# The Itô Integral with respect to an Infinite Dimensional Lévy Process: A Series Approach

Authors: Stefan Tappe
Journal: International Journal of Stochastic Analysis
 (2013)

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## Abstract

We present an alternative construction of the infinite dimensional Itô integral with respect to a Hilbert space valued Lévy process. This approach is based on the well-known theory of
real-valued stochastic integration, and the respective Itô integral is given by a series of Itô integrals with respect to standard Lévy processes. We also prove that this stochastic integral coincides with the Itô integral that has been developed in the literature.


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## Body

## 1. Introduction

The Itô integral with respect to an infinite dimensional Wiener process has been developed in [1–3], and for the more general case of an infinite dimensional square-integrable martingale, it has been defined in [4, 5]. In these references, one first constructs the Itô integral for elementary processes and then extends it via the Itô isometry to a larger space, in which the space of elementary processes is dense.For stochastic integrals with respect to a Wiener process, series expansions of the Itô integral have been considered, for example, in [6–8]. Moreover, in [9], series expansions have been used in order to define the Itô integral with respect to a Wiener process for deterministic integrands with values in a Banach space. Later, in [10], this theory has been extended to general integrands with values in UMD Banach spaces.To the best of the author's knowledge, a series approach for the construction of the Itô integral with respect to an infinite dimensional Lévy process does not exist in the literature so far. The goal of the present paper is to provide such a construction, which is based on the real-valued Itô integral; see, for example, [11–13], and where the Itô integral is given by a series of Itô integrals with respect to real-valued Lévy processes. This approach has the advantage that we can use results from the finite dimensional case, and it might also be beneficial for lecturers teaching students who are already aware of the real-valued Itô integral and have some background in functional analysis. In particular, it avoids the tedious procedure of proving that elementary processes are dense in the space of integrable processes.In [14], the stochastic integral with respect to an infinite dimensional Lévy process is defined as a limit of Riemannian sums, and a series expansion is provided. A particular feature of [14] is that stochastic integrals are considered as L2-curves. The connection to the usual Itô integral for a finite dimensional Lévy process has been established in [15]; see also Appendix B in [16]. Furthermore, we point out [17, 18], where the theory of stochastic integration with respect to Lévy processes has been extended to Banach spaces.The idea to use series expansions for the definition of the stochastic integral has also been utilized in the context of cylindrical processes; see [19] for cylindrical Wiener processes and [20] for cylindrical Lévy processes.The construction of the Itô integral, which we present in this paper, is divided into the following steps.(i)
For anH-valued process X (with H denoting a separable Hilbert space) and a real-valued square-integrable martingale M, we define the Itô integral
(1)X·M:=∑k∈ℕ‍(〈X,fk〉H·M)fk,
where (fk)k∈ℕ denotes an orthonormal basis of H, and 〈X,fk〉H·M denotes the real-valued Itô integral. We will show that this definition does not depend on the choice of the orthonormal basis.(ii)
Based on the just defined integral, for anℓ2(H)-valued process X and a sequence (Mj)j∈ℕ of standard Lévy processes, we define the Itô integral as
(2)∑j∈ℕ‍Xj·Mj.
For this, we will ensure convergence of the series.(iii)
In the next step, letL denote an ℓλ2-valued Lévy process, where ℓλ2 is a weighted space of sequences (cf. [21]). From the Lévy process L, we can construct a sequence (Mj)j∈ℕ of standard Lévy processes, and for a ℓ2(H)-valued process X, we define the Itô integral
(3)X·L:=∑j∈ℕ‍Xj·Mj.(iv)
Finally, letL be a general Lévy process on some separable Hilbert space U with covariance operator Q. Then, there exist sequences of eigenvalues (λj)j∈ℕ and eigenvectors, which diagonalize the operator Q. Denoting by L20(H) an appropriate space of Hilbert Schmidt operators from U to H, our idea is to utilize the integral from the previous step and to define the Itô integral for a L20(H)-valued process X as
(4)X·L:=Ψ(X)·Φ(L),
where Φ:U→ℓλ2 and Ψ:L20(H)→ℓ2(H) are isometric isomorphisms such that Φ(L) is an ℓλ2-valued Lévy process. We will show that this definition does not depend on the choice of the eigenvalues and eigenvectors.The remainder of this text is organized as follows. In Section2, we provide the required preliminaries and notation. After that, we start with the construction of the Itô integral as outlined earlier. In Section 3, we define the Itô integral for H-valued processes with respect to a real-valued square-integrable martingale, and in Section 4, we define the Itô integral for ℓ2(H)-valued processes with respect to a sequence of standard Lévy processes. Section 5 gives a brief overview about Lévy processes in Hilbert spaces, together with the required results. Then, in Section 6, we define the Itô integral for ℓ2(H)-valued processes with respect to an ℓλ2-valued Lévy process, and in Section 7, we define the Itô integral in the general case, where the integrand is an L20(H)-valued process and the integrator a general Lévy process on some separable Hilbert space U. We also prove the mentioned series representation of the stochastic integral and show that it coincides with the usual Itô integral, which has been developed in [5].


