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

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.

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 an 𝐻-valued process 𝑋 (with 𝐻 denoting a separable Hilbert space) and a real-valued square-integrable martingale 𝑀, we define the Itô integral 𝑋⋅𝑀∶=_𝑘∈ℕ𝑋,𝑓_𝑘𝐻⋅𝑀𝑓_𝑘, (1) where (𝑓_𝑘)_𝑘∈ℕ denotes an orthonormal basis of 𝐻, and ⟨𝑋,𝑓_𝑘⟩_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 an ℓ²(𝐻)-valued process 𝑋 and a sequence (𝑀^𝑗)_𝑗∈ℕ of standard Lévy processes, we define the Itô integral as _𝑗∈ℕ𝑋^𝑗⋅𝑀^𝑗. (2) For this, we will ensure convergence of the series.(iii)In the next step, let 𝐿 denote an ℓ²_𝜆-valued Lévy process, where ℓ²_𝜆 is a weighted space of sequences (cf. [21]). From the Lévy process 𝐿, we can construct a sequence (𝑀^𝑗)_𝑗∈ℕ of standard Lévy processes, and for a ℓ²(𝐻)-valued process 𝑋, we define the Itô integral 𝑋⋅𝐿∶=_𝑗∈ℕ𝑋^𝑗⋅𝑀^𝑗. (3)(iv)Finally, let 𝐿 be a general Lévy process on some separable Hilbert space 𝑈 with covariance operator 𝑄. Then, there exist sequences of eigenvalues (𝜆_𝑗)_𝑗∈ℕ and eigenvectors, which diagonalize the operator 𝑄. Denoting by 𝐿⁰₂(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 𝐿⁰₂(𝐻)-valued process 𝑋 as 𝑋⋅𝐿∶=Ψ(𝑋)⋅Φ(𝐿), (4) where Φ∶𝑈→ℓ²_𝜆 and Ψ∶𝐿⁰₂(𝐻)→ℓ²(𝐻) are isometric isomorphisms such that Φ(𝐿) is an ℓ²_𝜆-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 Section 2, 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 ℓ²_𝜆-valued Lévy process, and in Section 7, we define the Itô integral in the general case, where the integrand is an 𝐿⁰₂(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,(ℱ_𝑡)_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. Let 𝑝≥1 be arbitrary. (1)We define the Lebesgue space ℒ^𝑝_𝑇(𝐸)∶=ℒ^𝑝Ω,ℱ_𝑇,ℙ;𝔻([0,𝑇];𝐸), (5) 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 by 𝒜^𝑝_𝑇(𝐸) the space of all 𝐸-valued adapted processes 𝑋∈ℒ^𝑝_𝑇(𝐸).(3)We denote by ℳ^𝑝_𝑇(𝐸) the space of all 𝐸-valued martingales 𝑀∈ℒ^𝑝_𝑇(𝐸).(4)We define the factor spaces 𝑀^𝑝_𝑇(𝐸)ℳ∶=^𝑝𝑇(𝐸)𝑁,𝐴^𝑝_𝑇(𝐸)𝒜∶=^𝑝𝑇(𝐸)𝑁,𝐿^𝑝_𝑇(𝐸)ℒ∶=^𝑝𝑇(𝐸)𝑁, (6)(7) where 𝑁⊂ℳ^𝑝_𝑇(𝐸) denotes the subspace consisting of all 𝑀∈ℳ^𝑝_𝑇(𝐸) with 𝑀=0 up to indistinguishability.

Remark 2. Let us emphasize the following. (1)Since the Skorokhod space 𝔻([0,𝑇];𝐸) equipped with the supremum norm is a Banach space, the Lebesgue space 𝐿^𝑝_𝑇(𝐸) equipped with the standard norm ‖𝑋‖_𝐿^𝑝_𝑇(𝐸)∶=𝔼‖𝑋‖^𝑝_𝐸^1/𝑝 (8) is a Banach space too.(2)By the completeness of the filtration (ℱ_𝑡)_𝑡≥0, adaptedness of an element 𝑋∈𝐿^𝑝_𝑇(𝐸) does not depend on the choice of the representative. This ensures that the factor space 𝐴^𝑝_𝑇(𝐸) of adapted processes is well defined.(3)The definition of 𝐸-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 𝑀^𝑝_𝑇(𝐸)⊂𝐴^𝑝_𝑇(𝐸)⊂𝐿^𝑝_𝑇(𝐸). (9)

The following auxiliary result shows that these inclusions are closed.

Lemma 3. Let 𝑝≥1 be arbitrary. Then, the following statements are true: (1)𝑀^𝑝_𝑇(𝐸) is closed in 𝐴^𝑝_𝑇(𝐸);(2)𝐴^𝑝_𝑇(𝐸) is closed in 𝐿^𝑝_𝑇(𝐸).

Proof. Let (𝑀^𝑛)_𝑛∈ℕ⊂𝑀^𝑝_𝑇(𝐸) be a sequence, and let 𝑀∈𝐴^𝑝_𝑇(𝐸) be such that 𝑀^𝑛→𝑀 in 𝐿^𝑝_𝑇(𝐸). Furthermore, let 𝜏≤𝑇 be a bounded stopping time. Then, we have 𝔼𝑀_𝜏^𝑝_𝐸≤𝔼sup_{𝑡∈[0,𝑇]}𝑀_𝑡^𝑝_𝐸<∞, (10) showing that 𝑀_𝜏∈𝐿^𝑝(Ω,ℱ_𝜏,ℙ;𝐸). Furthermore, we have 𝔼𝑀^𝑛_𝜏−𝑀_𝜏^𝑝_𝐸≤𝔼sup_{𝑡∈[0,𝑇]}𝑀^𝑛_𝑡−𝑀_𝑡^𝑝_𝐸⟶0. (11) By Doob’s optional stopping theorem (which also holds true for 𝐸-valued martingales; see [2, Remark 2.2.5]), it follows that 𝔼𝑀_𝜏=lim_𝑛→∞𝔼𝑀^𝑛_𝜏=lim_𝑛→∞𝔼𝑀^𝑛₀=𝔼𝑀₀. (12) Using Doob’s optional stopping theorem again, we conclude that 𝑀∈𝑀^𝑝_𝑇(𝐸), proving the first statement.
Now, let (𝑋^𝑛)_𝑛∈ℕ⊂𝐴^𝑝_𝑇(𝐸) be a sequence, and let 𝑋∈𝐿^𝑝_𝑇(𝐸) be such that 𝑋^𝑛→𝑋 in 𝐿^𝑝_𝑇(𝐸). Then, for each 𝑡∈[0,𝑇], we have 𝔼𝑋^𝑛_𝑡−𝑋_𝑡^𝑝_𝐸≤𝔼sup_{𝑠∈[0,𝑇]}𝑋^𝑛_𝑠−𝑋_𝑠^𝑝_𝐸⟶0, (13) and, hence, ℙ-almost surely 𝑋^𝑛𝑘_𝑡→𝑋_𝑡 for some subsequence (𝑛_𝑘)_𝑘∈ℕ, showing that 𝑋_𝑡 is ℱ_𝑡-measurable. This proves that 𝑋∈𝐴^𝑝_𝑇(𝐸), providing the second statement.

Note that, by Doob's martingale inequality [2, Theorem 2.2.7], for 𝑝>1, an equivalent norm on 𝑀^𝑝_𝑇(𝐸) is given by ‖𝑀‖_𝑀^𝑝_𝑇(𝐸)∶=𝔼𝑀_𝑇^𝑝_𝐸^1/𝑝. (14) Furthermore, if 𝐸=𝐻 is a separable Hilbert space, then 𝑀²_𝑇(𝐻) is a separable Hilbert space equipped with the inner product ⟨𝑀,𝑁⟩_𝑀²_𝑇(𝐻)∶=𝔼𝑀_𝑇,𝑁_𝑇𝐻. (15) Finally, we recall the following result about series of pairwise orthogonal vectors in Hilbert spaces.

