapping

intervals on [0, T ] with

n⋃
i=1

[ui, vi] = [0, T ] and each [ui, vi] is δ-fine belated McShane,

that is, [ui, vi] ⊂ [ξi, ξi + δ(ξi))

(iv) δ-fine belated McShane partial division of [0, T ] if {[ui, vi]}ni=1 is a collection of non-
overlapping intervals on [0, T ] and each [ui, vi] is δ-fine belated McShane.

We note that each ξi in Definition 1 does not necessarily belong to [ui, vi]. The term
partial division is used in Definition 1 since the finite collection of non-overlapping intervals
of [0, T ] may not cover the entire interval [0, T ].

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

∑
(v − u)

∣∣∣ ≤ η.
To define the Itô-McShane integral, we shall use the definition of belated partial divi-

sion in Definition 1, employed by the authors in [17, p.499].

Definition 3. Let f : [0, T ] × Ω → L(U, V ) be an adapted process. Then f is said to
be Itô-McShane integrable, or IM-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 number
η > 0 such that for any (δ, η)-fine belated McShane partial division D = {([ui, 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 −Wu) :=
n∑
i=1

fξi(Wvi −Wui).

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

will be denoted by (IM)

∫ T

0
ft dWt or (IM)

∫ T

0
f dW . We shall denote (IM)

∫ 0

0
f dW

by the zero random variable 0 from Ω to V and denote by ΛIM, the collection of all
Itô-McShane integrable processes on [0, T ].

Refer to [6, Lemma 3.5 and Lemma 3.6] for the proofs of the following two lemmas.
Denote by J , the collection of all closed intervals [u, v] ⊂ [0, T ].



J.D. Cagubcob, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 101-117 105

Lemma 1. Let f : [0, T ] × Ω → L(U, V ) be an adapted process and {([ui, vi], ξi)}ni=1 be
a finite collection such that {[ui, vi]} is a collection of non-overlapping intervals in J ,
ξ1 < ξ2 < · · · < ξn, and ξi ≤ ui for each i = 1, 2, . . . , n. Then

E

∑
i<j

〈
fξi(Wvi −Wui), fξj (Wvj −Wuj )

〉
V

 = 0.

Lemma 2. Let f : [0, T ] × Ω → L(U, V ) be an adapted process and {([ui, vi], ξi)}ni=1 be
a finite collection such that {(ui, vi]} is a collection of non-overlapping intervals in J ,
ξ1 < ξ2 < · · · < ξn, and ξi ≤ ui for each i = 1, 2, . . . , n. Then

E

∥∥∥∥∥
n∑
i=1

fξi(Wvi −Wui)

∥∥∥∥∥
2

V

 =

n∑
i=1

E
[
‖fξi(Wvi −Wui)‖

2
V

]

=
n∑
i=1

(vi − ui)E
[
‖fξi‖

2
L2(UQ,V )

]
.

Refer to [6, Example 3.7] for the proof of the following example.

Example 1. Let g : Ω → L(U, V ) be a random variable bounded in L2(UQ, V ), that is,
there exists M > 0 such that ‖g(ω)‖L2(UQ,V ) ≤ M for all ω ∈ Ω and let 0̂ : Ω → L(U, V )

be a random variable such that for all ω ∈ Ω, 0̂(ω) is the zero operator in L(U, V ). Let
s ∈ [0, T ] be fixed. Let f : [0, T ] × Ω → L(U, V ) be an adapted process on a filtered
probability space (Ω,F , {Ft},P) such that for t ∈ [0, T ],

ft =

{
g if t = s

0̂ if t 6= s.

Then f is IM-integrable to the zero random variable 0 ∈ L2(Ω, V ) on [0, T ].

In the following proofs, denote by Leb∗ and Leb, the Lebesgue outer measure and
Lebesgue measure, respectively.

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

]
=

0 almost everywhere (abbrev. as a.e.) on [0, T ]. Then f is IM-integrable to 0 on [0, T ].

