Backwards Itunhbox voidb@x �group let unhbox voidb@x setbox @tempboxa hbox {oglobal mathchardef accent@spacefactor spacefactor }accent 94 oegroup spacefactor accent@spacefactor -Henstock Integral for the Hilbert-Schmidt-Valued Stochastic Process


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 1, 2019, 58-78
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Backwards Itô-Henstock Integral for the
Hilbert-Schmidt-Valued Stochastic Process

Ricky F. Rulete1, Mhelmar A. Labendia2,∗

1 Department of Mathematics and Statistics, College of Arts and Sciences,
University of Southeastern Philippines, Bo. Obrero, 8000 Davao City, Philippines
2 Department of Mathematics and Statistics, College of Science and Mathematics,
Mindanao State University-Iligan Institute of Technology, 9200 Iligan City, Philippines

Abstract. In this paper, a definition of backwards Itô-Henstock integral for the Hilbert-Schmidt-
valued stochastic process is introduced. We formulate the Itô isometry for this integral. Moreover,
an equivalent definition for this integral is given using the concept of AC2[0, T ]-property, a version
of absolute continuity.

2010 Mathematics Subject Classifications: 60H30, 60H05

Key Words and Phrases: Backwards Itô-Henstock integral, Itô Isometry, AC2-property

1. Introduction

The most well-known integral is the Riemann integral. It was formulated by Bernhard
Riemann in 1850. This is the first integral introduced to most students in the study of
elementary calculus. However, the class of Riemann-integrable functions is quite limited.
Henri Lebesgue attempts to solve some of the shortcomings of the Riemann integral.
However, for non-mathematicians the Lebesgue integral is difficult to understand and
requires enough background of measure theory. In 1950s, a Riemann-type integral was
discovered independently by R. Henstock and J. Kurzwiel. This integral includes Riemann
and that of Lebesgue. This integral is now known as Henstock-Kurzwiel or HK integral. In
this paper, however, we will call this integral simply as Henstock integral. The Henstock
integral used non-uniform meshes in contrast to Riemann. Such technique turns out to
encompass the classical stochastic integral, see([7], [8], [9], [13] and [14]]). This technique
is now known as the Henstock approach.

In stochastic calculus, the stochastic integral of a real-valued adapted process is ob-
tained from the mean square limit of stochastic integrals of simple processes, see [16].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i1.3342

Email addresses: ricky rulete@yahoo.com.ph (R. Rulete),
mhelmar.labendia@g.msuiit.edu.ph (M. Labendia)

http://www.ejpam.com 58 c© 2019 EJPAM All rights reserved.



R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 59

This is the classical approach to stochastic integration which is almost similar in defining
the Lebesgue integral of a measurable function. Hence, Henstock approach to stochastic
integration have been studied in several papers see([15], [17], [21], [22] and [23]) since it
gives more explicit definition, reduces the technicalities in the classical way of defining the
stochastic integral and is less measure theoretic.

In [6], [19], and [18], the concept of stochastic integral has been extended to infinite-
dimensional spaces, namely Hilbert and Banach spaces. In a Hilbert space, the stochastic
integral is presented in a manner similar to the real-valued case. The integrator is Q-
Wiener process, a Hilbert space-valued Wiener process which is dependent on a symmetric
nonnegative trace-class operator Q and the integrand is an operator-valued stochastic
process. In a general Banach space, however, there seems to be no unifying treatment of
stochastic integration.

In 2018, Labendia, et.al. [11], introduced the (forward) Itô-Henstock integral of an
operator-valued stochastic process with respect to a Hilbert space-valued Q-Wiener pro-
cess. This integral uses (forward) filtration. Moreover, the δ-fine partial division is belated
in the sense that the associated points (or tags) are always on the left endpoints of the
subintervals. They formulated a version of Itô’s formula and gave an alternative defini-
tion of the classical Itô integral of an L(U, V )-valued stochastic process using Henstock
approach, where U and V are separable Hilbert spaces and L(U, V ) is the space of all
bounded linear operators Q : U → V . In [10], the (forward) Itô-Henstock integral has
been characterized using AC2[0, T ]-property, a version of absolute continuity.

The backwards Itô integral with respect to a Brownian motion was defined by Arcede
and Cabral in 2011, see [3]. In this integral, all processes start at a fix time T > 0 and
then proceed backwards to some earlier time s. Henstock approach was used together with
the notions of backwards δ-fine partial division (backwards in the sense that the tags are
the right endpoints of the disjoint left-open subintervals) and backwards filtration. One
of their results are the fundamental theorem of calculus, integration-by-parts and the Itô
formula for backwards Itô integral see([4], [5]).

In this paper, we define the backwards Itô-Henstock integral of an operator-valued
stochastic process with respect to a Hilbert space-valued Q-Wiener process which is ac-
tually an extension of the work of Arcede and Cabral in [3]. Here, we formulate the Itô
isometry and give an equivalent definition using the concept of AC2 property, a version of
absolute continuity.

2. Preliminaries

Throughout this paper, R denotes the set of real numbers, R+
0 denotes the set of

nonnegative real numbers, N the set of positive integers and {Ω,G,P} denotes a probability
space.

Let {Gt : 0 ≤ t ≤ T} be a family of sub σ-field of G. Then {Gt : 0 ≤ t ≤ T} is called
a backwards filtration if Gt ⊆ Gs for all 0 ≤ s ≤ t ≤ T . If in addition, {Gt : 0 ≤ t ≤ T}
satisfies the following condition: (1) GT contains all sets of P-measure zero in G; and (2)
for each t ∈ [0, T ], Gt = Gt− :=

⋂
s<t Gs. Then {Gt : 0 ≤ t ≤ T} is called a standard



R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 60

backwards filtration. We often write {Gt} instead of {Gt : 0 ≤ t ≤ T}. See [1].
Let H be a separable Banach space. A stochastic process f or simply process is a

function f : [0, T ]× Ω→ H, where [0, T ] is an interval in R+
0 and f(·, t) is Gt-measurable

for each t ∈ [0, T ]. A process f = {ft : t ∈ [0, T ]} is said to be backwards adapted to a
standard backwards filtration {Gt} if ft is Gt-measurable for each t ∈ [0, T ].