Lemma 4. Let 𝐻 be a separable Hilbert space, and let (ℎ_𝑛)_𝑛∈ℕ⊂𝐻 be a sequence with ⟨ℎ_𝑛,ℎ_𝑚⟩_𝐻=0 for 𝑛≠𝑚. Then, the following statements are equivalent. (1)The series ∑^∞_𝑛=1ℎ_𝑛 converges in 𝐻.(2)The series ∑_𝑛∈ℕℎ_𝑛 converges unconditionally in 𝐻.(3)One has ∑^∞_𝑛=1‖ℎ_𝑛‖²_𝐻<∞.
If the previous conditions are satisfied, then one has ^∞_𝑛=1ℎ_𝑛²_𝐻=^∞_𝑛=1ℎ_𝑛²_𝐻. (16)

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, let 𝐻 be a separable Hilbert space, and let 𝑇>0 be a finite time horizon. Furthermore, let 𝑀∈ℳ²_𝑇(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. Let 𝑋 be an 𝐻-valued, predictable process with 𝔼^𝑇₀𝑋_𝑠²_𝐻𝑑⟨𝑀,𝑀⟩_𝑠<∞. (17) Then, for every orthonormal basis (𝑓_𝑘)_𝑘∈ℕ of 𝐻, the series _𝑘∈ℕ𝑋,𝑓_𝑘𝐻⋅𝑀𝑓_𝑘 (18) converges unconditionally in 𝑀²_𝑇(𝐻), and its value does not depend on the choice of the orthonormal basis (𝑓_𝑘)_𝑘∈ℕ.

Proof. Let (𝑓_𝑘)_𝑘∈ℕ be an orthonormal basis of 𝐻. For 𝑗,𝑘∈ℕ with 𝑗≠𝑘, we have 𝑋,𝑓_𝑗𝐻⋅𝑀𝑓_𝑗,𝑋,𝑓_𝑘𝐻⋅𝑀𝑓_{𝑘𝑀2𝑇(𝐻)}=𝔼^𝑇₀𝑋_𝑠,𝑓_𝑗𝐻𝑑𝑀_𝑠𝑓_𝑗,^𝑇₀𝑋_𝑠,𝑓_𝑘𝐻𝑑𝑀_𝑠𝑓_𝑘𝐻=𝔼^𝑇₀𝑋_𝑠,𝑓_𝑗𝐻𝑑𝑀_𝑠^𝑇₀𝑋_𝑠,𝑓_𝑘𝐻𝑑𝑀_𝑠𝑓_𝑗,𝑓_𝑘𝐻=0. (19) Moreover, by the Itô isometry for the real-valued Itô integral and the monotone convergence theorem, we obtain ^∞_𝑘=1𝑋,𝑓_𝑘𝐻⋅𝑀𝑓_𝑘²_{𝑀2𝑇(𝐻)}=^∞_𝑘=1𝔼^𝑇₀𝑋_𝑠,𝑓_𝑘𝐻𝑑𝑀_𝑠𝑓_𝑘²_𝐻=^∞_𝑘=1𝔼^𝑇₀𝑋_𝑠,𝑓_𝑘𝐻𝑑𝑀_𝑠²=^∞_𝑘=1𝔼^𝑇₀𝑋_𝑠,𝑓_𝑘𝐻²𝑑⟨𝑀,𝑀⟩_𝑠=𝔼^𝑇₀^∞_𝑘=1𝑋_𝑠,𝑓_𝑘𝐻²𝑑⟨𝑀,𝑀⟩_𝑠=𝔼^𝑇₀𝑋_𝑠²_𝐻𝑑⟨𝑀,𝑀⟩_𝑠. (20) Therefore, by (17) and Lemma 4, the series (18) converges unconditionally in 𝑀²_𝑇(𝐻).
Now, let (𝑔_𝑘)_𝑘∈ℕ be another orthonormal basis of 𝐻. We define 𝕁^𝑓,𝕁^𝑔∈𝑀²_𝑇(𝐻) by 𝕁^𝑓∶=^∞_𝑘=1𝑋,𝑓_𝑘𝐻⋅𝑀𝑓_𝑘,𝕁^𝑔∶=^∞_𝑘=1𝑋,𝑔_𝑘𝐻⋅𝑀𝑔_𝑘. (21) Let ℎ∈𝐻 be arbitrary. Then, we have ℎ,𝕁^𝑓_𝐻,ℎ,𝕁^𝑔_𝐻∈𝑀²_𝑇(ℝ) (22) and the identity ℎ,𝕁^𝑓_𝐻−⟨ℎ,𝑋⟩_𝐻⋅𝑀²_{𝑀2𝑇(ℝ)}=ℎ,^∞_𝑘=1(𝑋,𝑓_𝑘𝐻⋅𝑀)𝑓_𝑘𝐻−⟨ℎ,𝑋⟩_𝐻⋅𝑀²_{𝑀2𝑇(ℝ)}=^∞_𝑘=1(ℎ,𝑓_𝑘𝐻𝑓_𝑘,𝑋_𝐻⋅𝑀)−⟨ℎ,𝑋⟩_𝐻⋅𝑀²_{𝑀2𝑇(ℝ)}=lim_𝑛→∞^𝑛_𝑘=1(ℎ,𝑓_𝑘𝐻𝑓_𝑘,𝑋_𝐻⋅𝑀)−⟨ℎ,𝑋⟩_𝐻⋅𝑀²_{𝑀2𝑇(ℝ)}=lim_𝑛→∞^𝑛_𝑘=1ℎ,𝑓_𝑘𝐻𝑓_𝑘,𝑋_𝐻−⟨ℎ,𝑋⟩_𝐻⋅𝑀²_{𝑀2𝑇(ℝ)}. (23) For all 𝑥∈𝐻, we have ^𝑛_𝑘=1𝑥,𝑓_𝑘𝐻𝑓_𝑘,ℎ_𝐻−⟨𝑥,ℎ⟩_𝐻²⟶0as𝑛⟶∞, (24) and, by the Cauchy-Schwarz inequality, ^𝑛_𝑘=1𝑥,𝑓_𝑘𝐻𝑓_𝑘,ℎ_𝐻−⟨𝑥,ℎ⟩_𝐻²=^∞_𝑘=𝑛+1𝑥,𝑓_𝑘𝐻𝑓_𝑘,ℎ_𝐻²≤^∞_𝑘=1𝑥,f_𝑘𝐻^2∞_𝑘=1𝑓_𝑘,ℎ_𝐻²=‖𝑥‖²_𝐻‖ℎ‖²_𝐻foreach𝑛∈ℕ. (25) Therefore, by the Itô isometry for the real-valued Itô integral and Lebesgue's dominated convergence theorem together with (17), we obtain ℎ,𝕁^𝑓_𝐻−⟨ℎ,𝑋⟩_𝐻⋅𝑀²_{𝑀2𝑇(ℝ)}=lim_𝑛→∞𝔼^𝑇₀^𝑛_𝑘=1ℎ,𝑓_𝑘𝐻𝑓_𝑘,𝑋_𝑠𝐻−ℎ,𝑋_𝑠𝐻𝑑𝑀_𝑠²=lim_𝑛→∞𝔼^𝑇₀^𝑛_𝑘=1ℎ,𝑓_𝑘𝐻𝑓_𝑘,𝑋_𝐻−⟨ℎ,𝑋⟩_𝐻²𝑑⟨𝑀,𝑀⟩_𝑠=0. (26) Analogously, we prove that ℎ,𝕁^𝑔_𝐻−⟨ℎ,𝑋⟩_𝐻⋅𝑀²_{𝑀2𝑇(ℝ)}=0. (27) Therefore, denoting by 𝕁̃^𝑓,𝕁̃^𝑔∈ℳ²_𝑇(𝐻) representatives of 𝕁^𝑓, 𝕁^𝑔, we obtain ℎ,𝕁̃^𝑓_𝑇𝐻=ℎ,𝕁̃^𝑔_𝑇𝐻∀ℎ∈𝐻,ℙ-almostsurely. (28) By separability of 𝐻, we deduce that ℎ,𝕁̃^𝑓_𝑇𝐻=ℎ,𝕁̃^𝑔_𝑇𝐻ℙ-almostsurely,∀ℎ∈𝐻. (29) Consequently, we have 𝕁̃^𝑓_𝑇=𝕁̃^𝑔_𝑇ℙ-almostsurely, (30) Implying that 𝕁^𝑓=𝕁^𝑔. This proves that the value of the series (18) does not depend on the choice of the orthonormal basis.