## 2. Preliminaries and Notation

In this section, we provide the required preliminary results and some basic notation. Throughout this text, let(Ω,ℱ,(ℱt)t≥0,ℙ) be a filtered probability space satisfying the usual conditions. For the upcoming results, let E be a separable Banach space, and let T>0 be a finite time horizon.Definition 1.
Letp≥1 be arbitrary.  (1)
We define the Lebesgue space(5)ℒTp(E):=ℒp(Ω,ℱT,ℙ;𝔻([0,T];E)),
where 𝔻([0,T];E) denotes the Skorokhod space consisting of all càdlàg functions from [0,T] to E, equipped with the supremum norm.(2)
We denote by𝒜Tp(E) the space of all E-valued adapted processes X∈ℒTp(E).(3)
We denote byℳTp(E) the space of all E-valued martingales M∈ℒTp(E).(4)
We define the factor spaces(6)MTp(E):=ℳTp(E)N,ATp(E):=𝒜Tp(E)N,(7)LTp(E):=ℒTp(E)N,
where N⊂ℳTp(E) denotes the subspace consisting of all M∈ℳTp(E) with M=0 up to indistinguishability.Remark 2.
Let us emphasize the following.(1)
Since the Skorokhod space𝔻([0,T];E) equipped with the supremum norm is a Banach space, the Lebesgue space LTp(E) equipped with the standard norm
(8)∥X∥LTp(E):=𝔼[∥X∥Ep]1/p
is a Banach space too.(2)
By the completeness of the filtration(ℱt)t≥0, adaptedness of an element X∈LTp(E) does not depend on the choice of the representative. This ensures that the factor space ATp(E) of adapted processes is well defined.(3)
The definition ofE-valued martingales relies on the existence of conditional expectation in Banach spaces, which has been established in [1, Proposition 1.10].Note that we have the inclusions(9)MTp(E)⊂ATp(E)⊂LTp(E).The following auxiliary result shows that these inclusions are closed.Lemma 3.
Letp≥1 be arbitrary. Then, the following statements are true:  (1)
MTp(E) is closed in ATp(E);(2)
ATp(E) is closed in LTp(E).Proof.
Let(Mn)n∈ℕ⊂MTp(E) be a sequence, and let M∈ATp(E) be such that Mn→M in LTp(E). Furthermore, let τ≤T be a bounded stopping time. Then, we have
(10)𝔼[∥Mτ∥Ep]≤𝔼[supt∈[0,T]∥Mt∥Ep]<∞,
showing that Mτ∈Lp(Ω,ℱτ,ℙ;E). Furthermore, we have
(11)𝔼[∥Mτn-Mτ∥Ep]≤𝔼[supt∈[0,T]∥Mtn-Mt∥Ep]⟶0.
By Doob’s optional stopping theorem (which also holds true for E-valued martingales; see [2, Remark 2.2.5]), it follows that
(12)𝔼[Mτ]=limn→∞𝔼[Mτn]=limn→∞𝔼[M0n]=𝔼[M0].
Using Doob’s optional stopping theorem again, we conclude that M∈MTp(E), proving the first statement.
Now, let(Xn)n∈ℕ⊂ATp(E) be a sequence, and let X∈LTp(E) be such that Xn→X in LTp(E). Then, for each t∈[0,T], we have
(13)𝔼[∥Xtn-Xt∥Ep]≤𝔼[sups∈[0,T]∥Xsn-Xs∥Ep]⟶0,
and, hence, ℙ-almost surely Xtnk→Xt for some subsequence (nk)k∈ℕ, showing that Xt is ℱt-measurable. This proves that X∈ATp(E), providing the second statement.Note that, by Doob's martingale inequality [2, Theorem 2.2.7], for p>1, an equivalent norm on MTp(E) is given by
(14)∥M∥MTp(E):=𝔼[∥MT∥Ep]1/p.
Furthermore, if E=H is a separable Hilbert space, then MT2(H) is a separable Hilbert space equipped with the inner product
(15)〈M,N〉MT2(H):=𝔼[〈MT,NT〉H].
Finally, we recall the following result about series of pairwise orthogonal vectors in Hilbert spaces.Lemma 4.
LetH be a separable Hilbert space, and let (hn)n∈ℕ⊂H be a sequence with 〈hn,hm〉H=0 for n≠m. Then, the following statements are equivalent. (1)
The series∑n=1∞‍hn converges in H.(2)
The series∑n∈ℕ‍hn converges unconditionally in H.(3)
One has∑n=1∞‍∥hn∥H2<∞.
If the previous conditions are satisfied, then one has(16)∥∑n=1∞‍hn∥H2=∑n=1∞‍∥hn∥H2.Proof.
This follows from [22, Theorem12.6] and [23, SatzV.4.8].