Let U and V be separable Hilbert spaces. Denote 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.

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 ofQ is defined by trQ =

∑∞
j=1 〈Qej , ej〉U .

It is shown in [20] that tr Q is well-defined and may be defined in terms of an arbitrary

ONB. Moreover, there exists a unique operator Q
1
2 ∈ L(U) such that Q

1
2 is nonnegative

definite and (Q
1
2 )2 = Q. An operator Q : U → U is said to be trace-class if tr [Q] :=

tr (QQ∗)
1
2 < ∞. 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, and λj → 0 as j → ∞ [20, 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 and 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 [18, p.90], [6, 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 [19, 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 [18, p.418],

[19, p.111]. It is shown in [6, p.25] that L(U, V ) is properly contained in L2(UQ, V ).
We fix an element Q ∈ L(U), symmetric nonnegative definite trace-class operator.

A U -valued stochastic process Wt, t ∈ [0, T ], on a probability space (Ω,G,P) is called a
Q-Wiener process in U if:

(i) W (0, ω) = 0U for each ω ∈ Ω,

(ii) W has P-almost surely (abbrev. as P-a.s.) continuous trajectories, i.e.,

W (·, ω) : [0, T ]→ U is P-a.s. continuous

(iii) the increments of W are independent, i.e. the random variables

Wt1 ,Wt2 −Wt1 ,Wt3 −Wt2 , . . . ,Wtn −Wtn−1

are independent for all 0 ≤ t1 < · · · < tn ≤ T , n ∈ N ,



R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 61

(iv) the increments have the following Gaussian laws:

P ◦ (Wt −Ws)
−1 = N (0, (t− s)Q) for all 0 ≤ s ≤ t ≤ T .

By Proposition 4.2 (see [18, p.88]), such a Q-Wiener process exists.
We define N := {A ∈ G | P(A) = 0}, G̃t := σ(WT −Ws | t ≤ s ≤ T ), G̃0

t := σ(G̃t ∪ N )
and

Gt :=
⋂
s<t

G̃0
s , t ∈ [0, T ]. (1)

Since N ⊆ G̃0
s for all s ∈ [0, T ] and {Gt}0≤t≤T is decreasing, we have the following result:

Proposition 1. Let t ∈ [0, T ]. Then the filtration Gt given in (1) is a standard backwards
filtration.

We note that the distance from an element u ∈ U to a nonempty subset A ⊂ U ,
denoted by dist(u,A) is defined to be

dist(u,A) = inf
a∈A
||u− a||U .

Proposition 2. Let Wt, t ∈ [0, T ], be an arbitrary U -valued Q-Wiener process on a
probability space (Ω,G,P). Then Wt −Ws is independent of Gt for all 0 ≤ s ≤ t ≤ T ,
where Gt is given in (1).

Proof. Let 0 ≤ s ≤ t ≤ T . Since a U -valued Q-Wiener process has independent
increments, Wt−Ws and WT −Wt are independent. It follows that Wt−Ws and WT −W ′t
are independent for all t ≤ t′ ≤ T . Hence, Wt−Ws is independent of σ(WT −W ′t : t ≤ t′ ≤
T ) = G̃t. Also, Wt−Ws is independent of G̃0

t . To prove now that Wt−Ws is independent
of Gt it is enough to show that

P ({Wt −Ws ∈ A} ∩B) = P({Wt −Ws ∈ A}) · P(B)

for any B ∈ Gt and any closed subset A ∈ U as {A ⊂ U | A closed} generates B(U) and is
stable under finite intersection. But we have

P ({Wt −Ws ∈ A} ∩B) = E
[
1{Wt−Ws∈A} · 1B

]
= E [1A ◦ (Wt −Ws) · 1B] .

Let F : Ω→ {0, 1} be defined by

F (ω) = lim
n→∞

(1− ndist((Wt −Ws)(ω), A)) ∨ 0, ω ∈ Ω

where ∨ denotes “maximum”. Let ω ∈ Ω. If (Wt − Ws)(ω) ∈ A, then dist((Wt −
Ws)(ω), A) = 0 and F (ω) = 1 = (1A ◦Wt−Ws)(ω). Otherwise, dist((Wt−Ws)(ω), A) > 0
and F (ω) = 0 = (1A ◦Wt −Ws)(ω). Hence, 1A ◦ (Wt −Ws) = F . So, we have

P ({Wt −Ws ∈ A} ∩B) = E
[(

lim
n→∞

(1− ndist(Wt −Ws, A)) ∨ 0
)
· 1B

]



R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 62

= E
[

lim
n→∞

((1− ndist(Wt −Ws, A)) ∨ 0) · 1B
]

= lim
n→∞

E [((1− ndist(Wt −Ws, A)) ∨ 0) · 1B] .

Moreover, for each ω ∈ Ω, we have

dist((Wt −Ws)(ω), A) = inf
a∈A
||(Wt −Ws)(ω)− a||U

= inf
a∈A

∥∥∥ lim
m→∞

Wt− 1
m

(ω)−Ws(ω)− a
∥∥∥
U

= lim
m→∞

inf
a∈A

∥∥∥Wt− 1
m

(ω)−Ws(ω)− a
∥∥∥
U

= lim
m→∞

dist((Wt− 1
m
−Ws)(ω), A).