Proof. Let ε > 0 be given. Let G = {t ∈ [0, T ] : E
[
‖ft‖L2(UQ,V )2

]
6= 0}. Then,

Leb(G) = 0 and so G is measurable. Let ξ ∈ G. For any [u, v] ⊂ [ξ, T ],

E
[
‖fξ(Wv −Wu)‖2V

]
= (v − u)E

[
‖fξ‖2L2(UQ,V )

]
.



J.D. Cagubcob, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 101-117 106

Let Am =

m∑
k=1

〈fξ(Wv −Wu), gk〉2, where {gk} is an ONB in V . Since Am → G :=

∞∑
k=1

〈fξ(Wv −Wu), gk〉2 asm→∞ andAm ≤ Am+1, by the monotone convergence theorem

for Lebesgue integral,

lim
m→∞

E

[
m∑
k=1

〈fξ(Wv −Wu), gk〉2
]

= E

[ ∞∑
k=1

〈fξ(Wv −Wu), gk〉2
]

= E
[
‖fξ(Wv −Wu)‖2V

]
<∞.

(3)

Thus, there exists N ∈ N such that E
[
‖fξ‖2L2(UQ,V )

]
< N . Now, since G is measurable,

there exists an open set O containing G such that Leb(O) < ε
2N . Thus, for all ξ ∈ G, there

exists δ1(ξ) > 0 such that [ξ, ξ + δ1(ξ)) ⊂ O. Let D′ = {([u, v], ξ)} be a δ1-fine belated
McShane partial division such that each ξ ∈ G. Then, (D′)

∑
(v − u) < ε

2N and so

E
[∥∥∥(D′)

∑
fξ(Wv −Wu)− 0

∥∥∥2
V

]
≤ 2(D′)

∑
(v − u)E

[
‖fξ‖2L2(UQ,V )

]
< 2N · ε

2N
= ε.

(4)

Thus, for any δ-fine belated McShane partial division D = {([u, v], ξ)} where δ(·) > 0
on [0, T ] and δ(ξ) ≥ δ1(ξ) for ξ ∈ G, we have

E
[∥∥∥(D)

∑
fξ(Wv −Wu)− 0

∥∥∥2
V

]
≤ 2(Dξ∈G)

∑
(v − u)E

[
‖fξ‖2L2(UQ,V )

]
+ 2(Dξ∈[0,T ]\G)

∑
(v − u)E

[
‖fξ‖2L2(UQ,V )

]
< ε.

(5)

The above inequality also holds for (δ, η)-fine belated McShane partial division of [0, T ].
Thus, f is IM-integrable to 0 on [0, T ].

It is worth noting that the Itô-McShane integral possesses some of the standard prop-
erties of an integral namely, uniqueness of an integral, linearity, integrability on every
subinterval of [0, T ], the Cauchy criterion, and the Saks-Henstock Lemma. The proofs of
these results are standard in Henstock-Kurzweil integration, hence omitted.

(i) The IM integral is uniquely determined, in the sense that if A1 and A2 are two IM
integrals of f , then ‖A1 −A2‖L2(U,V ) = 0.



J.D. Cagubcob, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 101-117 107

(ii) Let f, g ∈ ΛIM and let α, β ∈ R. Then αf + βg ∈ ΛIM and

(IM)

∫ T

0
(αf + βg) dW = α · (IM)

∫ T

0
f dW + β · (IM)

∫ T

0
g dW.

(iii) Cauchy criterion. A process f is IM-integrable on [0, T ] if and only if for every
ε > 0, there exist a positive function δ on [0, T ] and a number η > 0 such that for
any two (δ, η)-fine belated McShane partial divisions D1 and D2 of [0, T ], we have

E
[
‖S(f,D1, δ, η)− S(f,D2, δ, η)‖2V

]
< ε.

(iv) If f is IM-integrable on [0, T ], then f is IM-integrable on [c, d] ⊂ [0, T ].

(v) If f is IM-integrable on [0, c] and [c, T ] where c ∈ (0, T ), then f is IM-integrable
on [0, T ] and

(IM)

∫ T

0
f dW = (IM)

∫ c

0
f dW + (IM)

∫ T

c
f dW.

(vi) Sequential definition. A process f is IM-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
McShane partial division Dn of [0, T ], we have

E
[
‖S(f,Dn, δn, ηn)−A‖2V

]
→ 0 as n→∞.