## 3. The Itô Integral with respect to a Real-Valued Square-Integrable Martingale

In this section, we define the Itô integral for Hilbert space valued processes with respect to a real-valued, square-integrable martingale, which is based on the real-valued Itô integral.In what follows, letH be a separable Hilbert space, and let T>0 be a finite time horizon. Furthermore, let M∈ℳT2(ℝ) be a square-integrable martingale. Recall that the quadratic variation 〈M,M〉 is the (up to indistinguishability) unique real-valued, nondecreasing, predictable process with 〈M,M〉0=0 such that M2-〈M,M〉 is a martingale.Proposition 5.
LetX be an H-valued, predictable process with
(17)𝔼[∫0T‍∥Xs∥H2d〈M,M〉s]<∞.
Then, for every orthonormal basis (fk)k∈ℕ of H, the series
(18)∑k∈ℕ‍(〈X,fk〉H·M)fk
converges unconditionally in MT2(H), and its value does not depend on the choice of the orthonormal basis (fk)k∈ℕ.Proof.
Let(fk)k∈ℕ be an orthonormal basis of H. For j,k∈ℕ with j≠k, we have
(19)〈(〈X,fj〉H·M)fj,(〈X,fk〉H·M)fk〉MT2(H)=𝔼[〈(∫0T‍〈Xs,fj〉HdMs)fj,(∫0T‍〈Xs,fk〉HdMs)fk〉H]=𝔼[(∫0T‍〈Xs,fj〉HdMs)(∫0T‍〈Xs,fk〉HdMs)〈fj,fk〉H]=0.
Moreover, by the Itô isometry for the real-valued Itô integral and the monotone convergence theorem, we obtain
(20)∑k=1∞‍∥(〈X,fk〉H·M)fk∥MT2(H)2=∑k=1∞‍𝔼[∥(∫0T‍〈Xs,fk〉HdMs)fk∥H2]=∑k=1∞‍𝔼[|∫0T〈Xs,fk〉H‍dMs|2]=∑k=1∞‍𝔼[∫0T‍|〈Xs,fk〉H|2d〈M,M〉s]=𝔼[∫0T‍∑k=1∞‍|〈Xs,fk〉H|2d〈M,M〉s]=𝔼[∫0T‍∥Xs∥H2d〈M,M〉s].
Therefore, by (17) and Lemma 4, the series (18) converges unconditionally in MT2(H).
Now, let(gk)k∈ℕ be another orthonormal basis of H. We define 𝕁f,𝕁g∈MT2(H) by
(21)𝕁f:=∑k=1∞‍(〈X,fk〉H·M)fk,𝕁g:=∑k=1∞‍(〈X,gk〉H·M)gk.
Let h∈H be arbitrary. Then, we have
(22)〈h,𝕁f〉H,〈h,𝕁g〉H∈MT2(ℝ)
and the identity
(23)∥〈h,𝕁f〉H-〈h,X〉H·M∥MT2(ℝ)2=∥〈h,∑k=1∞‍(〈X,fk〉H·M)fk〉H-〈h,X〉H·M∥MT2(ℝ)2=∥∑k=1∞‍(〈h,fk〉H〈fk,X〉H·M)-〈h,X〉H·M∥MT2(ℝ)2=limn→∞∥∑k=1n‍(〈h,fk〉H〈fk,X〉H·M)-〈h,X〉H·M∥MT2(ℝ)2=limn→∞∥(∑k=1n‍〈h,fk〉H〈fk,X〉H-〈h,X〉H)·M∥MT2(ℝ)2.
For all x∈H, we have
(24)|∑k=1n‍〈x,fk〉H〈fk,h〉H-〈x,h〉H|2⟶0asn⟶∞,
and, by the Cauchy-Schwarz inequality,
(25)|∑k=1n‍〈x,fk〉H〈fk,h〉H-〈x,h〉H|2=|∑k=n+1∞‍〈x,fk〉H〈fk,h〉H|2≤(∑k=1∞‍|〈x,fk〉H|2)(∑k=1∞‍|〈fk,h〉H|2)=∥x∥H2∥h∥H2foreachn∈ℕ.
Therefore, by the Itô isometry for the real-valued Itô integral and Lebesgue's dominated convergence theorem together with (17), we obtain
(26)∥〈h,𝕁f〉H-〈h,X〉H·M∥MT2(ℝ)2=limn→∞𝔼[|∫0T(∑k=1n‍〈h,fk〉H〈fk,Xs〉H-〈h,Xs〉H)dMs‍|2]=limn→∞𝔼[∫0T‍|∑k=1n‍〈h,fk〉H〈fk,X〉H-〈h,X〉H|2d〈M,M〉s]=0.
Analogously, we prove that
(27)∥〈h,𝕁g〉H-〈h,X〉H·M∥MT2(ℝ)2=0.
Therefore, denoting by 𝕁~f,𝕁~g∈ℳT2(H) representatives of 𝕁f, 𝕁g, we obtain
(28)〈h,𝕁~Tf〉H=〈h,𝕁~Tg〉H∀h∈H,ℙ-almostsurely.
By separability of H, we deduce that
(29)〈h,𝕁~Tf〉H=〈h,𝕁~Tg〉Hℙ-almostsurely,∀h∈H.
Consequently, we have
(30)𝕁~Tf=𝕁~Tgℙ-almostsurely,
Implying that 𝕁f=𝕁g. This proves that the value of the series (18) does not depend on the choice of the orthonormal basis.Now, Proposition5 gives rise to the following definition.Definition 6.
For everyH-valued, predictable process X satisfying (17), we define the Itô integral X·M=(∫0t‍XsdMs)t∈[0,T] as
(31)X·M:=∑k∈ℕ‍(〈X,fk〉H·M)fk,
where (fk)k∈ℕ denotes an orthonormal basis of H.According to Proposition5, definition (31) of the Itô integral is independent of the choice of the orthonormal basis (fk)k∈ℕ, and the integral process X·M belongs to MT2(H).Remark 7.
As the proof of Proposition5 shows, the components of the Itô integral X·M are pairwise orthogonal elements of the Hilbert space MT2(H).Proposition 8.
For everyH-valued, predictable process X satisfying (17), one has the Itô isometry
(32)𝔼[∥∫0T‍XsdMs∥H2]=𝔼[∫0T‍∥Xs∥H2d〈M,M〉s].Proof.
Let(fk)k∈ℕ be an orthonormal basis of H. According to (19), we have
(33)〈(〈X,fj〉H·M)fj,(〈X,fk〉H·M)fk〉MT2(H)=0forj≠k.
Thus, by Lemma 4 and (20), we obtain
(34)𝔼[∥∫0T‍XsdMs∥H2]=∥X·M∥MT2(H)2=∥∑k=1∞‍(〈X,fk〉H·M)fk∥MT2(H)2=∑k=1∞‍∥(〈X,fk〉H·M)fk∥MT2(H)2=𝔼[∫0T‍∥Xs∥H2d〈M,M〉s],