This implies that

P ({Wt −Ws ∈ A} ∩B) = lim
n→∞

E
[(

(1− n lim
m→∞

dist(Wt− 1
m
−Ws, A)) ∨ 0

)
· 1B

]
= lim

n→∞
E
[

lim
m→∞

(
(1− ndist(Wt− 1

m
−Ws, A)) ∨ 0

)
· 1B

]
= lim

n→∞
lim
m→∞

E
[(

(1− ndist(Wt− 1
m
−Ws, A)) ∨ 0

)
· 1B

]
.

Since Wt− 1
m
−Ws is independent of G̃0

t− 1
m

⊇ Gt if m is large, we have

P ({Wt −Ws ∈ A} ∩B) = lim
n→∞

lim
m→∞

E
[
(1− ndist(Wt− 1

m
−Ws, A)) ∨ 0

]
· E [1B]

= P({Wt −Ws ∈ A}) · P(B).

This completes the proof.
�

From now onwards, the backwards filtered probability shall mean a filtered probability
space such that Wt is adapted to Gt and Wt−Ws is independent of Gt for all 0 ≤ s ≤ t ≤ T .

3. Backwards Itô-Henstock Integral

In this section, we shall present the backwards Itô-Henstock integral and some related
results.

Let δ be a positive function on (0, T ]. A finite collection D = {((ui, ξi], ξi)}ni=1 of
interval-point pairs is said to be a backwards partial division of [0, T ] if {(ui, ξi]}ni=1 is a
finite collection of disjoint subintervals of (0, T ]. An interval-point pair ((u, ξ], ξ) is said
to be backwards δ-fine if (u, ξ] ⊆ (ξ − δ(ξ), ξ], whenever (u, ξ] ⊆ (0, T ] and ξ ∈ (0, T ]. We
call D = {((ui, ξi], ξi)}ni=1 a backwards δ-fine partial division of [0, T ] if D is a backwards
partial division of [0, T ] and for each i, the interval-point pair ((ui, ξi], ξi) is backwards
δ-fine.



R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 63

We note that given any positive function δ, one may not be able to find a full division
that covers the entire interval (0, T ]. For instance, let δ(ξ) = ξ/2. Then the interval (0, T ]
cannot be covered by any finite collection of backwards δ-fine intervals.

Given η > 0, a given backwards δ-fine partial division D = {((ui, ξi], ξi)}ni=1 is said to
be backwards (δ, η)-fine partial division of [0, T ] if it fails to cover (0, T ] by at most length
η, that is, ∣∣∣∣∣T − (D)

n∑
i=1

(ξi − ui)

∣∣∣∣∣ ≤ η.
We are now ready to define the backwards Itô-Henstock integral. Throughout the

following 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-Weiner process.

Definition 1. Let f : [0, T ] × Ω → L2(UQ, V ) be a backwards adapted process. Then f
is said to be backwards Itô-Henstock integrable, or IHB-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 such that for any backwards (δ, η)-fine partial division
D = {((ui, ξi], 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(Wξ −Wu) :=
n∑
i=1

fξi(Wξi −Wui).

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

which will be denoted by (IHB)
∫ T

0 ft dWt or (IHB)
∫ T

0 f dW .

Refer to [11, Lemma 3.5 and Lemma 3.6] for the proofs of the following two lemmas.
When we speak of a subinterval of [0, T ], we shall mean that the subinterval is either a
closed interval [v, ξ] or half-open interval (v, ξ].

Lemma 1. Let f : [0, T ]×Ω→ L2(UQ, V ) be a backwards adapted process and {[vi, ξi]}ni=1

be a finite collection of disjoint subintervals of [0, T ]. Then

E

∑
i<p

〈
fξi(Wξi −Wvi , ), fξp(Wξp −Wvp)

〉
V

 = 0.

Lemma 2. Let f : [0, T ]×Ω→ L2(UQ, V ) be a backwards adapted process and {[vi, ξi]}ni=1

be a finite collection of disjoint subintervals of [0, T ]. Then

E

∥∥∥∥∥
n∑
i=1

fξi(Wξi −Wvi)

∥∥∥∥∥
2

V

 =

n∑
i=1

E
[
‖fξi(Wξi −Wvi)‖

2
V

]
=

n∑
i=1

(ξi− vi)E
[
‖fξi‖

2
L2(UQ,V )

]
.



R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 64

Theorem 1. Let v, ξ ∈ [0, T ] with v < ξ. Then

(i) E
[
||Wξ −Wv||2U

]
= (ξ − v)trQ;

(ii) E
[
||Wξ −Wv||4U

]
= (ξ − v)2

2

∞∑
j=1

λ2
j + (trQ)2

.

Proof. Let v, ξ ∈ [0, T ] with v < ξ. Then

(i) E
[
||Wξ −Wv||2U

]
= E

 ∞∑
j=1

〈Wξ −Wv, ej〉2U

 =
∞∑
j=1

E
[
〈Wξ −Wv, ej〉2U

]
. Since

〈Wt, ej〉U√
λj

is a Brownian motion,

E

(〈Wξ −Wv, ej〉U√
λj

)2
 = ξ − v.

Hence,

E
[
||Wξ −Wv||2U

]
=
∞∑
j=1

(ξ − v)λj = (ξ − v)
∞∑
j=1

λj = (ξ − v)trQ.