In this case, A := (IM)

∫ T

0
ft dWt.

(vii) Saks-Henstock lemma (Weak version). Let f be IM-integrable on [0, T ] and F [u, v] :=

(IM)

∫ v

u
f dW for any [u, v] ⊂ [0, T ]. Then for every ε > 0, there exists a posi-

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

E
[∥∥∥(D)

∑
{fξ(Wv −Wξ)− F [u, v]}

∥∥∥2
V

]
< ε.

Next, we define the concept of AC2[0, T ]-property, a version of absolute continuity.

Definition 4. A function F : J × Ω→ V is said to be belated McShane differentiable at
ξ ∈ [0, T ) if there exists a random variable fξ : Ω→ L(U, V ) such that for all ε > 0, there
exists a positive function δ on [0, T ] such that for all δ-fine belated McShane interval-point
pair ([u, v], ξ) of [0, T ],

E
[
‖fξ(Wv −Wu)− F [u, v]‖2V

]
< ε(v − u).

The random variable fξ is called the belated McShane derivative of F at the point ξ ∈ [0, T )
and is denoted by DFξ.



J.D. Cagubcob, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 101-117 108

We note that we write F [u, v] instead of F ([u, v]).

Definition 5. A function F : J × Ω→ V

(i) is said to be AC2[0, T ] if for every ε > 0, there exists η > 0 such that for any finite

collection D = {[u, v]} of non-overlapping intervals [u, v] ∈ J with (D)
∑

(v− u) ≤

η, we have E
[∥∥∥(D)

∑
F [u, v]

∥∥∥2
V

]
< ε;

(ii) has the orthogonal increment property if for all non-overlapping intervals [a, b], [u, v] ⊂
[0, T ], E [〈F [a, b], F [u, v]〉] = 0.

The proof of the following theorem is parallel to the proof in [5].

Theorem 1. [5] Let f be IM-integrable on [0, T ] and define

F [u, v] := (IM)

∫ v

u
fsdWs for all [u, v] ⊂ [0, T ].

Then F is AC2[0, T ] and has the orthogonal increment property.

Lemma 3. [5] Let f : [0, T ] × Ω → L(U, V ) be an adapted process, F : J × Ω → V with
orthogonal increment property and {[ui, vi]}ni=1 be a finite collection of non-overlapping
subintervals of [0, T ]. Then

E

∥∥∥∥∥
n∑
i=1

{fξi(Wvi −Wui)− F [ui, vi]}

∥∥∥∥∥
2

V

 =
n∑
i=1

E
[
‖fξi(Wvi −Wui)− F [ui, vi]‖2V

]
.

Lemma 4. Let f ∈ ΛIM. Then for every ε > 0, there exist a positive function δ on [0, T ]
and a positive number η such that

E
[∥∥∥(D)

∑
fξ(Wv −Wu)

∥∥∥2
V

]
< ε

for any δ-fine belated McShane partial division D = {([u, v], ξ)} of [0, T ] with

(D)
∑

(v − u) ≤ η.

Proof. Let ε > 0 be given. Then there exist a positive function δ on [0, T ] and a
number η > 0 such that for any (δ, η)-fone belated McShane partial division P of [0, T ],
wehave

E

[∥∥∥∥S(f, P, δ, η)− (IM)

∫ T

0
ftdWt

∥∥∥∥2
V

]
<
ε

4
.



J.D. Cagubcob, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 101-117 109

Let D = {([u, v], ξ)} be a δ-fine belated McShane partial division of [0, T ] with (D)
∑

(v−
u) ≤ η. Construct a (δ, η)-fine belated McShane partial division of [0, T ]. By assumption,

E

[∥∥∥∥(D ∪D1)
∑

fξ(Wv −Wu)− (IM)

∫ T

0
ftdWt

∥∥∥∥2
V

]
<
ε

4
.