Now, Proposition 5 gives rise to the following definition.

Definition 6. For every 𝐻-valued, predictable process 𝑋 satisfying (17), we define the Itô integral 𝑋⋅𝑀=(∫^𝑡₀𝑋_𝑠𝑑𝑀_𝑠)_{𝑡∈[0,𝑇]} as 𝑋⋅𝑀∶=_𝑘∈ℕ𝑋,𝑓_𝑘𝐻⋅𝑀𝑓_𝑘, (31) where (𝑓_𝑘)_𝑘∈ℕ denotes an orthonormal basis of 𝐻.

According to Proposition 5, definition (31) of the Itô integral is independent of the choice of the orthonormal basis (𝑓_𝑘)_𝑘∈ℕ, and the integral process 𝑋⋅𝑀 belongs to 𝑀²_𝑇(𝐻).

Remark 7. As the proof of Proposition 5 shows, the components of the Itô integral 𝑋⋅𝑀 are pairwise orthogonal elements of the Hilbert space 𝑀²_𝑇(𝐻).

Proposition 8. For every 𝐻-valued, predictable process 𝑋 satisfying (17), one has the Itô isometry 𝔼^𝑇₀𝑋_𝑠𝑑𝑀_𝑠²_𝐻=𝔼^𝑇₀𝑋_𝑠²_𝐻𝑑⟨𝑀,𝑀⟩_𝑠. (32)

Proof. Let (𝑓_𝑘)_𝑘∈ℕ be an orthonormal basis of 𝐻. According to (19), we have 𝑋,𝑓_𝑗𝐻⋅𝑀𝑓_𝑗,𝑋,𝑓_𝑘𝐻⋅𝑀𝑓_{𝑘𝑀2𝑇(𝐻)}=0for𝑗≠𝑘. (33) Thus, by Lemma 4 and (20), we obtain 𝔼^𝑇₀𝑋_𝑠𝑑𝑀_𝑠²_𝐻=‖𝑋⋅𝑀‖²_{𝑀2𝑇(𝐻)}=^∞_𝑘=1(𝑋,𝑓_𝑘𝐻⋅𝑀)𝑓_𝑘²_{𝑀2𝑇(𝐻)}=^∞_𝑘=1(𝑋,𝑓_𝑘𝐻⋅𝑀)𝑓_𝑘²_{𝑀2𝑇(𝐻)}=𝔼^𝑇₀𝑋_𝑠²_𝐻𝑑⟨𝑀,𝑀⟩_𝑠, (34) finishing the proof.

Proposition 9. Let 𝑋 be a 𝐻-valued simple process of the form 𝑋=𝑋₀𝟙_{0}+^𝑛_𝑖=1𝑋_𝑖𝟙_{(𝑡𝑖,𝑡𝑖+1]} (35) with 0=𝑡₁<⋯<𝑡_𝑛+1=𝑇 and ℱ_𝑡𝑖-measurable random variables 𝑋_𝑖∶Ω→𝐻 for 𝑖=0,…,𝑛. Then, one has 𝑋⋅𝑀=^𝑛_𝑖=1𝑋_𝑖𝑀^𝑡𝑖+1−𝑀^𝑡𝑖. (36)

Proof. Let (𝑓_𝑘)_𝑘∈ℕ be an orthonormal basis of 𝐻. Then, for each 𝑘∈ℕ, the process ⟨𝑋,𝑓_𝑘⟩ is a real-valued simple process with representation 𝑋,𝑓_𝑘𝐻=𝑋₀,𝑓_𝑘𝐻𝟙_{0}+^𝑛_𝑖=1𝑋_𝑖,𝑓_𝑘𝐻𝟙_{(𝑡𝑖,𝑡𝑖+1]}. (37) Thus, by the definition of the real-valued Itô integral for simple processes, we obtain 𝑋⋅𝑀=_𝑘∈ℕ𝑋,𝑓_𝑘𝐻⋅𝑀𝑓_𝑘=_𝑘∈ℕ^𝑛_𝑖=1𝑋_𝑖,𝑓_𝑘𝐻𝑀^t𝑖+1−𝑀^𝑡𝑖𝑓_𝑘=^𝑛_{𝑖=1𝑘∈ℕ}𝑋_𝑖,𝑓_𝑘𝐻𝑓_𝑘𝑀^𝑡𝑖+1−𝑀^𝑡𝑖=^𝑛_𝑖=1𝑋_𝑖𝑀^𝑡𝑖+1−𝑀^𝑡𝑖, (38) finishing the proof.

Lemma 10. Let 𝑋 be a 𝐻-valued, predictable process satisfying (17). Then, for every orthonormal basis (𝑓_𝑘)_𝑘∈ℕ of 𝐻, one has ^∞_𝑘=1𝑋,𝑓_𝑘𝐻²⋅⟨𝑀,𝑀⟩=‖𝑋‖²_𝐻⋅⟨𝑀,𝑀⟩, (39) where the convergence takes place in 𝐴¹_𝑇(ℝ).

Proof. We define the integral process 𝕀∶=‖𝑋‖²_𝐻⋅⟨𝑀,𝑀⟩ (40) and the sequence (𝕀^𝑛)_𝑛∈ℕ of partial sums by 𝕀^𝑛∶=^𝑛_𝑘=1𝑋,𝑓_𝑘𝐻²⋅⟨𝑀,𝑀⟩. (41) By (17) we have 𝕀∈𝐴¹_𝑇(ℝ) and (𝕀^𝑛)_𝑛∈ℕ⊂𝐴¹_𝑇(ℝ). Furthermore, by Lebesgue's dominated convergence theorem, we have 𝕀−𝕀^𝑛_{𝐿1𝑇(ℝ)}=𝔼sup_{[]𝑡∈0,𝑇}𝕀_𝑡−𝕀^𝑛_𝑡=𝔼sup_{[]𝑡∈0,𝑇}^𝑡₀^∞_𝑘=𝑛+1X_𝑠,𝑓_𝑘²𝑑⟨𝑀,𝑀⟩_𝑠=𝔼^𝑇₀^∞_𝑘=𝑛+1𝑋_𝑠,𝑓_𝑘²𝑑⟨𝑀,𝑀⟩_𝑠⟶0for𝑛⟶∞, (42) which concludes the proof.

