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

In this paper, we formulate a descriptive definition or a version of fundamental theorem 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 differentiability and a version of absolute continuity of a Hilbert space-valued stochastic process. 2010 Mathematics Subject Classifications: 60H30, 60H05


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 [2][3][4][7][8][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 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 [10,11,[15][16][17].
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.

Preliminaries
Throughout this paper, let (Ω, F, {F t }, 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 {f t } 0≤t≤T , is said to be adapted to a filtration {F t } if f t is F t -measurable for all t ∈ [0, T ].When no confusion arises, we may refer to a process adapted to {F t } 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 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. Using the Square-root Lemma [14, p.196 j=1 , or simply {e j }, 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 Qe j , e j 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 L 1 (U ) the space of all trace-class operators on U , which is known [14, p.209] to be a Banach space with norm 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, {λ j } ∈ 1 , and λ j → 0 as j → ∞ [14, 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.Let {λ j , e j } be an eigensequence defined by Q.Then the subspace is a separable Hilbert space with λ j e j as its ONB, see [13, p.90 ) the space of all Hilbert-Schmidt operators from U Q to V , which is known [12, 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 [13, p.418], [12, p.111].It is shown in [1, p.25] that L(U, V ) is properly contained in L 2 (U Q , V ).We also note that L 2 (U Q , V ) contains genuinely unbounded linear operators from U to V .
Let Q : U → U be a symmetric nonnegative definite trace-class operator, {λ j , e j } be an eigensequence defined by Q, and {B j } be a sequence of independent Brownian motions (abbrev.as BM ) defined on (Ω, F, {F t }, P).The process is called a Q-Wiener process in U .The series in (1) converges in L 2 (Ω, U ).For each λ j B j (t) e j , u U , with the series converging in L 2 (Ω, R).
Since the operator Q is assumed to be symmetric nonnegative definite trace-class, there exists a U -valued process W such that Wt (u)(ω) = W t (ω), u U P-almost surely (abbrev.as P-a.s.). ( 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 λ j > 0 for all j, Wt(e j ) √ λ j , j = 1, 2, . . ., is a sequence of real-valued BM defined on (Ω, F, {F t }, P), see [13, p.87].
A filtration {F t } on a probability space (Ω, F, P) is called normal if (i) F 0 contains all elements A ∈ F such that P(A) = 0, and (ii) [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, {F t }, P) shall mean a probability space equipped with a normal filtration.

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 derivative 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, {λ j , e j } is an eigensequence defined by Q, and We note that each ξ i in Definition 1 does not necessarily belong to [u i , v i ].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 ].
To define the Itô-McShane integral, we shall use the definition of belated partial division in Definition 1, employed by the authors in [17, p.499].
) such that for every > 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 where 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) Refer to [6,Example 3.7] for the proof of the following example.
) be an adapted process on a filtered probability space (Ω, F, {F t }, P) such that for t ∈ [0, T ], 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.
, by the monotone convergence theorem for Lebesgue integral, Thus, there exists Thus, for any δ-fine belated McShane partial division < . ( The above inequality also holds for (δ, η) It is worth noting that the Itô-McShane integral possesses some of the standard properties 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 A 1 and A 2 are two IM integrals of f , then (iii) Cauchy criterion.A process f is IM-integrable on [0, T ] if and only if for every > 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 D 1 and D 2 of [0, T ], we have (IM) (vi) Sequential definition.A process f is IM-integrable on [0, T ] if and only if there exist A ∈ L 2 (Ω, V ), 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 (δ n , η n )-fine belated McShane partial division D n of [0, T ], we have In this case, A := (IM) Next, we define the concept of AC 2 [0, T ]-property, a version of absolute continuity.
) such that for all > 0, there exists a positive function δ on [0, T ] such that for all δ-fine belated McShane interval-point pair The random variable f ξ is called the belated McShane derivative of F at the point ξ ∈ [0, T ) and is denoted by DF ξ .
We note that we write Definition 5. A function F : J × Ω → V (i) is said to be AC 2 [0, T ] if for every > 0, there exists η > 0 such that for any finite collection (ii) has the orthogonal increment property if for all non-overlapping intervals The proof of the following theorem is parallel to the proof in [5].
[5] Let f be IM-integrable on [0, T ] and define Then F is AC 2 [0, T ] and has the orthogonal increment property.
Lemma 4. Let f ∈ Λ IM .Then for every > 0, there exist a positive function δ on [0, T ] and a positive number η such that Proof.Let > 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 Hence, This proves the lemma.
(ii) for every > 0, there exist a positive function δ on [0, T ] such that whenever we have Proof.Suppose that f ∈ Λ IM .By the Saks-Henstock lemma for IM integral, (ii) holds.Next we show that F is AC 2 [0, T ].Let > 0 be given.By Lemma 4, there exist a positive function δ on [0, T ] and a number η > 0 such that for all j.This means that for all j, there exist a positive function δ j on [a j , b j ] and a number η j > 0 such that for any (δ j , η j )-fine belated McShane 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 m j=1 η j ≤ η.
This implies that Hence, Thus, F is AC 2 [0, T ].Conversely, assume that (i) and (ii) hold.Let > 0 be given.Since Hence, Thus, (i) F has the orthogonal increment property, 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 {δ n } of positive functions defined on [0, T ] and a decreasing sequence {η n } of positive numbers such that for any , respectively, we have In view of Lemma 5, we have the following lemma.Lemma 6.Let f ∈ Λ IM and define F : The immediate consequence of Lemma 5.(i) is the strong version of Saks-Henstock lemma.

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 4. Let f : [0, T ] × Ω → L(U, V ) be an adapted process and let F : J × Ω → V be AC 2 [0, T ], has the orthogonal increment property, and DF t = f t a.e. on [0, T ).Then f ∈ Λ IM and Then for every > 0, there exists a positive function δ 1 on [0, T ] such that for any δ If A = ∅, then we are done.Suppose that It follows that there exists N ∈ N such that N − 1 ≤ E f ξ 2 L 2 (U Q ,V ) < N .Since F is AC 2 [0, T ], there exists η > 0 with η < 4N such that for all finite collection {[u j , v j ]} p j=1 of non-overlapping intervals of [0, T ] with p j=1 (v j − u j ) < η, we have Let D = {([u, v], ξ)} be a δ-fine belated McShane partial division of [0, T ].Then using ( 11) and ( 12), we have 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 ∈ Λ IM if and only if there exists an AC 2 [0, T ] function F : J × Ω → V that satisfies the orthogonal increment property and DF t = f t a.e. on [0, T ).

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 process using the concept of belated McShane derivative and AC 2 [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.

Definition 1 .
e. 0 < T and can be replaced with any closed interval [a, b].If no confusion arises, we may write Let δ be a positive function defined on [0, T ].A finite collection D random variable 0 from Ω to V and denote by Λ IM , 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