Hence,

E
[∥∥∥(D)

∑
fξ(Wv −Wu)

∥∥∥2
V

]
≤ 2E

[∥∥∥∥(D ∪D1)
∑

fξ(Wv −Wu)− (IM)

∫ T

0
ftdWt

∥∥∥∥2
V

]

+ 2E

[∥∥∥∥(IM)

∫ T

0
ftdWt − (D1)

∑
fξ(Wv −Wu)

∥∥∥∥2
V

]
2
( ε

4

)
+ 2

( ε
4

)
= ε.

(6)

This proves the lemma.

Theorem 2. A process f : [0, T ]× Ω→ L(U, V ) is IM-integrable if and only if

(i) there exists an AC2[0, T ] function F : J × Ω→ V and

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

E
[∥∥∥(D)

∑
{fξ(Wv −Wu)− F [u, v]}

∥∥∥2
V

]
< ε.

Proof. Suppose that f ∈ ΛIM. By the Saks-Henstock lemma for IM integral, (ii)
holds. Next we show that F is AC2[0, T ]. Let ε > 0 be given. By Lemma 4, there exist a
positive function δ on [0, T ] and a number η > 0 such that

E
[∥∥∥(D)

∑
fξ(Wv −Wu)

∥∥∥2
V

]
<
ε

4

for any δ-fine belated McShane partial division D = {([u, v], ξ)} of [0, T ] with (D
∑

(v −
u)) ≤ η. Let {[aj , bj ]}mj=1 be a finite collection of disjoint subintervals [aj , bj ] ∈ J with∑m

j=1(bj −aj) ≤ η. Note that f is also IM-integrable on[aj , bj ] for all j. This means that
for all j, there exist a positive function δj on [aj , bj ] and a number ηj > 0 such that for
any (δj , ηj)-fine belated McShane partial division Dj of [aj , bj ], we have

E
[
‖S(f,Dj , δj , ηj)− F [aj , bj ]‖2V

]
<

ε

4 · 22j
.



J.D. Cagubcob, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 101-117 110

We can choose {δj}mj=1 and {ηj}mj=1 such that δj(ξ) ≤ δ(ξ) for all j and
∑m

j=1 ηj ≤ η. Let
P = D1 ∪D2 ∪ · · · ∪Dm, which is a δ-fine belated partial division of [0, T ] with

(P )
∑

(v − u) ≤
m∑
j=1

(bj − aj) ≤ η.

This implies that

E
[∥∥∥(P )

∑
fξ(Wv −Wu)

∥∥∥2
V

]
<
ε

4
.

Hence,

E

∥∥∥∥∥∥
m∑
j=1

F [aj , bj ]

∥∥∥∥∥∥
2

v


≤ 2E

∥∥∥∥∥∥
m∑
j=1

{F [aj , bj ]− S(f,Dj , δj , ηj)}

∥∥∥∥∥∥
2

V


+ 2E

∥∥∥∥∥∥
m∑
j=1

S(f,Dj , δj , ηj)

∥∥∥∥∥∥
2

V


≤ 2

 m∑
j=1

√
E
[
‖F [aj , bj ]− S(f,Dj , δj , ηj)‖2V

]2

2E
[∥∥∥(P )

∑
fξ(Wv −Wu)

∥∥∥2
V

]

< 2

 ∞∑
j=1

√
ε

2 · 2j

2

+ 2
( ε

4

)
≤ ε.

(7)

Thus, F is AC2[0, T ].
Conversely, assume that (i) and (ii) hold. Let ε > 0 be given. Since F is AC2[0, T ],

choose η > 0 such that whenever {[uj , vj ]}mj=1 is a finite collection of subintervals [uj , vj ] ∈
J with

∑m
j=1(vj − uj) ≤ η, we have

E

∥∥∥∥∥∥
m∑
j=1

F [uj , vj ]

∥∥∥∥∥∥
2

V

 < ε

4
.

Let D = {([u, v], ξ)} be a (δ, η)-fine belated McShane partial division of [0, T ] and let
Dc = {[u, v]} be the collection of all subintervals [u, v] ⊂ [0, T ] which are not included in
the set D. Since F is AC2[0, T ],

E
[∥∥∥(Dc)

∑
F [u, v]

∥∥∥2
v

]
<
ε

4
.