Remark 11. As a consequence of the Doob-Meyer decomposition theorem, for two square-integrable martingales 𝑋,𝑌∈ℳ²_𝑇(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 every 𝐻-valued, predictable process 𝑋 satisfying (17), one has ⟨𝑋⋅𝑀,𝑋⋅𝑀⟩=‖𝑋‖²_𝐻⋅⟨𝑀,𝑀⟩. (43)

Proof. Let (𝑓_𝑘)_𝑘∈ℕ be an orthonormal basis of 𝐻. We define the process 𝕁∶=𝑋⋅𝑀 and the sequence (𝕁^𝑛)_𝑛∈ℕ of partial sums by 𝕁^𝑛∶=^𝑛_𝑘=1𝑋,𝑓_𝑘𝐻⋅𝑀𝑓_𝑘. (44) By Proposition 5, we have 𝕁^𝑛⟶𝕁in𝑀²_T(𝐻). (45) Defining the integral process 𝕀 by (40) and the sequence (𝕀^𝑛)_𝑛∈ℕ of partial sums by (41), using Lemma 10, we have 𝕀^𝑛⟶𝕀in𝐴¹_T(ℝ). (46) Furthermore, we define the process 𝑀∈𝐴¹_𝑇(ℝ) and the sequence (𝑀^𝑛)_𝑛∈ℕ⊂𝐴¹_𝑇(ℝ) as 𝑀∶=‖𝕁‖²_𝐻𝑀−𝕀,_𝑛∶=𝕁^𝑛2_𝐻−𝕀^𝑛,𝑛∈ℕ. (47) Then, we have (𝑀^𝑛)_𝑛∈ℕ⊂𝑀¹_𝑇(ℝ). Indeed, for each 𝑛∈ℕ, we have 𝑀^𝑛=^𝑛_𝑘=1𝑋,𝑓_𝑘𝐻⋅𝑀𝑓_𝑘²_𝐻−^𝑛_𝑘=1𝑋,𝑓_𝑘𝐻²=⋅⟨𝑀,𝑀⟩_𝑛𝑘=1𝑋,𝑓_𝑘𝐻⋅𝑀𝑓_𝑘²_𝐻−^𝑛_𝑘=1𝑋,𝑓_𝑘𝐻²=⋅⟨𝑀,𝑀⟩_𝑛𝑘=1𝑋,𝑓_𝑘𝐻⋅𝑀²−𝑋,𝑓_𝑘𝐻²⋅⟨𝑀,𝑀⟩. (48) For every 𝑘∈ℕ, the quadratic variation of the real-valued process ⟨𝑋,𝑓_𝑘⟩_𝐻⋅𝑀 is given by 𝑋,𝑓_𝑘𝐻⋅𝑀,𝑋,𝑓_𝑘𝐻⋅𝑀=𝑋,𝑓_𝑘𝐻²⋅⟨𝑀,𝑀⟩ (49) see, for example, [12, TheoremI.4.40.d], which shows that 𝑀^𝑛 is a martingale. Since 𝑀^𝑛∈𝐴¹_𝑇(ℝ), we deduce that 𝑀^𝑛∈𝑀¹_𝑇(ℝ).
Next, we prove that 𝑀^𝑛→𝑀 in 𝐴¹_𝑇(ℝ). Indeed, since ‖𝕁‖²_𝐻−𝕁^𝑛2_𝐻≤𝕁−𝕁^𝑛2_𝐻+2‖𝕁‖_𝐻𝕁−𝕁^𝑛_𝐻, (50) by the Cauchy-Schwarz inequality and (45) we obtain ‖𝕁‖²_𝐻−𝕁^𝑛2_{𝐻𝐿1𝑇(ℝ)}=𝔼sup_{[]𝑡∈0,𝑇}𝕁_𝑡²_𝐻−𝕁^𝑛_𝑡²_𝐻≤𝔼sup_{[]𝑡∈0,𝑇}𝕁_𝑡−𝕁^𝑛_𝑡²_𝐻+2𝔼sup_{𝑡∈[0,𝑇]}𝕁_𝑡𝐻𝕁_𝑡−𝕁^𝑛_𝑡𝐻≤𝔼sup_{[]𝑡∈0,𝑇}𝕁_𝑡−𝕁^𝑛_𝑡²_𝐻+2𝔼sup_{[]𝑡∈0,𝑇}𝕁_𝑡²_𝐻^1/2𝔼sup_{[]𝑡∈0,𝑇}𝕁_𝑡−𝕁^𝑛_𝑡²_𝐻^1/2=𝕁−𝕁^𝑛2_{𝐿2𝑇(𝐻)}+2‖𝕁‖_𝐿²_𝑇(𝐻)×𝕁−𝕁^𝑛_{𝐿2𝑇(𝐻)}⟶0. (51) Therefore, together with (46), we get 𝑀−𝑀^𝑛_{𝐿1𝑇(ℝ)}≤‖𝕁‖²−𝕁^𝑛2_{𝐿1𝑇(ℝ)}+𝕀−𝕀^𝑛_{𝐿1𝑇(ℝ)}⟶0, (52) showing that 𝑀^𝑛→𝑀 in 𝐴¹_𝑇(ℝ). Now, Lemma 3 yields that 𝑀∈𝑀¹_𝑇(ℝ), which concludes the proof.

Theorem 13. Let 𝑁∈ℳ²_𝑇(ℝ) be another square-integrable martingale, and let 𝑋,𝑌 be two 𝐻-valued, predictable processes satisfying (17) and 𝔼^𝑇₀𝑌_𝑠²_𝐻𝑑⟨𝑁,𝑁⟩_𝑠<∞. (53) Then, one has ⟨𝑋⋅𝑀,𝑌⋅𝑁⟩=⟨𝑋,𝑌⟩_𝐻⋅⟨𝑀,𝑁⟩. (54)

Proof . Using Proposition 12 and the identities 𝑥,𝑦_𝐻=14𝑥+𝑦²_𝐻−𝑥−𝑦²_𝐻1,𝑥,𝑦∈𝐻,⟨𝑀,𝑁⟩=4(⟨𝑀+𝑁,𝑀+𝑁⟩−⟨𝑀−𝑁,𝑀−𝑁⟩), (55) identity (54) follows from a straightforward calculation.

Proposition 14. Let 𝑁∈ℳ²_𝑇(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 ⟨𝑋⋅𝑀,𝑌⋅𝑁⟩_𝑀²_𝑇(𝐻)=0. (56)

Proof. Using Remark 11, Theorem 13, and the hypothesis ⟨𝑀,𝑁⟩=0, we obtain ⟨𝑋⋅𝑀,𝑌⋅𝑁⟩_𝑀²_𝑇(𝐻)=𝔼^𝑇₀𝑋_𝑠𝑑𝑀_𝑠,^𝑇₀𝑌_𝑠𝑑𝑁_𝑠𝐻=𝔼^𝑇₀𝑋_𝑠𝑑𝑀_𝑠,^𝑇₀𝑌_𝑠𝑑𝑁_𝑠=𝔼^𝑇₀𝑋_𝑠,𝑌_𝑠𝐻𝑑⟨𝑀,𝑁⟩_𝑠=0, (57) completing the proof.

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

In this section, we introduce the Itô integral for ℓ²(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 ℓ²(𝐻)∶=ℎ^𝑗_𝑗∈ℕ⊂𝐻∶^∞_𝑗=1ℎ^𝑗2_𝐻<∞, (58) which, equipped with the inner product ℎ,𝑔_ℓ2(𝐻)=^∞_𝑗=1ℎ^𝑗,𝑔^𝑗_𝐻, (59) is a separable Hilbert space.

