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

In this paper, a definition of backwards Itô-Henstock integral for the Hilbert-Schmidtvalued 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 AC[0, T ]-property, a version of absolute continuity. 2010 Mathematics Subject Classifications: 60H30, 60H05


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 obtained from the mean square limit of stochastic integrals of simple processes, see [16].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 infinitedimensional 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 process.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 definition 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 AC 2 [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 actually 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 AC 2 property, a version of absolute continuity.

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 {G t : 0 ≤ t ≤ T } be a family of sub σ-field of G.
satisfies the following condition: (1) G T contains all sets of P-measure zero in G; and (2) for each t ∈ [0, T ], G t = G t− := s<t G s .Then {G t : 0 ≤ t ≤ T } is called a standard backwards filtration.We often write {G t } instead of {G t : 0 ≤ t ≤ T }.See [1].
Let H be a separable Banach space.A stochastic process f or simply process is a function 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 L 2 (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 {e j } ∞ j=1 , or simply {e j }, be an orthonormal basis (abbrev.as ONB) in 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 is a symmetric nonnegative definite trace-class operator, then there exists an ONB {e j } ⊂ U and a sequence of nonnegative real numbers {λ j } such that Qe j = λ j e j for all j ∈ N, and λ j → 0 as j → ∞ [20, p.203].We shall call the sequence of pairs {λ j , e j } an eigensequence defined by Q.
Let Q : U → U be a symmetric nonnegative definite trace-class operator and let {λ j , e j } be an eigensequence defined by Q.Then the subspace λ j e j as its ONB, see [18, p.90], [6, p.23].Let {f j } be an ONB in ) the space of all Hilbert-Schmidt operators from U Q to V , which is known [19, p.112] to be a separable Hilbert space and the norm S L 2 (U Q ,V ) may be defined in terms of an arbitrary ONB, see [18, p.418], [19, p.111].It is shown in [6, p.25 We fix an element Q ∈ L(U ), symmetric nonnegative definite trace-class operator.A U -valued stochastic process W t , t ∈ [0, T ], on a probability space (Ω, G, P) is called a Q-Wiener process in U if: (i) W (0, ω) = 0 U 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 (iv) the increments have the following Gaussian laws: By Proposition 4.2 (see [18, p.88]), such a Q-Wiener process exists.
We define Since N ⊆ G0 s for all s ∈ [0, T ] and {G t } 0≤t≤T is decreasing, we have the following result: Then the filtration G t given in (1) is a standard backwards filtration.
We note that the distance from an element u ∈ U to a nonempty subset for any B ∈ G t and any closed subset A ∈ U as {A ⊂ U | A closed} generates B(U ) and is stable under finite intersection.But we have Let F : Ω → {0, 1} be defined by This implies that This completes the proof.
From now onwards, the backwards filtered probability shall mean a filtered probability space such that W t is adapted to G t and W t −W s is independent of G t for all 0 ≤ s ≤ t ≤ T .

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 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 = {((u i , ξ i ], ξ i )} n i=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, 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, {λ j , e j } is an eigensequence defined by Q, and W is a U -valued Q-Weiner process. ) be a backwards adapted process.Then f is said to be backwards Itô-Henstock integrable, or IH B -integrable, on [0, T ] with respect to W if there exists A ∈ L 2 (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 In this case, f is IH B -integrable to A on [0, T ] and A is called the IH B -integral of f which will be denoted by (IH B ) T 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 Hence, (ii) Note that Thus, Thereby, completing the proof.
Proof.We shall consider first the following claims.
This proves Claim 2.
be the collection of all subintervals of [0, T ] which are not included in D. Then By Claim 1, Claim 2, and Lemma 2, we have 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 A 1 and A 2 are two backwards Itô-Henstock integrals of f in Definition 1, then and and ( ), a decreasing sequence {δ n } of positive functions defined on (0, T ], and a decreasing sequence of positive numbers {η n } such that for any backwards (δ n , η n )-fine partial division D n of [0, T ], we have In this case, 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 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

Itô Isometry and AC 2 [0, T ]-property
This section presents the Itô isometry and the equivalent definition of backwards Itô-Henstock using the notion of AC 2 [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 In this case, A is called the Lebesgue integral of f which will be denoted by (L) 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 {δ n (ξ)} on (0, T ] and a decreasing sequence of positive constants {η n } such that where D n is any backwards (δ n , η n )-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 δ n on (0, T ] and a positive number η such that for any backwards δ n -fine partial division

Hence, lim
n→∞ Conversely, let us assume that there exists A ∈ R and a decreasing sequence {δ n (ξ)} of positive functions on (0, T ] and a decreasing sequence of positive numbers {η n } such that lim 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 backwards δ-fine partial division Hence, for each δ n and η n , there exists a δ n -fine partial division D n of [0, T ] with leading to a contradiction.
We now state and prove the Itô isometry.
Proof.From property (5) section 3, there exists a decreasing sequence {δ n (ξ)} of positive functions defined on (0, T ], and a decreasing sequence of positive numbers {η n } such that for any backwards (δ n , η n )-fine partial division D n = {((v Let ε > 0 be given.Then there exists N ∈ N such that for all n ≥ N , .
Using Lemma 2, we have Since the above equality holds for any backwards (δ n , η n )-fine partial division of [0, T ], by is Lebesgue integrable on [0, T ] and 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 AC 2 [0, T ] if for every ε > 0, there exists η > 0 such that for any finite collection Then for every ε > 0, there exist a positive function δ on (0, T ] and a positive number η such that 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 Hence, This proves the lemma.Theorem 4. Let f be IH B -integrable on [0, T ] and define 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 Let {(a j , b j ]} m j=1 be a finite collection of disjoint subintervals (a j , b j ] ∈ J with m j=1 |b j − a j | ≤ η.By property (4) section 3, f is also IH B -integrable on [a j , b j ] for all j.This means that for all j, there exist positive function δ j on (a j , b j ] and a positive number η j such that for any backwards (δ j , η j )-fine partial division D j of [a j , b j ], we have We can choose {δ j } m j=1 and {η j } m j=1 such that δ j (ξ) ≤ δ(ξ) for all j and This implies that Hence, Thus, F is AC 2 [0, T ].
The following result provides an equivalent definition of an IH B -integrable process using AC 2 property.Theorem 5. Let f : [0, T ] × Ω → L 2 (U Q , V ) be a backwards process.Then f is IH Bintegrable on [0, T ] if and only if there exists an AC 2 [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 Proof.Suppose that f is IH B -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 AC 2 [0, T ], choose η > 0 such that whenever {(v j , ξ j ]} m j=1 is a finite collection of subintervals (v j , ξ j ] ∈ J with

4 .
Let D = {((v, ξ], ξ)} be a backwards (δ, η)-fine partial division of [0, T ] and let D c be the collection of all subintervals of [0, T ] which are not included in the set D.Since F is AC 2 [0, T ], E (D c ) F (v, ξ)