J.D. Cagubcob, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 101-117 111

Hence,

E
[∥∥∥(D)

∑
fξ(Wv −Wu)− F [0, T ]

∥∥∥2
V

]
≤ 2E

[∥∥∥(D)
∑
{fξ(Wv −Wu)− F [u, v]}

∥∥∥2
V

]
+ 2E

[∥∥∥(Dc)
∑

F [u, v]
∥∥∥2
V

]
< 2

( ε
4

)
+ 2

( ε
4

)
= ε.

(8)

Thus, f is IM-integrable to F [0, T ].

Lemma 5. Let f ∈ ΛIM and define F : J × Ω→ V by

F [u, v] := (IM)

∫ v

u
ft dWt.

(i) F has the orthogonal increment property,

(ii) E [〈fc(Wb −Wa), F [u, v]〉V ] = 0, where c ≤ a.

Proof. We shall only prove (i) since (ii) follows the same arguments in (i). By the
sequential definition of IM integral, there exists a decreasing sequence {δn} of positive
functions defined on [0, T ] and a decreasing sequence {ηn} of positive numbers such that

for any (δn, ηn)-fine belated McShane partial division Dn[a, b] = {([u(n)i , v
(n)
i , ξ

(n)
i ])}mi=1 and

Dn[u, v] = {([u(n)j , v
(n)
j , ξ

(n)
j ])}pj=1 of [a, b] and [u, v], respectively, we have

E
[
‖S(f,Dn[a, b], δn, ηn)− F [a, b]‖2V

]
→ 0 as n→∞

and
E
[
‖S(f,Dn[u, v], δn, ηn)− F [u, v]‖2V

]
→ 0 as n→∞.

By Lemma 2, for every n ∈ N

E

 m∑
i=1

p∑
j=1

〈
fξi(n)(W

v
(n)
i

−W
u
(n)
i

), f
ξ
(n)
j

(W
v
(n)
j

−W
u
(n)
j

)

〉
V

 = 0.

Since S(f,Dn[a, b], δn, ηm) → F [a, b] and S(f,Dn[u, v], δn, ηm) → F [u, v] in L2(Ω, V ) as
n→∞, it follows that

E [〈S(f,Dn[a, b], δn, ηm), S(f,Dn[u, v], δn, ηn)〉V ]→ E [〈F [a, b], F [u, v]〉V ]

as n→∞. Thus, E [〈F [a, b], F [u, v]〉V ] = 0.

In view of Lemma 5, we have the following lemma.



J.D. Cagubcob, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 101-117 112

Lemma 6. Let f ∈ ΛIM and define F : J × Ω→ V by

F [u, v] := (IM)

∫ v

u
ft dWt.

Let {([ui, vi], ξi)}ni=1 be a finite collection such that {[ui, vi]} is a collection of non-overlapping
intervals in J , ξ1 < ξ2 < · · · < ξn, and ξi ≤ ui for each i = 1, 2, . . . , n. Then

(i) E

∥∥∥∥∥
n∑
i=1

{fξi(Wvi −Wui)− F (ui, vi)}

∥∥∥∥∥
2

V


=

n∑
i=1

E
[
‖fξi(Wvi −Wui)− F (ui, vi)‖2V

]
;

(ii) E

∥∥∥∥∥
n∑
i=1

F (ui, vi)

∥∥∥∥∥
2

V

 =
n∑
i=1

E
[
‖F (ui, vi)‖2V

]
.

The immediate consequence of Lemma 5.(i) is the strong version of Saks-Henstock
lemma.

Lemma 7. (Saks-Henstock Lemma (Strong Version)). Let f ∈ ΛIM and let F : J ×

Ω → V be defined by F (u, v) := (IM)

∫ v

u
ft dWt. Then for every ε > 0, there exists

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

(D)
∑

E
[
‖fξ(Wv −Wu)− F (u, v)‖2V

]
< ε.

4. Descriptive Definition of Itô-McShane Integral

In this section, we present a version of Fundamental Theorem for the Itô-McShane
integral of an operator-valued stochastic process.

Theorem 3. Let f ∈ ΛIM and let

F [u, v] := (IM)

∫ v

u
f dW for all [u, v] ⊂ [0, T ].