(ii) Note that

E
[
||Wξ −Wv||4U

]
= E

 ∞∑
j=1

〈Wξ −Wv, ej〉2U

2
= E

 ∞∑
j=1

〈Wξ −Wv, ej〉4U +
∑
j 6=l
〈Wξ −Wv, ej〉2U 〈Wξ −Wv, el〉2U


=
∞∑
j=1

E 〈Wξ −Wv, ej〉4U +
∑
j 6=l

E
[
〈Wξ −Wv, ej〉2U

]
E
[
〈Wξ −Wv, el〉2U

]
.

Since
〈Wt, ej〉U√

λj
is a Brownian motion,

E

(〈Wξ −Wv, ej〉U√
λj

)4
 = 3(ξ − v)2 and E

(〈Wξ −Wv, ej〉U√
λj

)2
 = ξ − v.



R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 65

Thus,

E
[
||Wξ −Wv||4U

]
=
∞∑
j=1

3λ2
j (ξ − v)2 +

∑
j 6=l

λjλl(ξ − v)2

= (ξ − v)2

2

∞∑
j=1

λ2
j +

 ∞∑
j=1

λ2
j +

∑
j 6=l

λjλl


= (ξ − v)2

2
∞∑
j=1

λ2
j +

 ∞∑
j=1

λj

2
= (ξ − v)2

2
∞∑
j=1

λ2
j + (trQ)2

 .
Thereby, completing the proof. �

Example 1. Let W : [0, T ] × Ω → U be a Q-Weiner process. Then 〈Wt, ·〉U is IHB-
integrable on [0, T ] and

(IHB)

∫ T

0
〈Wt, ·〉U dWt =

1

2
(||WT ||2U + T (trQ)).

Proof. We shall consider first the following claims.
Claim 1. Let D = {[vi, ξi]}ni=1 be a finite collection of disjoint subintervals of [0, T ].

Then

E

∣∣∣∣∣(D)
n∑
i=1

{
||Wξi −Wvi ||2U − (ξi − vi)trQ

}∣∣∣∣∣
2
 = 2M

[
(D)

n∑
i=1

(ξi − vi)2

]

where M =
∑∞

j=1 λ
2
j <∞.

To verify the first claim,

E

∣∣∣∣∣(D)
n∑
i=1

{
||Wξi −Wvi ||2U − (ξi − vi)trQ

}∣∣∣∣∣
2


=
n∑
i=1

E
[(
||Wξi −Wvi ||2U − (ξi − vi)trQ

)2]
+ 2

∑
i<p

E
[(
||Wξi −Wvi ||2U − (ξi − vi)trQ

)
(
||Wξp −Wvp ||2U − (ξp − vp)trQ

)]
.



R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 66

Note that∑
i<p

E
[(
||Wξi −Wvi ||2U − (ξi − vi)trQ

) (
||Wξp −Wvp ||2U − (ξp − vp)trQ

)]
=
∑
i<p

E
[
E
[(
||Wξi −Wvi ||2U − (ξi − vi)trQ

)
(
||Wξp −Wvp ||2U − (ξp − vp)trQ

) ∣∣∣∣Gξi]]
=
∑
i<p

[(ξi − vi)(ξp − vp)(trQ)2 − (ξi − vi)(ξp − vp)(trQ)2

− (ξp − vp)(ξi − vi)(trQ)2 + (ξi − vi)(ξp − vp)(trQ)2]

= 0.

It follows that

E

∣∣∣∣∣(D)
n∑
i=1

{
||Wξi −Wvi ||2U − (ξi − vi)trQ

}∣∣∣∣∣
2


=
n∑
i=1

E
[(
||Wξi −Wvi ||2U − (ξi − vi)trQ

)2]
.

By Theorem 1,

n∑
i=1

E
[(
||Wξi −Wvi ||2U − (ξi − vi)trQ

)2]
=

n∑
i=1

E
[
||Wξi −Wvi ||4U − 2(ξi − vi)||Wξi −Wvi ||2U trQ+ (ξi − vi)2(trQ)2

]
=

n∑
i=1

(ξi − vi)2

2
∞∑
j=1

λ2
j + (trQ)2

− 2(ξi − vi)2(trQ)2 + (ξi − vi)2(trQ)2


=

n∑
i=1

(ξi − vi)2

2

∞∑
j=1

λ2
j + (trQ)2

− (ξi − vi)2(trQ)2


=

n∑
i=1

(ξi − vi)2

2

∞∑
j=1

λ2
j + (trQ)2 − (trQ)2


=

n∑
i=1

(ξi − vi)2

2

∞∑
j=1

λ2
j


=

2
∞∑
j=1

λ2
j

( n∑
i=1

(ξi − vi)2

)
.



R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 67

This proves claim 1.

Claim 2. E
[
‖〈Wt, ·〉U‖

2
L2(UQ,R)

]
= tM .

E
[
‖〈Wt, ·〉U‖

2
L2(UQ,R)

]
= E

 ∞∑
j=1

∣∣∣〈Wt,
√
λjej

〉
U

∣∣∣2


=
∞∑
j=1

λjE
[
〈Wt −W0, ej〉2U

]
= t

∞∑
j=1

λj 〈λjej , ej〉2U

= t
∞∑
j=1

λ2
j

= tM.

This proves Claim 2.

Claim 3. 〈Wt, ·〉U is IHB-integrable to
1

2
(||WT ||2U + T trQ) on [0, T ].