finishing the proof.Proposition 9.
LetX be a H-valued simple process of the form
(35)X=X01{0}+∑i=1n‍Xi1(ti,ti+1]
with 0=t1<⋯<tn+1=T and ℱti-measurable random variables Xi:Ω→H for i=0,…,n. Then, one has
(36)X·M=∑i=1n‍Xi(Mti+1-Mti).Proof.
Let(fk)k∈ℕ be an orthonormal basis of H. Then, for each k∈ℕ, the process 〈X,fk〉 is a real-valued simple process with representation
(37)〈X,fk〉H=〈X0,fk〉H1{0}+∑i=1n‍〈Xi,fk〉H1(ti,ti+1].
Thus, by the definition of the real-valued Itô integral for simple processes, we obtain
(38)X·M=∑k∈ℕ‍(〈X,fk〉H·M)fk=∑k∈ℕ‍(∑i=1n‍〈Xi,fk〉H(Mti+1-Mti))fk=∑i=1n‍(∑k∈ℕ‍〈Xi,fk〉Hfk)(Mti+1-Mti)=∑i=1n‍Xi(Mti+1-Mti),
finishing the proof.Lemma 10.
LetX be a H-valued, predictable process satisfying (17). Then, for every orthonormal basis (fk)k∈ℕ of H, one has
(39)∑k=1∞‍|〈X,fk〉H|2·〈M,M〉=∥X∥H2·〈M,M〉,
where the convergence takes place in AT1(ℝ).Proof.
We define the integral process(40)𝕀:=∥X∥H2·〈M,M〉
and the sequence (𝕀n)n∈ℕ of partial sums by
(41)𝕀n:=∑k=1n‍|〈X,fk〉H|2·〈M,M〉.
By (17) we have 𝕀∈AT1(ℝ) and (𝕀n)n∈ℕ⊂AT1(ℝ). Furthermore, by Lebesgue's dominated convergence theorem, we have
(42)∥𝕀-𝕀n∥LT1(ℝ)=𝔼[supt∈[0,T]|𝕀t-𝕀tn|]=𝔼[supt∈[0,T]|∫0t‍∑k=n+1∞‍|〈Xs,fk〉|2d〈M,M〉s|]=𝔼[∫0T‍∑k=n+1∞‍|〈Xs,fk〉|2d〈M,M〉s]⟶0forn⟶∞,
which concludes the proof.Remark 11.
As a consequence of the Doob-Meyer decomposition theorem, for two square-integrable martingalesX,Y∈ℳT2(H), there exists (up to indistinguishability) a unique real-valued, predictable process 〈X,Y〉 with finite variation paths and 〈X,Y〉0=0 such that 〈X,Y〉H-〈X,Y〉 is a martingale.Proposition 12.
For everyH-valued, predictable process X satisfying (17), one has
(43)〈X·M,X·M〉=∥X∥H2·〈M,M〉.Proof.
Let(fk)k∈ℕ be an orthonormal basis of H. We define the process 𝕁:=X·M and the sequence (𝕁n)n∈ℕ of partial sums by
(44)𝕁n:=∑k=1n‍(〈X,fk〉H·M)fk.
By Proposition 5, we have
(45)𝕁n⟶𝕁inMT2(H).
Defining the integral process 𝕀 by (40) and the sequence (𝕀n)n∈ℕ of partial sums by (41), using Lemma 10, we have
(46)𝕀n⟶𝕀inAT1(ℝ).
Furthermore, we define the process M∈AT1(ℝ) and the sequence (Mn)n∈ℕ⊂AT1(ℝ) as
(47)M:=∥𝕁∥H2-𝕀,Mn:=∥𝕁n∥H2-𝕀n,n∈ℕ.
Then, we have (Mn)n∈ℕ⊂MT1(ℝ). Indeed, for each n∈ℕ, we have
(48)Mn=∥∑k=1n‍(〈X,fk〉H·M)fk∥H2-∑k=1n‍|〈X,fk〉H|2·〈M,M〉=∑k=1n‍∥(〈X,fk〉H·M)fk∥H2-∑k=1n‍|〈X,fk〉H|2·〈M,M〉=∑k=1n‍(|〈X,fk〉H·M|2-|〈X,fk〉H|2·〈M,M〉).
For every k∈ℕ, the quadratic variation of the real-valued process 〈X,fk〉H·M is given by
(49)〈〈X,fk〉H·M,〈X,fk〉H·M〉=|〈X,fk〉H|2·〈M,M〉
see, for example, [12, TheoremI.4.40.d], which shows that Mn is a martingale. Since Mn∈AT1(ℝ), we deduce that Mn∈MT1(ℝ).
Next, we prove thatMn→M in AT1(ℝ). Indeed, since
(50)|∥𝕁∥H2-∥𝕁n∥H2|≤∥𝕁-𝕁n∥H2+2∥𝕁∥H∥𝕁-𝕁n∥H,
by the Cauchy-Schwarz inequality and (45) we obtain
(51)∥∥𝕁∥H2-∥𝕁n∥H2∥LT1(ℝ)=𝔼[supt∈[0,T]|∥𝕁t∥H2-∥𝕁tn∥H2|]≤𝔼[supt∈[0,T]∥𝕁t-𝕁tn∥H2]+2𝔼[supt∈[0,T]∥𝕁t∥H∥𝕁t-𝕁tn∥H]≤𝔼[supt∈[0,T]∥𝕁t-𝕁tn∥H2]+2𝔼[supt∈[0,T]∥𝕁t∥H2]1/2𝔼[supt∈[0,T]∥𝕁t-𝕁tn∥H2]1/2=∥𝕁-𝕁n∥LT2(H)2+2∥𝕁∥LT2(H)×∥𝕁-𝕁n∥LT2(H)⟶0.
Therefore, together with (46), we get
(52)∥M-Mn∥LT1(ℝ)≤∥∥𝕁∥2-∥𝕁n∥2∥LT1(ℝ)+∥𝕀-𝕀n∥LT1(ℝ)⟶0,
showing that Mn→M in AT1(ℝ). Now, Lemma 3 yields that M∈MT1(ℝ), which concludes the proof.Theorem 13.
LetN∈ℳT2(ℝ) be another square-integrable martingale, and let X,Y be two H-valued, predictable processes satisfying (17) and
(53)𝔼[∫0T‍∥Ys∥H2d〈N,N〉s]<∞.
Then, one has
(54)〈X·M,Y·N〉=〈X,Y〉H·〈M,N〉.Proof .
Using Proposition12 and the identities
(55)〈x,y〉H=14(∥x+y∥H2-∥x-y∥H2),x,y∈H,〈M,N〉=14(〈M+N,M+N〉-〈M-N,M-N〉),
identity (54) follows from a straightforward calculation.Proposition 14.
LetN∈ℳT2(ℝ) be another square-integrable martingale such that 〈M,N〉=0, and let X,Y be two H-valued, predictable processes satisfying (17) and (53). Then, one has
(56)〈X·M,Y·N〉MT2(H)=0.Proof.
Using Remark11, Theorem 13, and the hypothesis 〈M,N〉=0, we obtain
(57)〈X·M,Y·N〉MT2(H)=𝔼[〈∫0T‍XsdMs,∫0T‍YsdNs〉H]=𝔼[〈∫0T‍XsdMs,∫0T‍YsdNs〉]=𝔼[∫0T‍〈Xs,Ys〉Hd〈M,N〉s]=0,
completing the proof.