Then DFt = ft a.e. on [0, T ).

Proof. Let A = {t ∈ [0, T ) : DFt does not exist or DFt 6= ft}. Let ξ ∈ A. Then there
exists γ(ξ) > 0 such that for every positive function δ on [0, T ], there exists a δ-fine belated
McShane interval-point pair ([u, v], ξ) of [0, T ] with

E
[
‖fξ(Wv −Wu)− F [u, v]‖2V

]
≥ γ(ξ)(v − u). (9)



J.D. Cagubcob, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 101-117 113

Let ε > 0. By the strong version of Saks-Henstock lemma (Lemma 7), there exists a
positive function δ1 on [0, T ] such that for any δ1-fine belated McShane partial division
D = {([u, v], ξ)} of [0, T ], we have

(D)
∑

E
[
‖fξ(Wv −Wu)− F [u, v]‖2V

]
< ε. (10)

Let ξ1, ξ2, . . . , ξn ∈ A. By (9), each ξi corresponds a δ1-fine belated interval [ui, vi], for
i = 1, 2, . . . , n. Thus,

n∑
i=1

γ(ξi)(vi − ui) ≤
n∑
i=1

E
[
‖fξi(Wvi −Wui)− F [ui, vi]‖2V

]
< ε.

For m ∈ N, let Am =

{
t ∈ A : γ(t) ≥ 1

m

}
. Then A =

⋃
m∈N

Am. Hence, for all i ∈

{1, 2, . . . , n}, ξi ∈ Ami for some mi ∈ N. Let m : max{mi : i = 1, 2, . . . , n}. Then

n∑
i=1

1

m
(vi − ui) ≤

n∑
i=1

γ(ξi)(vi − ui) < ε

which implies that
n∑
i=1

(vi−ui) < mε. Let V be the family of interval-point pairs {([u, v], ξ)}

induced from all δ1-fine belated McShane partial division of [0, T ] such that ξ ∈ Am. Then
V is a Vitali cover of Am. Applying the Vitali Covering Lemma, there exists a finite
collection {([ui, vi], ξi)}ni=1 in V such that

Leb∗(Am) <
n∑
i=1

(vi − ui) + ε < (m+ 1)ε.

Since ε is arbitrary, Leb(Am) = 0. Thus, Leb(A) = 0.

Theorem 4. Let f : [0, T ]× Ω→ L(U, V ) be an adapted process and let F : J × Ω→ V
be AC2[0, T ], has the orthogonal increment property, and DFt = ft a.e. on [0, T ). Then
f ∈ ΛIM and

F [u, v] := (IM)

∫ v

u
fs dWs for all [u, v] ⊂ [0, T ].

Proof. Let A = {t ∈ [0, T ) : DFt does not exist or DFt 6= ft}. Then Leb(A) = 0. Let
ξ ∈ Ac := [0, T ]\A. Then for every ε > 0, there exists a positive function δ1 on [0, T ] such
that for any δ1-fine belated McShane interval-point pair ([u, v], ξ) of [0, T ], we have

E
[
‖fξ(Wv −Wu)− F [u, v]‖2V

]
<

ε

4T
(v − u).

Let D1 = {([ui, vi], ξi)}ni=1 be a δ1-fine belated McShane partial division on [0, T ] with
ξi ∈ Ac. Then by Lemma 3,



J.D. Cagubcob, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 101-117 114

E

∥∥∥∥∥
n∑
i=1

{fξi(Wvi −Wui)− F [ui, vi]}

∥∥∥∥∥
2

V


=

n∑
i=1

E
[
‖fξi(Wvi −Wui)− F [ui, vi]‖2V

]
<

ε

4T

n∑
i=1

(vi − ui)

≤ ε

4
.

(11)

If A = ∅, then we are done. Suppose that A 6= ∅. Let ξ ∈ A. Then for any [u, v] ⊂ [ξ, T ],

E
[
‖fξ(Wv −Wu)‖2V

]
= (v − u)E

[
‖fξ‖2L2(UQ,V )

]
.