Let ε > 0 be given. Let M =
∞∑
j=1

λ2
j . Choose a constant function δ on [0, T ] defined by

δ(t) =
ε

2MT
and a number η =

ε

12MT
. Let D = {((v, ξ], ξ)} be a backwards (δ, η)-fine

partial division of [0, T ]. Let Dc be the collection of all subintervals of [0, T ] which are not
included in D. Then

E

[∣∣∣∣(D)
∑
〈Wξ, ·〉U (Wξ −Wv)−

1

2
||WT ||2U −

1

2
T (trQ)

∣∣∣∣2
]

= E

[∣∣∣∣(D)
∑
〈Wξ,Wξ −Wv〉U −

1

2
||WT ||2U −

1

2
T (trQ)

∣∣∣∣2
]

= E
[∣∣∣∣(D)

∑{
〈Wξ,Wξ −Wv〉U −

1

2
(||Wξ||2U − ||Wv||2U )− 1

2
(ξ − v)trQ

}
+(Dc)

∑{
−1

2
(||Wξ||2U − ||Wv||2U )− 1

2
(ξ − v)trQ

}∣∣∣∣2
]

≤ 2E

[∣∣∣∣(D)
∑{

〈Wξ,Wξ −Wv〉U −
1

2
(||Wξ||2U − ||Wv||2U )− 1

2
(ξ − v)trQ

}∣∣∣∣2
]

+ 2E

[∣∣∣∣(Dc)
∑{

−1

2
(||Wξ||2U − ||Wv||2U )− 1

2
(ξ − v)trQ

}∣∣∣∣2
]

=
1

2
E
[∣∣∣(D)

∑{
−2 〈Wξ,Wξ −Wv〉U + ||Wξ||2U − ||Wv||2U + (ξ − v)trQ

}∣∣∣2]



R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 68

+
1

2
E
[∣∣∣(Dc)

∑{
||Wξ||2U − ||Wv||2U + (ξ − v)trQ

}∣∣∣2]
=

1

2
E
[∣∣∣(D)

∑{
−||Wξ −Wv||2U + (ξ − v)trQ

}∣∣∣2]
+

1

2
E
[∣∣∣(Dc)

∑{
||Wξ||2U − ||Wv||2U − 2 〈Wξ,Wξ −Wv〉U

+2 〈Wξ,Wξ −Wv〉U + (ξ − v)trQ
}∣∣2]

≤ 1

2
E
[∣∣∣(D)

∑{
||Wξ −Wv||2U − (ξ − v)trQ

}∣∣∣2]
+ E

[∣∣∣(Dc)
∑{

||Wξ −Wv||2U − (ξ − v)trQ
}∣∣∣2]

+ 4E
[∣∣∣(Dc)

∑
〈Wξ,Wξ −Wv〉U

∣∣∣2] .
By Claim 1, Claim 2, and Lemma 2, we have

E

[∣∣∣∣(D)
∑
〈Wξ, ·〉U (Wξ −Wv)−

1

2
||WT ||2U −

1

2
T (trQ)

∣∣∣∣2
]

≤M
[
(D)

∑
(ξ − v)2

]
+ 2M

[
(Dc)

∑
(ξ − v)2

]
+ 4 · (Dc)

∑
(ξ − v)ξM

< MTδ + 2MTη + 4MTη

= MT
( ε

2MT

)
+ 6MT

( ε

12MT

)
= ε.

Thus, 〈Wt, ·〉U is IHB-integrable on [0, T ] and

(IHB)

∫ T

0
〈Wt, ·〉U dWt =

1

2
(||WT ||2U + T trQ).

�

The following statements show that the backwards Itô-Henstock integral possesses the
standard properties of an integral. Refer to [12] for analogous proofs.

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

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

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

(IHB)

∫ T

0
(f + g) dW = (IHB)

∫ T

0
f dW + (IHB)

∫ T

0
g dW ;



R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 69

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

(IHB)

∫ T

0
(αf) dW = α · (IHB)

∫ T

0
f dW.

(3) If f : [0, T ]× Ω→ L2(UQ, V ) is IHB-integrable on [0, c] and [c, T ] where c ∈ (0, T ),
then f is IHB-integrable on [0, T ] and

(IHB)

∫ T

0
f dW = (IHB)

∫ c

0
f dW + (IHB)

∫ T

c
f dW.

(4) If f : [0, T ]×Ω→ L2(UQ, V ) is IHB-integrable on [0, T ], then f is also IHB-integrable
on every subinteval [c, d] of [0, T ].

(5) A process f : [0, T ] × Ω → L2(UQ, V ) is IHB-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
backwards (δn, ηn)-fine partial division Dn of [0, T ], we have

lim
n→∞

E
[
‖S(f,Dn, δn, ηn)−A‖2V

]
= 0.

In this case,

A = (IHB)

∫ T

0
ft dWt.

(6) (Cauchy criterion). A process f : [0, T ] × Ω → L2(UQ, V ) is IHB-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 backwards (δ, η)-fine 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 IHB-integrable on [0, T ]

and F (u, v) := (IHB)

∫ v

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

there exist a positive function δ on (0, T ] and a positive number η such that for any
backwards (δ, η)-fine partial division D of [0, T ], we have

E
[∥∥∥(D)

∑
{fξ(Wξ −Wv)− F (v, ξ)}

∥∥∥2

V

]
< ε.



R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 70

4. Itô Isometry and AC2[0, T ]-property

This section presents the Itô isometry and the equivalent definition of backwards Itô-
Henstock using the notion of AC2[0, T ]-property. Before we proceed with the Itô isometry,
we need to define the backwards Henstock integral which is equivalent to the Lebesgue
integral (see [2]).

Definition 2. A real-valued function f defined on [0, T ] is said to be Lebesgue integrable
to A ∈ R if given ε > 0, there exists a positive function δ on (0, T ] and a real constant
η > 0 such that ∣∣∣(D)

∑
f(ξ)(ξ − v)−A

∣∣∣ < ε

whenever D is a backwards δ-fine partial division of [0, T ] with (D)
∑

(ξ − v) > T − η. In

this case, A is called the Lebesgue integral of f which will be denoted by (L)

∫ T

0
f(t) dt.