## 4.  The Itô Integral with respect to a Sequence of Standard Lévy Processes

In this section, we introduce the Itô integral forℓ2(H)-valued processes with respect to a sequence of standard Lévy processes, which is based on the Itô integral (31) from the previous section. We define the space of sequences
(58)ℓ2(H):={(hj)j∈ℕ⊂H:∑j=1∞‍∥hj∥H2<∞},
which, equipped with the inner product
(59)〈h,g〉ℓ2(H)=∑j=1∞‍〈hj,gj〉H,
is a separable Hilbert space.Definition 15.
A sequence(Mj)j∈ℕ of real-valued Lévy processes is called a sequence of standard Lévy processes if it consists of square-integrable martingales with 〈Mj,Mk〉t=δjk·t for all j,k∈ℕ. Here, δjk denotes the Kronecker delta
(60)δjk={1,ifj=k,0,ifj≠k.
For the rest of this section, let(Mj)j∈ℕ be a sequence of standard Lévy processes.Proposition 16.
For everyℓ2(H)-valued, predictable process X with
(61)𝔼[∫0T‍∥Xs∥ℓ2(H)2ds]<∞,
the series
(62)∑j∈ℕ‍Xj·Mj
converges unconditionally in MT2(H).Proof.
Forj,k∈ℕ with j≠k, we have 〈Mj,Mk〉=0, and, hence, by Proposition 14, we obtain
(63)〈Xj·Mj,Xk·Mk〉MT2(H)=0.
Moreover, by the Itô isometry (Proposition 8) and the monotone convergence theorem, we have
(64)∑j=1∞‍∥Xj·Mj∥MT2(H)2=∑j=1∞‍𝔼[∥∫0T‍XsjdMsj∥H2]=∑j=1∞‍𝔼[∫0T‍∥Xsj∥H2ds]=𝔼[∫0T‍∑j=1∞‍∥Xsj∥H2ds]=𝔼[∫0T‍∥Xs∥ℓ2(H)2ds].
Thus, by (61) and Lemma 4, the series (62) converges unconditionally in MT2(H).Therefore, for aℓ2(H)-valued, predictable process X satisfying (61) we can define the Itô integral as the series (62).Remark 17.
As the proof of Proposition16 shows, the components of the Itô integral ∑j∈ℕ‍Xj·Mj are pairwise orthogonal elements of the Hilbert space MT2(H).Proposition 18.
For eachℓ2(H)-valued, predictable process X satisfying (61), one has the Itô isometry
(65)𝔼[∥∑j∈ℕ‍∫0T‍XsjdMsj∥H2]=𝔼[∫0T‍∥Xs∥ℓ2(H)2ds].Proof .
Using (63), Lemma 4, and identity (64), we obtain
(66)𝔼[∥∑j=1∞‍∫0T‍XsjdMsj∥H2]=∥∑j=1∞‍Xj·Mj∥MT2(H)2=∑j=1∞‍∥Xj·Mj∥MT2(H)2=𝔼[∫0T‍∥Xs∥ℓ2(H)2ds],
completing the proof.Proposition 19.
LetX be a ℓ2(H)-valued simple process of the form
(67)X=X01{0}+∑i=1n‍Xi1(ti,ti+1]
with 0=t1<⋯<tn+1=T and ℱti-measurable random variables Xi:Ω→ℓ2(H) for i=0,…,n. Then, one has
(68)X·M=∑i=1n‍∑j∈ℕ‍Xij((Mj)ti+1-(Mj)ti).Proof .
For eachj∈ℕ, the process Xj is a H-valued simple process having the representation
(69)Xj=X0j1{0}+∑i=1n‍Xij1(ti,ti+1].
Hence, by Proposition 9, we obtain
(70)X·M=∑j∈ℕ‍Xj·Mj=∑j∈ℕ‍∑i=1n‍Xij((Mj)ti+1-(Mj)ti)=∑i=1n‍∑j∈ℕ‍Xij((Mj)ti+1-(Mj)ti),
which finishes the proof.