Definition 15. A sequence (𝑀^𝑗)_𝑗∈ℕ of real-valued Lévy processes is called a sequence of standard Lévy processes if it consists of square-integrable martingales with ⟨𝑀^𝑗,𝑀^𝑘⟩_𝑡=𝛿_𝑗𝑘⋅𝑡 for all 𝑗,𝑘∈ℕ. Here, 𝛿_𝑗𝑘 denotes the Kronecker delta 𝛿_𝑗𝑘=1,if𝑗=𝑘,0,if𝑗≠𝑘. (60)
For the rest of this section, let (𝑀^𝑗)_𝑗∈ℕ be a sequence of standard Lévy processes.

Proposition 16. For every ℓ²(𝐻)-valued, predictable process 𝑋 with 𝔼^𝑇₀𝑋_𝑠²_ℓ2(𝐻)𝑑𝑠<∞, (61) the series _𝑗∈ℕ𝑋^𝑗⋅𝑀^𝑗 (62) converges unconditionally in 𝑀²_𝑇(𝐻).

Proof. For 𝑗,𝑘∈ℕ with 𝑗≠𝑘, we have ⟨𝑀^𝑗,𝑀^𝑘⟩=0, and, hence, by Proposition 14, we obtain 𝑋^𝑗⋅𝑀^𝑗,𝑋^𝑘⋅𝑀^𝑘_{𝑀2𝑇(𝐻)}=0. (63) Moreover, by the Itô isometry (Proposition 8) and the monotone convergence theorem, we have ^∞_𝑗=1𝑋^𝑗⋅𝑀^𝑗2_{𝑀2𝑇(𝐻)}=^∞_𝑗=1𝔼^𝑇₀𝑋^𝑗_𝑠𝑑𝑀^𝑗_𝑠²_𝐻=^∞_𝑗=1𝔼^𝑇₀𝑋^𝑗_𝑠²_𝐻𝑑𝑠=𝔼^𝑇₀^∞_𝑗=1𝑋^𝑗_𝑠²_𝐻𝑑𝑠=𝔼^𝑇₀𝑋_𝑠²_ℓ2(𝐻)𝑑𝑠. (64) Thus, by (61) and Lemma 4, the series (62) converges unconditionally in 𝑀²_𝑇(𝐻).

Therefore, for a ℓ²(𝐻)-valued, predictable process 𝑋 satisfying (61) we can define the Itô integral as the series (62).

Remark 17. As the proof of Proposition 16 shows, the components of the Itô integral ∑_𝑗∈ℕ𝑋^𝑗⋅𝑀^𝑗 are pairwise orthogonal elements of the Hilbert space 𝑀²_𝑇(𝐻).

Proposition 18. For each ℓ²(𝐻)-valued, predictable process 𝑋 satisfying (61), one has the Itô isometry 𝔼_𝑗∈ℕ^𝑇₀𝑋^𝑗_𝑠𝑑𝑀^𝑗_𝑠²_𝐻=𝔼^𝑇₀𝑋_𝑠²_ℓ2(𝐻)𝑑𝑠. (65)

Proof . Using (63), Lemma 4, and identity (64), we obtain 𝔼^∞_𝑗=1^𝑇₀𝑋^𝑗_𝑠𝑑𝑀^𝑗_𝑠²_𝐻=^∞_𝑗=1𝑋^𝑗⋅𝑀^𝑗2_{𝑀2𝑇(𝐻)}=^∞_𝑗=1𝑋^𝑗⋅𝑀^𝑗2_{𝑀2𝑇(𝐻)}=𝔼^𝑇₀𝑋_𝑠²_ℓ2(𝐻)𝑑𝑠, (66) completing the proof.

Proposition 19. Let 𝑋 be a ℓ²(𝐻)-valued simple process of the form 𝑋=𝑋₀𝟙_{0}+^𝑛_𝑖=1𝑋_𝑖𝟙_{(𝑡𝑖,𝑡𝑖+1]} (67) with 0=𝑡₁<⋯<𝑡_𝑛+1=𝑇 and ℱ_𝑡𝑖-measurable random variables 𝑋_𝑖∶Ω→ℓ²(𝐻) for 𝑖=0,…,𝑛. Then, one has 𝑋⋅𝑀=^𝑛_{𝑖=1𝑗∈ℕ}𝑋^𝑗_𝑖𝑀^{𝑗𝑡𝑖+1}−𝑀^𝑗𝑡𝑖. (68)

Proof . For each 𝑗∈ℕ, the process 𝑋^𝑗 is a 𝐻-valued simple process having the representation 𝑋^𝑗=𝑋^𝑗₀𝟙_{0}+^𝑛_𝑖=1𝑋^𝑗_𝑖𝟙_{(𝑡𝑖,𝑡𝑖+1]}. (69) Hence, by Proposition 9, we obtain 𝑋⋅𝑀=_j∈ℕ𝑋^𝑗⋅𝑀^𝑗=^𝑛_{𝑗∈ℕ𝑖=1}𝑋^𝑗_𝑖𝑀^{𝑗𝑡𝑖+1}−𝑀^𝑗𝑡𝑖=^𝑛_{𝑖=1𝑗∈ℕ}𝑋^𝑗_𝑖𝑀^{𝑗𝑡𝑖+1}−𝑀^𝑗𝑡𝑖, (70) which finishes the proof.

5. Lévy Processes in Hilbert Spaces

In this section, we provide the required results about Lévy processes in Hilbert spaces. Let 𝑈 be a separable Hilbert space.

Definition 20. A 𝑈-valued càdlàg, adapted process 𝐿 is called a Lévy process if the following conditions are satisfied. (1)We have 𝐿₀=0.(2)𝐿_𝑡−𝐿_𝑠 is independent of ℱ_𝑠 for all 𝑠≤𝑡.(3)We have 𝐿_𝑡−𝐿=_𝑠^d𝐿_𝑡−𝑠 for all 𝑠≤𝑡.

Definition 21. A 𝑈-valued Lévy process 𝐿 with 𝔼[‖𝐿_𝑡‖²_𝑈]<∞ and 𝔼[𝐿_𝑡]=0 for all 𝑡≥0 is called a square-integrable Lévy martingale.

Note that any square-integrable Lévy martingale 𝐿 is indeed a martingale; that is, 𝔼𝑋_𝑡∣ℱ_𝑠=𝑋_𝑠∀𝑠≤t (71) 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 𝔼𝐿_𝑡,𝑢_1𝑈𝐿_𝑠,𝑢_2𝑈=(𝑡∧𝑠)𝑄𝑢₁,𝑢_2𝑈. (72) Moreover, for all 𝑢₁,𝑢₂∈𝑈, the angle bracket process is given by 𝐿,𝑢_1𝑈,𝐿,𝑢_2𝑈𝑡=𝑡𝑄𝑢₁,𝑢_2𝑈,𝑡≥0; (73) see [5, Theorem4.49].

Lemma 22. Let 𝐿 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 𝑄_Φ∶=Φ𝑄Φ⁻¹.