Let Gm =
m∑
k=1

〈fξ(Wv −Wu), gk〉2, where {gk} is an ONB in V . Since Gm → G :=

∞∑
k=1

〈fξ(Wv −Wu), gk〉2 as m → ∞ and Gm ≤ Gm+1, by the monotone convergence theo-

rem for Lebesgue integral,

lim
m→∞

E

[
m∑
k=1

〈fξ(Wv −Wu), gk〉2
]

= E

[ ∞∑
k=1

〈fξ(Wv −Wu), gk〉2
]

= E
[
‖fξ(Wv −Wu)‖2V

]
<∞.

It follows that there exists N ∈ N such that N − 1 ≤ E
[
‖fξ‖2L2(UQ,V )

]
< N .

Since F is AC2[0, T ], there exists η > 0 with η < ε
4N such that for all finite collection

{[uj , vj ]}pj=1 of non-overlapping intervals of [0, T ] with
∑p

j=1(vj − uj) < η, we have

E

∥∥∥∥∥∥
p∑
j=1

F [uj , vj ]

∥∥∥∥∥∥
2

V

 < ε

4
.

Since Leb(A) = 0, there exists an open set O containing A such that Leb(O) < η.
Hence, for all ξ ∈ A ⊆ O. Thus, there exists δ2(ξ) > 0 such that [ξi, ξi + δ2(ξ)] ⊂ O. Let
D2 = {([u, v], ξ)} be a δ-fine belated McShane partial division such that ξ ∈ A. Then,
(D2)

∑
(v − u) ≤ Leb(O) < η. Then by Lemma 3,



J.D. Cagubcob, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 101-117 115

E
[∥∥∥(D2)

∑
{fξ(Wv −Wu)− F [u, v]}

∥∥∥2
V

]
= (D2)

∑
E
[
‖fξ(Wv −Wu)− F [u, v]‖2V

]
≤ 2(D2)

∑
E
[
‖fξ(Wv −Wu)‖2V

]
+ 2(D2)

∑
E
[
‖F [u, v]‖2V

]
= 2(D2)

∑
(v − u)E

[
‖fξ‖2L2(UQ,V )

]
+ 2(D2)

∑
E
[
‖F [u, v]‖2V

]
< 2N(D2)

∑
(v − u) + 2(D2)

∑
E
[
‖F [u, v]‖2V

]
≤ 2N · ε

4N
+ 2 · ε

4
= ε.

(12)

Let D = {([u, v], ξ)} be a δ-fine belated McShane partial division of [0, T ]. Then using
(11) and (12), we have

E
[∥∥∥(D)

∑
{fξ(Wv −Wu)− F [u, v]}

∥∥∥2
V

]

≤ 2E

∥∥∥∥∥∥
∑
ξ∈Ac

{fξ(Wv −Wu)− F [u, v]}

∥∥∥∥∥∥
2

V


+2E

∥∥∥∥∥∥
∑
ξ∈A
{fξ(Wv −Wu)− F [u, v]}

∥∥∥∥∥∥
2

V


< 2

( ε
4

)
+ 2

( ε
4

)
=
ε

2
+
ε

2
= ε.

By Theorem 2, f ∈ ΛIM and

F [u, v] := (IM)

∫ v

u
fs dWs for all [u, v] ⊂ [0, T ].

Combining Theorem 1, Theorem 3, and Theorem 4, we get the following result, which
is referred to as the Fundamental Theorem or the descriptive definition of the Itô-McShane
integral for the Hilbert-Schmidt-valued stochastic process.

Theorem 5. Let f : [0, T ] × Ω → L(U, V ) be an adapted process. Then f ∈ ΛIM if and
only if there exists an AC2[0, T ] function F : J × Ω → V that satisfies the orthogonal
increment property and DFt = ft a.e. on [0, T ).

5. Conclusion and Recommendation

In this paper, we formulate an equivalent definition of the Itô-McShane integral of
a operator-valued stochastic process with respect to a Hilbert space-valued Q-Wiener



REFERENCES 116

process using the concept of belated McShane derivative 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 authors would like to acknowledge the financial support from the Department of
Science and Technology-Accelerated Science and Technology Human Resource Develop-
ment Program (DOST-ASTHRDP) and to thank the unknown referee for reviewing this
paper.

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