Note that the backwards δ-fine partial division D of [0, T ] in Definition 2 is also a
backwards (δ, η)-fine partial division of [0, T ].

Theorem 2. The function f : [0, T ] → R is Lebesgue integrable to A ∈ R if and only if
there exists a decreasing sequence of positive functions {δn(ξ)} on (0, T ] and a decreasing
sequence of positive constants {ηn} such that

lim
n→∞

∣∣∣(Dn)
∑

f(ξ(n))(ξ(n) − v(n))−A
∣∣∣ = 0,

where Dn is any backwards (δn, ηn)-fine partial division of [0, T ].

Proof. Suppose that f : [0, T ] → R is Lebesgue integrable to A ∈ R. Then,
by Definition 2, for every ε = 1

n , n = 1, 2, 3, . . ., there exists a positive function δn
on (0, T ] and a positive number η such that for any backwards δn-fine partial division
Dn = {((v(n), ξ(n)], ξ(n))} of [0, T ] with (Dn)

∑
(ξ(n) − v(n)) > T − ηn we have∣∣∣(Dn)

∑
f(ξ(n))(ξ(n) − v(n))−A

∣∣∣ ≤ 1

n
.

Hence,

lim
n→∞

∣∣∣(Dn)
∑

f(ξ(n))(ξ(n) − v(n))−A
∣∣∣ = 0,

for any backwards (δn, ηn)-fine partial division Dn of [0, T ].
Conversely, let us assume that there exists A ∈ R and a decreasing sequence {δn(ξ)}

of positive functions on (0, T ] and a decreasing sequence of positive numbers {ηn} such
that

lim
n→∞

∣∣∣(Dn)
∑

f(ξ(n))(ξ(n) − v(n))−A
∣∣∣ = 0

Suppose that f is not Lebesgue integrable to A on [0, T ]. Then there exists ε > 0 such
that for every positive function δ on (0, T ] and every positive number η there exists a



R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 71

backwards δ-fine partial division D = {((v, ξ], ξ)} of [0, T ] with (D)
∑

(ξ−v) > T −η such
that ∣∣∣(D)

∑
f(ξ)(ξ − v)−A

∣∣∣ ≥ ε.
Hence, for each δn and ηn, there exists a δn-fine partial division Dn of [0, T ] with

(Dn)
∑

(ξ − v) > T − ηn such that
∣∣∣(Dn)

∑
f(ξ)(ξ − v)−A

∣∣∣ ≥ ε,
leading to a contradiction. �

We now state and prove the Itô isometry.

Theorem 3 (Itô Isometry). Let f be IHB-integrable on [0, T ]. Then E
[
‖ft‖2L2(UQ,V )

]
is

Lebesgue integrable on [0, T ] and

E

[∥∥∥∥(IHB)

∫ T

0
ft dWt

∥∥∥∥2

V

]
= (L)

∫ T

0
E
[
‖ft‖2L2(UQ,V )

]
dt <∞.

Proof. From property (5) section 3, there exists a decreasing sequence {δn(ξ)} of
positive functions defined on (0, T ], and a decreasing sequence of positive numbers {ηn}
such that for any backwards (δn, ηn)-fine partial division Dn = {((v(n)

i , ξ
(n)
i )], ξ

(n)
i )}p(n)

i=1 of
[0, T ], we have

lim
n→∞

E

[∥∥∥∥S(f,Dn, δn, ηn)− (IHB)

∫ T

0
ft dWt

∥∥∥∥2

V

]
= 0.

This means that

lim
n→∞

S(f,Dn, δn, ηn) = (IHB)

∫ T

0
ft dWt in L2(Ω, V ).

Let ε > 0 be given. Then there exists N ∈ N such that for all n ≥ N ,∥∥∥∥S(f,Dn, δn, ηn)− (IHB)

∫ T

0
ft dWt

∥∥∥∥
L2(Ω,V )

< ε.

Note that ∣∣∣∣∣‖S(f,Dn, δn, ηn)‖L2(Ω,V ) −
∥∥∥∥(IHB)

∫ T

0
ft dWt

∥∥∥∥
L2(Ω,V )

∣∣∣∣∣
≤
∥∥∥∥S(f,Dn, δn, ηn)− (IHB)

∫ T

0
ft dWt

∥∥∥∥
L2(Ω,V )

.

This implies that

lim
n→∞

‖S(f,Dn, δn, ηn)‖L2(Ω,V ) =

∥∥∥∥(IHB)

∫ T

0
ft dWt

∥∥∥∥
L2(Ω,V )



R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 72

lim
n→∞

√
E
[
‖S(f,Dn, δn, ηn)‖2V

]
=

√√√√E

[∥∥∥∥(IHB)

∫ T

0
ft dWt

∥∥∥∥2

V

]
.

Using Lemma 2, we have√√√√E

[∥∥∥∥(IHB)

∫ T

0
ft dWt

∥∥∥∥2

V

]
= lim

n→∞

√
E
[
‖S(f,Dn, δn, ηn)‖2V

]

= lim
n→∞

√√√√√E

∥∥∥∥∥∥
p(n)∑
i=1

f
ξ
(n)
i

(
W
ξ
(n)
i

−W
v
(n)
i

)∥∥∥∥∥∥
2

V


= lim

n→∞

√√√√p(n)∑
i=1

(
ξ

(n)
i − v(n)

i

)
E
[∥∥∥f

ξ
(n)
i

∥∥∥2

L2(UQ,V )

]
.

This implies that

lim
n→∞

p(n)∑
i=1

(
ξ

(n)
i − v(n)

i

)
E
[∥∥∥f

ξ
(n)
i

∥∥∥2

L2(UQ,V )

]
= E

[∥∥∥∥(IHB)

∫ T

0
ft dWt

∥∥∥∥2

V

]
.