## 5. Lévy Processes in Hilbert Spaces

In this section, we provide the required results about Lévy processes in Hilbert spaces. LetU be a separable Hilbert space.Definition 20.
AU-valued càdlàg, adapted process L is called a Lévy process if the following conditions are satisfied.  (1)
We haveL0=0.(2)
Lt-Ls is independent of ℱs for all s≤t.(3)
We haveLt-Ls=dLt-s for all s≤t.Definition 21.
AU-valued Lévy process L with 𝔼[∥Lt∥U2]<∞ and 𝔼[Lt]=0 for all t≥0 is called a square-integrable Lévy martingale.Note that any square-integrable Lévy martingaleL is indeed a martingale; that is,
(71)𝔼[Xt∣ℱs]=Xs∀s≤t
see [5, Proposition3.25]. According to [5, Theorem4.44], for each square-integrable Lévy martingale L, there exists a unique self-adjoint, nonnegative definite trace class operator Q∈L(U), called the covariance operator of L, such that for all t,s∈ℝ+ and u1,u2∈U, we have
(72)𝔼[〈Lt,u1〉U〈Ls,u2〉U]=(t∧s)〈Qu1,u2〉U.
Moreover, for all u1,u2∈U, the angle bracket process is given by
(73)〈〈L,u1〉U,〈L,u2〉U〉t=t〈Qu1,u2〉U,t≥0;
see [5, Theorem4.49].Lemma 22.
LetL be a U-valued square-integrable Lévy martingale with covariance operator Q, let V be another separable Hilbert space, and let Φ:U→V be an isometric isomorphism. Then, the process Φ(L) is a V-valued square-integrable Lévy martingale with covariance operator QΦ:=ΦQΦ-1.Proof .
The processΦ(L) is a V-valued càdlàg, adapted process with Φ(L0)=Φ(0)=0. Let s≤t be arbitrary. Then, the random variable Φ(Lt)-Φ(Ls)=Φ(Lt-Ls) is independent of ℱs, and we have
(74)Φ(Lt)-Φ(Ls)=Φ(Lt-Ls)=dΦ(Lt-s).
Moreover, for each t∈ℝ+, we have
(75)𝔼[∥Φ(Lt)∥V2]=𝔼[∥Lt∥U2]<∞,𝔼[Φ(Lt)]=Φ𝔼(Lt)=0,
showing that Φ(L) is a V-valued square-integrable Lévy martingale.
Lett,s∈ℝ+ and vi∈V, i=1,2 be arbitrary, and set ui:=Φ-1vi∈U,  i=1,2. Then, we have
(76)𝔼[〈Φ(Lt),v1〉V〈Φ(Ls),v2〉V]=𝔼[〈Φ(Lt),Φ(u1)〉V〈Φ(Ls),Φ(u2)〉V]=𝔼[〈Lt,u1〉U〈Ls,u2〉U]=(t∧s)〈Qu1,u2〉U=(t∧s)〈QΦ-1v1,Φ-1v2〉U=(t∧s)〈ΦQΦ-1v1,v2〉V=(t∧s)〈QΦv1,v2〉V,
showing that the Lévy martingale Φ(L) has the covariance operator QΦ.Now, letQ∈L(U) be a self-adjoint, positive definite trace class operator. Then, there exists a sequence (λj)j∈ℕ⊂(0,∞) with ∑j=1∞‍λj<∞ and an orthonormal basis (ej(λ))j∈ℕ of U such that
(77)Qej(λ)=λjej(λ)∀j∈ℕ.
We define the sequence of pairwise orthogonal vectors (ej)j∈ℕ as
(78)ej:=λjej(λ),j∈ℕ.Proposition 23.
LetL be a U-valued square-integrable Lévy martingale with covariance operator Q. Then, the sequence (Mj)j∈ℕ given by
(79)Mj:=1λj〈L,ej(λ)〉U,j∈ℕ,
is a sequence of standard Lévy processes.Proof.
For eachj∈ℕ, the process Mj is a real-valued square-integrable Lévy martingale. By (73), for all j,k∈ℕ, we obtain
(80)〈Mj,Mk〉t=1λjλk〈〈L,ej(λ)〉U,〈L,ek(λ)〉U〉t=t〈Qej(λ),ek(λ)〉Uλjλk=tλj〈ej(λ),ek(λ)〉Uλjλk=δjk·t,
showing that (Mj)j∈ℕ is a sequence of standard Lévy processes.