Proof . The process Φ(𝐿) is a 𝑉-valued càdlàg, adapted process with Φ(𝐿₀)=Φ(0)=0. Let 𝑠≤𝑡 be arbitrary. Then, the random variable Φ(𝐿_𝑡)−Φ(𝐿_𝑠)=Φ(𝐿_𝑡−𝐿_𝑠) is independent of ℱ_𝑠, and we have Φ𝐿_𝑡−Φ𝐿_𝑠=Φ𝐿_𝑡−𝐿=_𝑠^dΦ𝐿_𝑡−𝑠. (74) Moreover, for each 𝑡∈ℝ₊, we have 𝔼Φ𝐿_𝑡²_𝑉=𝔼𝐿_𝑡²_𝑈𝔼Φ𝐿<∞,_𝑡=Φ𝔼𝐿_𝑡=0, (75) showing that Φ(𝐿) is a 𝑉-valued square-integrable Lévy martingale.
Let 𝑡,𝑠∈ℝ₊ and 𝑣_𝑖∈𝑉, 𝑖=1,2 be arbitrary, and set 𝑢_𝑖∶=Φ⁻¹𝑣_𝑖∈𝑈,  𝑖=1,2. Then, we have 𝔼Φ(𝐿_𝑡),𝑣_1𝑉Φ(𝐿_𝑠),𝑣_2𝑉=𝔼Φ(𝐿_𝑡),Φ(𝑢₁)_𝑉Φ(𝐿_𝑠),Φ(𝑢₂)_𝑉=𝔼𝐿_𝑡,𝑢_1𝑈𝐿_𝑠,𝑢_2𝑈=(𝑡∧𝑠)𝑄𝑢₁,u_2𝑈=(𝑡∧𝑠)𝑄Φ⁻¹𝑣₁,Φ⁻¹𝑣_2𝑈=(𝑡∧𝑠)Φ𝑄Φ⁻¹𝑣₁,𝑣_2𝑉=(𝑡∧𝑠)𝑄_Φ𝑣₁,𝑣_2𝑉, (76) showing that the Lévy martingale Φ(𝐿) has the covariance operator 𝑄_Φ.

Now, let 𝑄∈𝐿(𝑈) be a self-adjoint, positive definite trace class operator. Then, there exists a sequence (𝜆_𝑗)_𝑗∈ℕ⊂(0,∞) with ∑^∞_𝑗=1𝜆_𝑗<∞ and an orthonormal basis (𝑒_𝑗^(𝜆))_𝑗∈ℕ of 𝑈 such that 𝑄𝑒_𝑗^(𝜆)=𝜆_𝑗𝑒_𝑗^(𝜆)∀𝑗∈ℕ. (77) We define the sequence of pairwise orthogonal vectors (𝑒_𝑗)_𝑗∈ℕ as 𝑒_𝑗∶=𝜆_𝑗𝑒_𝑗^(𝜆),𝑗∈ℕ. (78)

Proposition 23. Let 𝐿 be a 𝑈-valued square-integrable Lévy martingale with covariance operator 𝑄. Then, the sequence (𝑀^𝑗)_𝑗∈ℕ given by 𝑀^𝑗1∶=𝜆_𝑗𝐿,𝑒_𝑗^(𝜆)_𝑈,𝑗∈ℕ, (79) is a sequence of standard Lévy processes.

Proof. For each 𝑗∈ℕ, the process 𝑀^𝑗 is a real-valued square-integrable Lévy martingale. By (73), for all 𝑗,𝑘∈ℕ, we obtain 𝑀^𝑗,𝑀^𝑘_𝑡=1𝜆_𝑗𝜆_𝑘𝐿,𝑒_𝑗^(𝜆)_𝑈,𝐿,𝑒_𝑘^(𝜆)_𝑈𝑡=𝑡𝑄𝑒_𝑗^(𝜆),𝑒_𝑘^(𝜆)_𝑈𝜆_𝑗𝜆_𝑘=𝑡𝜆_𝑗𝑒_𝑗^(𝜆),𝑒_𝑘^(𝜆)_𝑈𝜆_𝑗𝜆_𝑘=𝛿_𝑗𝑘⋅𝑡, (80) showing that (𝑀^𝑗)_𝑗∈ℕ is a sequence of standard Lévy processes.

6. The Itô Integral with respect to an ℓ²_𝜆-Valued Lévy Process

In this section, we introduce the Itô integral for ℓ²(𝐻)-valued processes with respect to an ℓ²_𝜆-valued Lévy process, which is based on the Itô integral (62) from Section 4.

Let (𝜆_𝑗)_𝑗∈ℕ⊂(0,∞) be a sequence with ∑^∞_𝑗=1𝜆_𝑗<∞ and denote by ℓ²_𝜆 the weighted space of sequences ℓ²_𝜆∶=𝑣^𝑗_𝑗∈ℕ⊂ℝ∶^∞_𝑗=1𝜆_𝑗𝑣^𝑗2<∞, (81) which, equipped with the inner product ⟨𝑣,𝑤⟩_ℓ²_𝜆=^∞_𝑗=1𝜆_𝑗𝑣^𝑗𝑤^𝑗, (82) is a separable Hilbert space. Note that we have the strict inclusion ℓ²⫋ℓ²_𝜆, where ℓ² denotes the space of sequences ℓ²=𝑣^𝑗_𝑗∈ℕ⊂ℝ∶^∞_𝑗=1𝑣^𝑗2<∞. (83) We denote by (𝑔_𝑗)_𝑗∈ℕ the standard orthonormal basis of ℓ², which is given by 𝑔₁=(1,0,…),𝑔₂=(0,1,0,…),…. (84) Then, the system (𝑔_𝑗^(𝜆))_𝑗∈ℕ defined as 𝑔_𝑗^(𝜆)𝑔∶=^𝑗𝜆_𝑗,𝑗∈ℕ, (85) is an orthonormal basis of ℓ²_𝜆. Let 𝑄∈𝐿(ℓ²_𝜆) be a linear operator such that 𝑄𝑔_𝑗^(𝜆)=𝜆_𝑗𝑔_𝑗^(𝜆),∀𝑗∈ℕ. (86) Then, 𝑄 is a nuclear, self-adjoint, positive definite operator. Let 𝐿 be an ℓ²_𝜆-valued, square-integrable Lévy martingale with covariance operator 𝑄. According to Proposition 23, the sequence (𝑀^j)_𝑗∈ℕ given by 𝑀^𝑗1∶=𝜆_𝑗𝐿,𝑔_𝑗^(𝜆)_ℓ2𝜆,𝑗∈ℕ, (87) is a sequence of standard Lévy processes.

Definition 24. For every ℓ²(𝐻)-valued, predictable process 𝑋 satisfying (61), we define the Itô integral 𝑋⋅𝐿∶=(∫^𝑡₀𝑋_𝑠𝑑𝐿_𝑠)_{𝑡∈[0,𝑇]} as 𝑋⋅𝐿∶=_𝑗∈ℕ𝑋^𝑗⋅𝑀^𝑗. (88)

Remark 25. Note that 𝐿⁰₂(𝐻)≅ℓ²(𝐻), where 𝐿⁰₂(𝐻) denotes the space of Hilbert-Schmidt operators from ℓ² to 𝐻. In [21], the Itô integral for 𝐿⁰₂(𝐻)-valued processes with respect to an ℓ²_𝜆-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 (𝜇_𝑘)_𝑘∈ℕ be another sequence with ∑^∞_𝑘=1𝜇_𝑘<∞, and let Φ∶ℓ²_𝜆→ℓ²_𝜇 be an isometric isomorphism such that 𝑄_Φ𝑔_𝑘^(𝜇)=𝜇_𝑘𝑔_𝑘^(𝜇),∀𝑘∈ℕ. (89) By Lemma 22, the process Φ(𝐿) is a ℓ²_𝜇-valued, square integrable Lévy martingale with covariance operator 𝑄_Φ, and by Proposition 23, the sequence (𝑁^𝑘)_𝑘∈ℕ given by 𝑁^𝑘1∶=√𝜇_𝑘Φ(𝐿),𝑔_𝑘^(𝜇)_ℓ2𝜇,𝑘∈ℕ, (90)

is a sequence of standard Lévy processes.