Since the above equality holds for any backwards (δn, ηn)-fine partial division of [0, T ], by

Theorem 2, E
[∥∥∥f

ξ
(n)
i

∥∥∥2

L2(UQ,V )

]
is Lebesgue integrable on [0, T ] and

E

[∥∥∥∥(IHB)

∫ T

0
ft dWt

∥∥∥∥2

V

]
= (L)

∫ T

0
E
[
‖ft‖2L2(UQ,V )

]
dt <∞.

�

Throughout the following, denote by J the family of all left-open subintervals (v, ξ] of
[0, T ]. In the following, when no confusion arises, we may refer to F ((u, v], ·) or F ((u, v], ω)
as simply F (u, v).

Definition 3. 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 = {(v, ξ]} of disjoint subintervals
(v, ξ] ∈ J with

(D)
∑

(ξ − v) < η,

we have ∫
Ω

∥∥∥(D)
∑

F ((v, ξ), ω)
∥∥∥2

V
dP(ω) := E

[∥∥∥(D)
∑

F (v, ξ)
∥∥∥2

V

]
< ε.



R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 73

Lemma 3. Let f be IHB-integrable on [0, T ]. Then for every ε > 0, there exist a positive
function δ on (0, T ] and a positive number η such that

E
[∥∥∥(D)

∑
fξ(Wξ −Wv)

∥∥∥2

V

]
< ε

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

(D)
∑
|ξ − v| ≤ η.

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

E

[∥∥∥∥S(f, P, δ, η)− (IHB)

∫ T

0
ft dWt

∥∥∥∥2

V

]
<
ε

4
.

Let D = {((v, ξ], ξ) be a backwards δ-fine partial division of [0, T ] with

(D)
∑
|ξ − v| ≤ η.

Construct a backwards (δ, η)-fine partial division D1 of [0, T ] such that D and D1 are
disjoint and D ∪D1 is a backwards (δ, η)-fine partial division of [0, T ]. By assumption,

E

[∥∥∥∥(D ∪D1)
∑

fξ(Wξ −Wv)− (IHB)

∫ T

0
ft dWt

∥∥∥∥2

V

]
<
ε

4
.

Hence,

E
[∥∥∥(D)

∑
fξ(Wξ −Wv)

∥∥∥2

V

]
= E

[∥∥∥∥(D ∪D1)
∑

fξ(Wξ −Wv)− (IHB)

∫ T

0
ft dWt

+(IHB)

∫ T

0
ft dWt − (D1)

∑
fξ(Wξ −Wv)

∥∥∥∥2

V

]

≤ 2E

[∥∥∥∥(D ∪D1)
∑

fξ(Wξ −Wv)− (IHB)

∫ T

0
ft dWt

∥∥∥∥2

V

]

+ 2E

[∥∥∥∥(IHB)

∫ T

0
ft dWt − (D1)

∑
fξ(Wξ −Wv)

∥∥∥∥2

V

]
< 2

(ε
4

)
+ 2

(ε
4

)
= ε.

This proves the lemma. �



R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 74

Theorem 4. Let f be IHB-integrable on [0, T ] and define

F (v, ξ) := (IHB)

∫ ξ

v
ft dWt

for all (v, ξ] ∈ J . Then F is AC2[0, T ].

Proof. Let ε > 0 be given. By Lemma 3, there exist a positive function δ on (0, T ] and
a positive number η such that

E
[∥∥∥(D)

∑
fξ(Wξ −Wv)

∥∥∥2

V

]
<
ε

4

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

(D)
∑
|ξ − v| ≤ η.

Let {(aj , bj ]}mj=1 be a finite collection of disjoint subintervals (aj , bj ] ∈ J with
∑m

j=1 |bj −
aj | ≤ η. By property (4) section 3, f is also IHB-integrable on [aj , bj ] for all j. This
means that for all j, there exist positive function δj on (aj , bj ] and a positive number ηj
such that for any backwards (δj , ηj)-fine partial division Dj of [aj , bj ], we have

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

]
<

ε

4 · 22j
.

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 backwards δ-fine partial division of [0, T ] with

(P )
∑
|ξ − v| ≤

m∑
j=1

|bj − aj | ≤ η.

This implies that

E
[∥∥∥(P )

∑
fξ(Wξ −Wv)

∥∥∥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





R. Rulete, M. Labendia / Eur. J. Pure Appl. Math, 12 (1) (2019), 58-78 75

= 2

∥∥∥∥∥∥
m∑
j=1

{F (aj , bj)− S(f,Dj , δj , ηj)}

∥∥∥∥∥∥
2

L2(Ω,V )

+ 2E
[∥∥∥(P )

∑
fξ(Wξ −Wv)

∥∥∥2

V

]

= 2

 m∑
j=1

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

]2

+ 2E
[∥∥∥(P )

∑
fξ(Wξ −Wv)

∥∥∥2

V

]

< 2

 m∑
j=1

√
ε

2 · 2j

2

+ 2
(ε

4

)
< ε.

Thus, F is AC2[0, T ]. �

The following result provides an equivalent definition of an IHB-integrable process
using AC2 property.

Theorem 5. Let f : [0, T ] × Ω → L2(UQ, V ) be a backwards process. Then f is IHB-
integrable on [0, T ] if and only if there exists an AC2[0, T ] function F such that for every
ε > 0, there exist a positive function δ on (0, T ] such that whenever D = {((v, ξ], ξ)} is a
backwards δ-fine partial division of [0, T ], we have

E
[∥∥∥(D)

∑
{fξ(Wξ −Wv)− F (v, ξ)}

∥∥∥2

V

]
< ε.