## 6. The Itô Integral with respect to anℓλ2-Valued Lévy Process

In this section, we introduce the Itô integral forℓ2(H)-valued processes with respect to an ℓλ2-valued Lévy process, which is based on the Itô integral (62) from Section 4.Let(λj)j∈ℕ⊂(0,∞) be a sequence with ∑j=1∞‍λj<∞ and denote by ℓλ2 the weighted space of sequences
(81)ℓλ2:={(vj)j∈ℕ⊂ℝ:∑j=1∞‍λj|vj|2<∞},
which, equipped with the inner product
(82)〈v,w〉ℓλ2=∑j=1∞‍λjvjwj,
is a separable Hilbert space. Note that we have the strict inclusion ℓ2⫋ℓλ2, where ℓ2 denotes the space of sequences
(83)ℓ2={(vj)j∈ℕ⊂ℝ:∑j=1∞‍|vj|2<∞}.
We denote by (gj)j∈ℕ the standard orthonormal basis of ℓ2, which is given by
(84)g1=(1,0,…),g2=(0,1,0,…),….
Then, the system (gj(λ))j∈ℕ defined as
(85)gj(λ):=gjλj,j∈ℕ,
is an orthonormal basis of ℓλ2. Let Q∈L(ℓλ2) be a linear operator such that
(86)Qgj(λ)=λjgj(λ),∀j∈ℕ.
Then, Q is a nuclear, self-adjoint, positive definite operator. Let L be an ℓλ2-valued, square-integrable Lévy martingale with covariance operator Q. According to Proposition 23, the sequence (Mj)j∈ℕ given by
(87)Mj:=1λj〈L,gj(λ)〉ℓλ2,j∈ℕ,
is a sequence of standard Lévy processes.Definition 24.
For everyℓ2(H)-valued, predictable process X satisfying (61), we define the Itô integral X·L:=(∫0t‍XsdLs)t∈[0,T] as
(88)X·L:=∑j∈ℕ‍Xj·Mj.Remark 25.
Note thatL20(H)≅ℓ2(H), where L20(H) denotes the space of Hilbert-Schmidt operators from ℓ2 to H. In [21], the Itô integral for L20(H)-valued processes with respect to an ℓλ2-valued Wiener process has been constructed in the usual fashion (first for elementary and afterwards for general processes), and then the series representation (88) has been proven; see [21, Proposition  2.2.1].
Now, let(μk)k∈ℕ be another sequence with ∑k=1∞‍μk<∞, and let Φ:ℓλ2→ℓμ2 be an isometric isomorphism such that
(89)QΦgk(μ)=μkgk(μ),∀k∈ℕ.
By Lemma 22, the process Φ(L) is a ℓμ2-valued, square integrable Lévy martingale with covariance operator QΦ, and by Proposition 23, the sequence (Nk)k∈ℕ given by
(90)Nk:=1μk〈Φ(L),gk(μ)〉ℓμ2,k∈ℕ,is a sequence of standard Lévy processes.Theorem 26.
LetΨ∈L(ℓ2(H)) be an isometric isomorphism such that
(91)〈h,Ψ(w)〉H=Φ(〈h,w〉H)∀h∈H,w∈ℓ2(H).
Then, for every ℓ2(H)-valued, predictable process X satisfying (61), one has
(92)𝔼[∫0T‍∥Ψ(Xs)∥ℓ2(H)2ds]<∞
and the identity
(93)X·L=Ψ(X)·Φ(L).Proof.
SinceΨ is an isometry, by (61), we have
(94)𝔼[∫0T‍∥Ψ(Xs)∥ℓ2(H)2ds]=𝔼[∫0T‍∥Xs∥ℓ2(H)2ds]<∞,
showing (92). Moreover, by (89), we have
(95)ΦQΦ-1gk(μ)=QΦgk(μ)=μkgk(μ)∀k∈ℕ,
and, hence, we get
(96)Q(Φ-1gk(μ))=μk(Φ-1gk(μ))∀k∈ℕ.
By (86) and (96), the vectors (gj(λ))j∈ℕ and (Φ-1gk(μ))k∈ℕ are eigenvectors of Q with corresponding eigenvalues (λj)j∈ℕ and (μk)k∈ℕ. Therefore, and since Φ is an isometry, for j,k∈ℕ with λj≠μk, we obtain
(97)〈Φgj(λ),gk(μ)〉ℓμ2=〈gj(λ),Φ-1gk(μ)〉ℓλ2=0.
Let h∈H be arbitrary. Then, we have
(98)〈h,X·L〉H=〈h,∑j=1∞‍Xj·Mj〉H=∑j=1∞‍〈h,Xj·Mj〉H=∑j=1∞‍〈h,Xj〉H·Mj=∑j=1∞‍1λj〈〈h,X〉H,gj(λ)〉ℓλ2·1λj〈L,gj(λ)〉ℓλ2=∑j=1∞‍1λj〈Φ(〈h,X〉H),Φgj(λ)〉ℓμ2·1λj〈Φ(L),Φgj(λ)〉ℓμ2=∑j=1∞‍1λj〈Φ(〈h,X〉H),Φgj(λ)〉ℓμ2·1λj(∑k=1∞‍〈Φ(L),gk(μ)〉ℓμ2〈gk(μ),Φgj(λ)〉ℓμ2).
Since (λj)j∈ℕ and (μk)k∈ℕ are eigenvalues of Q, for each j∈ℕ, there are only finitely many k∈ℕ such that λj=μk. Therefore, by (97), and since (Φ(gj(λ)))j∈ℕ is an orthonormal basis of ℓμ2, we obtain
(99)〈h,X·L〉H=∑k=1∞‍(∑j=1∞‍1λj〈Φ(〈h,X〉H),Φgj(λ)〉ℓμ2〈Φgj(λ),gk(μ)〉ℓμ2)·〈Φ(L),gk(μ)〉ℓμ2=∑k=1∞‍1μk(∑j=1∞‍〈Φ(〈h,X〉H),Φgj(λ)〉ℓμ2〈Φgj(λ),gk(μ)〉ℓμ2)·〈Φ(L),gk(μ)〉ℓμ2=∑k=1∞‍1μk〈Φ(〈h,X〉H),gk(μ)〉ℓμ2·1μk〈Φ(L),gk(μ)〉ℓμ2=∑k=1∞‍Φ(〈h,X〉H)k·Nk.
Thus, taking into account (91) gives us
(100)〈h,X·L〉H=∑k=1∞‍〈h,Ψ(X)k〉H·Nk=∑k=1∞‍〈h,Ψ(X)k·Nk〉H=〈h,∑k=1∞‍Ψ(X)k·Nk〉H=〈h,Ψ(X)·Φ(L)〉H.
Since h∈H was arbitrary, using the separability of H as in the proof of Proposition 5, we arrive at (93).Remark 27.
From a geometric point of view, Theorem26 says that the “angle” measured by the Itô integral is preserved under isometries.