Theorem 26. Let Ψ∈𝐿(ℓ²(𝐻)) be an isometric isomorphism such that ⟨ℎ,Ψ(𝑤)⟩_𝐻=Φ⟨ℎ,𝑤⟩_𝐻∀ℎ∈𝐻,𝑤∈ℓ²(H). (91) Then, for every ℓ²(𝐻)-valued, predictable process 𝑋 satisfying (61), one has 𝔼^𝑇₀Ψ(𝑋_𝑠)²_ℓ2(𝐻)𝑑𝑠<∞ (92) and the identity 𝑋⋅𝐿=Ψ(𝑋)⋅Φ(𝐿). (93)

Proof. Since Ψ is an isometry, by (61), we have 𝔼^𝑇₀Ψ(𝑋_𝑠)²_ℓ2(𝐻)𝑑𝑠=𝔼^𝑇₀𝑋_𝑠²_ℓ2(𝐻)𝑑𝑠<∞, (94) showing (92). Moreover, by (89), we have Φ𝑄Φ⁻¹𝑔_𝑘^(𝜇)=𝑄_Φ𝑔_𝑘^(𝜇)=𝜇_𝑘𝑔_𝑘^(𝜇)∀𝑘∈ℕ, (95) and, hence, we get 𝑄Φ⁻¹𝑔_𝑘^(𝜇)=𝜇_𝑘Φ⁻¹𝑔_𝑘^(𝜇)∀𝑘∈ℕ. (96) By (86) and (96), the vectors (𝑔_𝑗^(𝜆))_𝑗∈ℕ and (Φ⁻¹𝑔_𝑘^(𝜇))_𝑘∈ℕ are eigenvectors of 𝑄 with corresponding eigenvalues (𝜆_𝑗)_𝑗∈ℕ and (𝜇_𝑘)_𝑘∈ℕ. Therefore, and since Φ is an isometry, for 𝑗,𝑘∈ℕ with 𝜆_𝑗≠𝜇_𝑘, we obtain Φ𝑔_𝑗^(𝜆),𝑔_𝑘^(𝜇)_ℓ2𝜇=𝑔_𝑗^(𝜆),Φ⁻¹𝑔_𝑘^(𝜇)_ℓ2𝜆=0. (97) Let ℎ∈𝐻 be arbitrary. Then, we have ⟨ℎ,𝑋⋅𝐿⟩_𝐻=ℎ,^∞_𝑗=1𝑋^𝑗⋅𝑀^𝑗_𝐻=^∞_𝑗=1ℎ,𝑋^𝑗⋅𝑀^𝑗_𝐻=^∞_𝑗=1ℎ,𝑋^𝑗_𝐻⋅𝑀^𝑗=^∞_𝑗=11𝜆_𝑗⟨ℎ,𝑋⟩_𝐻,𝑔_𝑗^(𝜆)_ℓ2𝜆⋅1𝜆_𝑗𝐿,𝑔_𝑗^(𝜆)_ℓ2𝜆=^∞_𝑗=11𝜆_𝑗Φ(⟨ℎ,𝑋⟩_𝐻),Φ𝑔_𝑗^(𝜆)_ℓ2𝜇⋅1𝜆_𝑗Φ(𝐿),Φ𝑔_𝑗^(𝜆)_ℓ2𝜇=^∞_𝑗=11𝜆_𝑗Φ(⟨ℎ,𝑋⟩_𝐻),Φ𝑔_𝑗^(𝜆)_ℓ2𝜇⋅1𝜆_𝑗^∞_𝑘=1Φ(𝐿),𝑔_𝑘^(𝜇)_ℓ2𝜇𝑔_𝑘^(𝜇),Φ𝑔_𝑗^(𝜆)_ℓ2𝜇. (98) Since (𝜆_𝑗)_𝑗∈ℕ and (𝜇_𝑘)_𝑘∈ℕ are eigenvalues of 𝑄, for each 𝑗∈ℕ, there are only finitely many 𝑘∈ℕ such that 𝜆_𝑗=𝜇_𝑘. Therefore, by (97), and since (Φ(𝑔_𝑗^(𝜆)))_𝑗∈ℕ is an orthonormal basis of ℓ²_𝜇, we obtain ⟨ℎ,𝑋⋅𝐿⟩_𝐻=^∞_𝑘=1^∞_𝑗=11𝜆_𝑗Φ(⟨ℎ,𝑋⟩_𝐻),Φ𝑔_𝑗^(𝜆)_ℓ2𝜇Φ𝑔_𝑗^(𝜆),𝑔_𝑘^(𝜇)_ℓ2𝜇⋅⟨Φ(𝐿),𝑔_𝑘^(𝜇)⟩_ℓ²_𝜇=^∞_𝑘=11𝜇_𝑘^∞_𝑗=1Φ(⟨ℎ,𝑋⟩_𝐻),Φ𝑔_𝑗^(𝜆)_ℓ2𝜇Φ𝑔_𝑗^(𝜆),𝑔_𝑘^(𝜇)_ℓ2𝜇⋅Φ(𝐿),𝑔_𝑘^(𝜇)_ℓ2𝜇=^∞_𝑘=11√𝜇_𝑘Φ(⟨ℎ,𝑋⟩_𝐻),𝑔_𝑘^(𝜇)_ℓ2𝜇⋅1√𝜇_𝑘Φ(𝐿),𝑔_𝑘^(𝜇)_ℓ2𝜇=^∞_𝑘=1Φ⟨ℎ,𝑋⟩_𝐻^𝑘⋅𝑁^𝑘. (99) Thus, taking into account (91) gives us ⟨ℎ,𝑋⋅𝐿⟩_𝐻=^∞_𝑘=1ℎ,Ψ(𝑋)^𝑘_𝐻⋅𝑁^𝑘=^∞_𝑘=1ℎ,Ψ(𝑋)^𝑘⋅𝑁^𝑘_𝐻=ℎ,^∞_𝑘=1Ψ(𝑋)^𝑘⋅𝑁^𝑘_𝐻=⟨ℎ,Ψ(𝑋)⋅Φ(𝐿)⟩_𝐻. (100) 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, Theorem 26 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.

Let 𝑈 be a separable Hilbert space, and let 𝑄∈𝐿(𝑈) be a nuclear, self-adjoint, positive definite linear operator. Then, there exist a sequence (𝜆_𝑗)_𝑗∈ℕ⊂(0,∞) with ∑^∞_𝑗=1𝜆_𝑗<∞ and an orthonormal basis (𝑒_𝑗^(𝜆))_𝑗∈ℕ of  𝑈 such that 𝑄𝑒_𝑗^(𝜆)=𝜆_𝑗𝑒_𝑗^(𝜆),∀𝑗∈ℕ; (101) namely, 𝜆_𝑗 are the eigenvalues of 𝑄, and each 𝑒_𝑗^(𝜆) is an eigenvector corresponding to 𝜆_𝑗. The space 𝑈₀∶=𝑄^1/2(𝑈), equipped with the inner product ⟨𝑢,𝑣⟩_𝑈0∶=𝑄^−1/2𝑢,𝑄^−1/2𝑣_𝑈, (102) is another separable Hilbert space, and the sequence (𝑒_𝑗)_𝑗∈ℕ given by 𝑒_𝑗=𝜆_𝑗𝑒_𝑗^(𝜆),𝑗∈ℕ, (103) is an orthonormal basis of 𝑈₀. We denote by 𝐿⁰₂(𝐻)∶=𝐿₂(𝑈₀,𝐻) the space of Hilbert-Schmidt operators from 𝑈₀ into 𝐻, which, endowed with the Hilbert-Schmidt norm ‖𝑆‖_𝐿⁰_2(𝐻)∶=^∞_𝑗=1𝑆𝑒_𝑗²_𝐻^1/2,𝑆∈𝐿⁰₂(𝐻), (104) itself is a separable Hilbert space. We define the isometric isomorphisms Φ_𝜆∶𝑈⟶ℓ²_𝜆,Φ_𝜆𝑒_𝑗^(𝜆)∶=𝑔_𝑗^(𝜆)Ψfor𝑗∈ℕ,_𝜆∶𝐿⁰₂(𝐻)⟶ℓ²(𝐻),Ψ_𝜆(𝑆)∶=𝑆𝑒_{𝑗𝑗∈ℕ}for𝑆∈𝐿⁰₂(𝐻). (105)(106) Recall that (𝑔_𝑗^(𝜆))_𝑗∈ℕ denotes the orthonormal basis of ℓ²_𝜆, 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 process Φ_𝜆(𝐿) is an ℓ²_𝜆-valued square-integrable Lévy martingale with covariance operator 𝑄_Φ𝜆.(2)One has 𝑄_Φ𝜆𝑔_𝑗^(𝜆)=𝜆_𝑗𝑔_𝑗^(𝜆),∀𝑗∈ℕ. (107)