Proof. Suppose that f is IHB-integrable on [0, T ]. By Theorem 4 and property (7)
section 3, the result follows.

For the converse, let ε > 0 be given. Since F is AC2[0, T ], choose η > 0 such that
whenever {(vj , ξj ]}mj=1 is a finite collection of subintervals (vj , ξj ] ∈ J with

m∑
j=1

|ξj − vj | ≤ η

we have

E

∥∥∥∥∥∥
m∑
j=1

F (vj , ξj)

∥∥∥∥∥∥
2

V

 < ε

4
.

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

E
[∥∥∥(Dc)

∑
F (v, ξ)

∥∥∥2

V

]
<
ε

4
.



REFERENCES 76

Hence,

E
[∥∥∥(D)

∑
fξ(Wξ −Wv)− F (0, T )

∥∥∥2

V

]
= E

[∥∥∥(D)
∑
{fξ(Wξ −Wv)− F (v, ξ)} − (Dc)

∑
F (v, ξ)

∥∥∥2

V

]
≤ 2E

[∥∥∥(D)
∑
{fξ(Wξ −Wv)− F (v, ξ)}

∥∥∥2

V

]
+ 2E

[∥∥∥(Dc)
∑

F (v, ξ)
∥∥∥2

V

]
< 2

(ε
4

)
+ 2

(ε
4

)
= ε.

Thus, f is IHB-integrable on [0, T ]. �

5. Conclusion and Recommendation

In this paper, we formulate the Itô isometry for the backwards Itô-Henstock integral
of an operator-valued stochastic process with respect to a Hilbert space-valued Q-Wiener
process and provide an equivalent definition for this integral using the concept AC2[0, T ]-
property, a version of absolute continuity. A worthwhile direction for further investigation
is to formulate a version of Itô’s formula for this type of integral.

Acknowledgement

The authors would like to acknowledge the financial support from the Department
of Science and Technology-Accelerated Science and Technology Human Resource Devel-
opment Program (DOST-ASTHRDP) and to thank the unknown referee for the helpful
comments for the improvement of this paper.

References

[1] D. Applebaum. Lévy Processes and Stochastic Calculus. Cambridge University Press,
New York, 2004.

[2] J. Arcede and E. Cabral. Backwards Henstock integral. MATIMYÁS MATEM-
ATIKA, Journal of the Mathematical Society of the Philippines, 32:1–12, 2009.

[3] J. Arcede and E. Cabral. An equivalent definition for the backwards Itô integral.
Thai Journal of Mathematics, 9:619–630, 2011.



REFERENCES 77

[4] J. Arcede and E. Cabral. Fundamental Theorem of Calculus for the backwards Itô
integral. MATIMYÁS MATEMATIKA, Journal of the Mathematical Society of the
Philippines, 34:1–9, 2011.

[5] J. Arcede and E. Cabral. On integration-by-parts and Itô formula for backwards Itô
integral. The MINDANAWAN Journal of Mathematics, 3:112–132, 2012.

[6] L. Gawarecki and V. Mandrekar. Stochastic Differential Equations in Infinite Dimen-
sions with Applications to Stochastic Partial Differential Equations. Springer, Berlin,
2011.

[7] R. A. Gordon. The Integrals of Lebesgue, Denjoy, Perron and Henstock. American
Mathematical Society, 1994.

[8] R. Henstock. Lectures on the Theory of Integration. World Scientific, Singapore,
1988.

[9] J. Kurzweil. Henstock-Kurzweil Integration: Its Relation to Topological Vector Spaces.
World Scientific, Singapore, 2000.

[10] M. Labendia and J. Benitez. Convergence theorems for the Itô-Henstock integrable
operator-valued stochastic process. Malaysian Journal of Mathematical Sciences, to
appear.

[11] M. Labendia E. De Lara-Tuprio and T. R. Teng. Itô-Henstock integral and Itô’s
formula for the operator-valued stochastic process. Mathematica Bohemica, 143:135–
160, 2018.

[12] P. Y. Lee. Lanzhou Lectures on Henstock Integration. World Scientific, Singapore,
1989.

[13] P. Y. Lee and R. Výborný. The Integral: An Easy Approach after Kurzweil and
Henstock. Cambridge University Press, Cambridge, 2000.

[14] T. Y. Lee. Henstock-Kurzweil Integration on Euclidean Spaces. World Scientific,
Singapore, 2011.

[15] E. J. McShane. Stochastic integrals and stochastic functional equations. SIAM J.
Appl. Math., 17:287–306, 1969.

[16] B. Øksendal. Stochastic Differential Equations (5th edition). Springer-Verlag, New
York, 1998.

[17] Z. R. Pop-Stojanovic. On Mcshane’s belated stochastic integral. SIAM J. Appl.
Math., 22:87–92, 1972.

[18] G. Da Prato and J. Zabczyk. Stochastic Equations in Infinite Dimensions. Cambridge
University Press, Cambridge, 1992.



REFERENCES 78

[19] C. Prévôt and M. Röckner. A Concise Course on Stochastic Partial Differential
Equations. 2007.

[20] M. Reed and B. Simon. Methods of Modern Mathematical Physics I: Functional
Analysis. 1980.

[21] T. L. Toh and T. S. Chew. The Riemann approach to stochastic integration using
non-uniform meshes. J. Math. Anal. Appl., 280:133–147, 2003.

[22] T. L. Toh and T. S. Chew. On the Henstock-Fubini theorem for multiple stochastic
integrals. Real Anal. Exchange, 30:295–310, 2004-2005.

[23] T. S. Chew T. L. Toh and J. Y. Tay. The non-uniform Riemann approach to Itô’s
integral. Real Anal. Exchange, 27:495–514, 2002-2003.