## 7.  The Itô Integral with respect to a General Lévy Process

In this section, we define the Itô integral with respect to a general Lévy process, which is based on the Itô integral (88) from the previous section.LetU be a separable Hilbert space, and let Q∈L(U) be a nuclear, self-adjoint, positive definite linear operator. Then, there exist a sequence (λj)j∈ℕ⊂(0,∞) with ∑j=1∞‍λj<∞ and an orthonormal basis (ej(λ))j∈ℕ of  U such that
(101)Qej(λ)=λjej(λ),∀j∈ℕ;
namely, λj are the eigenvalues of Q, and each ej(λ) is an eigenvector corresponding to λj. The space U0:=Q1/2(U), equipped with the inner product
(102)〈u,v〉U0:=〈Q-1/2u,Q-1/2v〉U,
is another separable Hilbert space, and the sequence (ej)j∈ℕ given by
(103)ej=λjej(λ),j∈ℕ,
is an orthonormal basis of U0. We denote by L20(H):=L2(U0,H) the space of Hilbert-Schmidt operators from U0 into H, which, endowed with the Hilbert-Schmidt norm
(104)∥S∥L20(H):=(∑j=1∞‍∥Sej∥H2)1/2,S∈L20(H),
itself is a separable Hilbert space. We define the isometric isomorphisms
(105)Φλ:U⟶ℓλ2,Φλej(λ):=gj(λ)forj∈ℕ,(106)Ψλ:L20(H)⟶ℓ2(H),Ψλ(S)≔(Sej)j∈ℕforS∈L20(H).
Recall that (gj(λ))j∈ℕ denotes the orthonormal basis of ℓλ2, which we have defined in (85). Let L be a U-valued square-integrable Lévy martingale with covariance operator Q.Lemma 28.
The following statements are true.(1)
The processΦλ(L) is an ℓλ2-valued square-integrable Lévy martingale with covariance operator QΦλ.(2)
One has(107)QΦλgj(λ)=λjgj(λ),∀j∈ℕ.Proof.
By Lemma22, the process Φλ(L) is an ℓλ2-valued square-integrable Lévy martingale with covariance operator QΦλ. Furthermore, by (105) and (101), for all j∈ℕ, we obtain
(108)QΦλgj(λ)=ΦλQΦλ-1gj(λ)=ΦλQej(λ)=Φλ(λjej(λ))=λjΦλej(λ)=λjgj(λ),
showing (107).Now, our idea is to the define the Itô integral for anL20(H)-valued, predictable process X with
(109)𝔼[∫0T‍∥Xs∥L20(H)2ds]<∞
by setting
(110)X·L:=Ψλ(X)·Φλ(L),
where the right-hand side of (110) denotes the Itô integral (88) from Definition 24. One might suspect that this definition depends on the choice of the eigenvalues (λj)j∈ℕ and eigenvectors (ej(λ))j∈ℕ. In order to prove that this is not the case, let (μk)k∈ℕ⊂(0,∞) be another sequence with ∑k=1∞‍μk<∞, and let (fk(μ))k∈ℕ be another orthonormal basis of  U such that
(111)Qfk(μ)=μkfk(μ),∀k∈ℕ.
Then, the sequence (fk)k∈ℕ given by
(112)fk=μkfk(μ),k∈ℕ,
is an orthonormal basis of U0. Analogous to (105) and (106), we define the isometric isomorphisms
(113)Φμ:U⟶ℓμ2,Φμfk(μ):=gk(μ)fork∈ℕ,Ψμ:L20(H)⟶ℓ2(H),Ψμ(S):=(Sfk)k∈ℕforS∈L20(H).
Furthermore, we define the isometric isomorphisms
(114)Φ:=Φμ∘Φλ-1:ℓλ2⟶ℓμ2,Ψ:=Ψμ∘Ψλ-1:ℓ2(H)⟶ℓ2(H).
The following diagram illustrates the situation:
(115)In order to show that the Itô integral (110) is well defined, we have to show that
(116)Ψλ(X)·Φλ(L)=Ψμ(X)·Φμ(L).
For this, we prepare the following auxiliary result.Lemma 29.
For allh∈H and w∈ℓ2(H), one has
(117)〈h,Ψ(w)〉H=Φ(〈h,w〉H).Proof.
By (101) and (111), the vectors (ej(λ))j∈ℕ and (fk(μ))k∈ℕ are eigenvectors of Q with corresponding eigenvalues (λj)j∈ℕ and (μk)k∈ℕ. Therefore, for j,k∈ℕ with λj≠μk, we have 〈ej(λ),fk(μ)〉U=0. For each v∈ℓλ2, we obtain
(118)Φ(v)=(Φμ∘Φλ-1)(v)=Φμ(∑k=1∞‍〈Φλ-1v,fk(μ)〉Ufk(μ))=∑k=1∞‍〈Φλ-1v,fk(μ)〉Ugk(μ)=(1μk〈Φλ-1v,fk(μ)〉U)k∈ℕ=(1μk∑j=1∞‍〈Φλ-1v,ej(λ)〉U〈ej(λ),fk(μ)〉U)k∈ℕ=(1μk∑j=1∞‍〈v,gj(λ)〉ℓλ2〈ej(λ),fk(μ)〉U)k∈ℕ=(∑j=1∞‍λjμk〈ej(λ),fk(μ)〉Uvj)k∈ℕ=(∑j=1∞‍〈ej(λ),fk(μ)〉Uvj)k∈ℕ.
Let w∈ℓ2(H) be arbitrary. By (106), we have
(119)w=Ψλ(Ψλ-1(w))=(Ψλ-1(w)ej)j∈ℕ,
and, hence,
(120)Ψ(w)=(Ψμ∘Ψλ-1)(w)=(Ψλ-1(w)fk)k∈ℕ=(Ψλ-1(w)(∑j=1∞‍〈fk,ej〉U0ej))k∈ℕ=(∑j=1∞‍〈fk,ej〉U0Ψλ-1(w)ej)k∈ℕ=(∑j=1∞‍〈fk,ej〉U0wj)k∈ℕ=(∑j=1∞‍〈ej(λ),fk(μ)〉Uwj)k∈ℕ.
Therefore, for all h∈H and w∈ℓ2(H), we obtain
(121)〈h,Ψ(w)〉H=(∑j=1∞‍〈ej(λ),fk(μ)〉U〈h,wj〉H)k∈ℕ=Φ(〈h,w〉H),
finishing the proof.Proposition 30.
The following statements are true.(1)
Φλ(L) is an ℓλ2-valued Lévy process with covariance operator QΦλ, and one has
(122)QΦλgj(λ)=λjgj(λ),∀j∈ℕ.(2)
Φμ(L) is an ℓμ2-valued Lévy process with covariance operator QΦμ, and one has
(123)QΦμgk(μ)=μkgk(μ)∀k∈ℕ.(3)
For everyL20(H)-valued, predictable process X with (109), one has
(124)𝔼[∫0T‍∥Ψλ(Xs)∥ℓ2(H)2ds]<∞,𝔼[∫0T‍∥Ψμ(Xs)∥ℓ2(H)2ds]<∞
and the identity (116).Proof.
The first two statements follow from Lemma28. Since Ψλ and Ψμ are isometries, we obtain
(125)𝔼[∫0T‍∥Ψλ(Xs)∥ℓ2(H)2ds]=𝔼[∫0T‍∥Xs∥L20(H)2ds]=𝔼[∫0T‍∥Ψμ(Xs)∥ℓ2(H)2ds],
which, together with (109), yields (124). Now, Theorem 26 applies by virtue of Lemma 29 and yields
(126)Ψλ(X)·Φλ(L)=Ψ(Ψλ(X))·Φ(Φλ(L))=Ψμ(X)·Φμ(L),
proving (116).Definition 31.
For everyL20(H)-valued process X satisfying (109), we define the Itô-Integral X·L=(∫0t‍XsdLs)t∈[0,T] by (110).By virtue of Proposition30, Definition (110) of the Itô integral neither depends on the choice of the eigenvalues (λj)j∈ℕ nor on the eigenvectors (ej(λ))j∈ℕ.Now, we will the prove the announced series representation of the Itô integral. According to Proposition23, the sequences (Mj)j∈ℕ and (Nj)j∈ℕ of real-valued processes given by
(127)Mj:=1λj〈L,ej(λ)〉U,Nj:=1λj〈Φλ(L),gj(λ)〉ℓλ2
are sequences of standard Lévy processes.Proposition 32.
For everyL20(H)-valued, predictable process X satisfying (109), the process (ξj)j∈ℕ given by
(128)ξj:=Xej,j∈ℕ,
is a ℓ2(H)-valued, predictable process, and, one has
(129)X·L=∑j∈ℕ‍ξj·Mj,
where the right-hand side of (129) converges unconditionally in MT2(H).Proof.
SinceΦλ is an isometry, for each j∈ℕ, we obtain
(130)Mj=1λj〈L,ej(λ)〉U=1λj〈Φλ(L),Φλej(λ)〉ℓλ2=1λj〈Φλ(L),gj(λ)〉ℓλ2=Nj.
Thus, by Definitions 31 and 24, we obtain
(131)X·L=Ψλ(X)·Φλ(L)=∑j∈ℕ‍Ψλ(X)j·Nj=∑j∈ℕ‍ξj·Mj,
and, by Proposition 16, the series converges unconditionally in MT2(H).Remark 33.
By Remark17 and the proof of Proposition 32, the components of the Itô integral ∑j∈ℕ‍ξj·Mj are pairwise orthogonal elements of the Hilbert space MT2(H).Proposition 34.
For everyL20(H)-valued process X satisfying (109), one has the Itô isometry
(132)𝔼[∥∫0T‍XsdLs∥H2]=𝔼[∫0T‍∥Ls∥L20(H)2ds].Proof.
By the Itô isometry (Proposition18), and since Ψλ is an isometry, we obtain
(133)𝔼[∥∫0T‍XsdLs∥H2]=𝔼[∥∫0T‍Ψλ(Xs)dΦλ(L)s∥H2]=𝔼[∫0T‍∥Ψλ(Xs)∥ℓ2(H)2ds]=𝔼[∫0T‍∥Xs∥L20(H)2ds],
completing the proof.We shall now prove that the stochastic integral, which we have defined so far, coincides with the Itô integral developed in [5]. For this purpose, it suffices to consider elementary processes. Note that for each operator S∈L(U,H), the restriction S|U0 belongs to L20(H), because
(134)∑j=1∞‍∥Sej∥H2≤∑j=1∞‍∥S∥L(U,H)2∥ej∥U2=∥S∥L(U,H)2∑j=1∞‍∥λjej(λ)∥U02=∥S∥L(U,H)2∑j=1∞‍λj<∞.Proposition 35.
LetX be a L(U,H)-valued simple process of the form
(135)X=X01{0}+∑i=1n‍Xi1(ti,ti+1]
with 0=t1<⋯<tn+1=T and ℱti-measurable random variables Xi:Ω→L(U,H) for i=0,…,n. Then, one has
(136)X|U0·L=∑i=1n‍Xi(Lti+1-Lti).Proof.
The processΨλ(X|U0) is an ℓ2(H)-valued simple process having the representation
(137)Ψλ(X|U0)=Ψλ(X0|U0)1{0}+∑i=1n‍Ψλ(Xi|U0)1(ti,ti+1].
Thus, by Proposition 19, and since Φλ is an isometry, we obtain
(138)X|U0·L=Ψλ(X|U0)·Φλ(L)=∑i=1n‍∑j∈ℕ‍Ψλ(Xi|U0)j((Nj)ti+1-(Nj)ti)=∑i=1n‍∑j∈ℕ‍Xiej〈Φλ(Lti+1-Lti),gj(λ)〉ℓλ2λj=∑i=1n‍∑j∈ℕ‍Xiej(λ)〈Lti+1-Lti,Φλ-1gj(λ)〉U=∑i=1n‍∑j∈ℕ‍Xiej(λ)〈Lti+1-Lti,ej(λ)〉U=∑i=1n‍Xi(∑j∈ℕ‍〈Lti+1-Lti,ej(λ)〉Uej(λ))=∑i=1n‍Xi(Lti+1-Lti),
completing the proof.Therefore, and since the space of simple processes is dense in the space of all predictable processes satisfying (109); see, for example, [5, Corollary8.17], the Itô integral (110) coincides with that in [5] for every L20(H)-valued, predictable process X satisfying (109). In particular, for a driving Wiener process, it coincides with the Itô integral from [1–3].By a standard localization argument, we can extend the definition of the Itô integral to all predictable processesX satisfying
(139)ℙ(∫0T‍∥Xs∥L20(H)2ds<∞)=1,∀T>0.
Since the respective spaces of predictable and adapted, measurable processes are isomorphic (see [24]), proceeding as in [24, Section3.2], we can further extend the definition of the Itô integral to all adapted, measurable processes X satisfying (139).



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*Source: 703769-2013-04-04.xml*