Proof. By Lemma 22, the process Φ_𝜆(𝐿) is an ℓ²_𝜆-valued square-integrable Lévy martingale with covariance operator 𝑄_Φ𝜆. Furthermore, by (105) and (101), for all 𝑗∈ℕ, we obtain 𝑄_Φ𝜆𝑔_𝑗^(𝜆)=Φ_𝜆𝑄Φ_𝜆⁻¹𝑔_𝑗^(𝜆)=Φ_𝜆𝑄𝑒_𝑗^(𝜆)=Φ_𝜆𝜆_𝑗𝑒_𝑗^(𝜆)=𝜆_𝑗Φ_𝜆𝑒_𝑗^(𝜆)=𝜆_𝑗𝑔_𝑗^(𝜆), (108) showing (107).

Now, our idea is to the define the Itô integral for an 𝐿⁰₂(𝐻)-valued, predictable process 𝑋 with 𝔼^𝑇₀𝑋_𝑠²_𝐿02(𝐻)𝑑𝑠<∞ (109) by setting 𝑋⋅𝐿∶=Ψ_𝜆(𝑋)⋅Φ_𝜆(𝐿), (110) 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 (𝜆_𝑗)_𝑗∈ℕ and eigenvectors (𝑒_𝑗^(𝜆))_𝑗∈ℕ. In order to prove that this is not the case, let (𝜇_𝑘)_𝑘∈ℕ⊂(0,∞) be another sequence with ∑^∞_𝑘=1𝜇_𝑘<∞, and let (𝑓_𝑘^(𝜇))_𝑘∈ℕ be another orthonormal basis of  𝑈 such that 𝑄𝑓_𝑘^(𝜇)=𝜇_𝑘𝑓_𝑘^(𝜇),∀𝑘∈ℕ. (111) Then, the sequence (𝑓_𝑘)_𝑘∈ℕ given by 𝑓_𝑘=√𝜇_𝑘𝑓_𝑘^(𝜇),𝑘∈ℕ, (112) is an orthonormal basis of 𝑈₀. Analogous to (105) and (106), we define the isometric isomorphisms Φ_𝜇∶𝑈⟶ℓ²_𝜇,Φ_𝜇𝑓_𝑘^(𝜇)∶=𝑔_𝑘^(𝜇)Ψfor𝑘∈ℕ,_𝜇∶𝐿⁰₂(𝐻)⟶ℓ²(𝐻),Ψ_𝜇(𝑆)∶=𝑆𝑓_{𝑘𝑘∈ℕ}for𝑆∈𝐿⁰₂(𝐻). (113) Furthermore, we define the isometric isomorphisms Φ∶=Φ_𝜇∘Φ_𝜆⁻¹∶ℓ²_𝜆⟶ℓ²_𝜇,Ψ∶=Ψ_𝜇∘Ψ_𝜆⁻¹∶ℓ²(𝐻)⟶ℓ²(𝐻). (114) The following diagram illustrates the situation: (115)

In order to show that the Itô integral (110) is well defined, we have to show that Ψ_𝜆(𝑋)⋅Φ_𝜆(𝐿)=Ψ_𝜇(𝑋)⋅Φ_𝜇(𝐿). (116) For this, we prepare the following auxiliary result.

Lemma 29. For all ℎ∈𝐻 and 𝑤∈ℓ²(𝐻), one has ⟨ℎ,Ψ(𝑤)⟩_𝐻=Φ⟨ℎ,𝑤⟩_𝐻. (117)

Proof. By (101) and (111), the vectors (𝑒_𝑗^(𝜆))_𝑗∈ℕ and (𝑓_𝑘^(𝜇))_𝑘∈ℕ are eigenvectors of 𝑄 with corresponding eigenvalues (𝜆_𝑗)_𝑗∈ℕ and (𝜇_𝑘)_𝑘∈ℕ. Therefore, for 𝑗,𝑘∈ℕ with 𝜆_𝑗≠𝜇_𝑘, we have ⟨𝑒_𝑗^(𝜆),𝑓_𝑘^(𝜇)⟩_𝑈=0. For each 𝑣∈ℓ²_𝜆, we obtain Φ(𝑣)=Φ_𝜇∘Φ_𝜆⁻¹(𝑣)=Φ_𝜇^∞_𝑘=1Φ_𝜆⁻¹𝑣,𝑓_𝑘^(𝜇)_𝑈𝑓_𝑘^(𝜇)=^∞_𝑘=1Φ_𝜆⁻¹𝑣,𝑓_𝑘^(𝜇)_𝑈𝑔_𝑘^(𝜇)=1√𝜇_𝑘Φ_𝜆⁻¹𝑣,𝑓_𝑘^(𝜇)_{𝑈𝑘∈ℕ}=1√𝜇_𝑘^∞_𝑗=1Φ_𝜆⁻¹𝑣,𝑒_𝑗^(𝜆)_𝑈𝑒_𝑗^(𝜆),𝑓_𝑘^(𝜇)_{𝑈𝑘∈ℕ}=1√𝜇_𝑘^∞_𝑗=1𝑣,𝑔_𝑗^(𝜆)_ℓ2𝜆𝑒_𝑗^(𝜆),𝑓_𝑘^(𝜇)_{𝑈𝑘∈ℕ}=^∞_𝑗=1𝜆_𝑗√𝜇_𝑘𝑒_𝑗^(𝜆),𝑓_𝑘^(𝜇)_𝑈𝑣^𝑗_𝑘∈ℕ=^∞_𝑗=1𝑒_𝑗^(𝜆),𝑓_𝑘^(𝜇)_𝑈𝑣^𝑗_𝑘∈ℕ. (118) Let 𝑤∈ℓ²(𝐻) be arbitrary. By (106), we have 𝑤=Ψ_𝜆Ψ_𝜆⁻¹(𝑤)=Ψ_𝜆⁻¹(𝑤)𝑒_{𝑗𝑗∈ℕ}, (119) and, hence, Ψ(𝑤)=Ψ_𝜇∘Ψ_𝜆⁻¹(𝑤)=Ψ_𝜆⁻¹(𝑤)𝑓_{𝑘𝑘∈ℕ}=Ψ_𝜆⁻¹(𝑤)^∞_𝑗=1𝑓_𝑘,𝑒_𝑗𝑈0𝑒_{𝑗𝑘∈ℕ}=^∞_𝑗=1𝑓_𝑘,𝑒_𝑗𝑈0Ψ_𝜆⁻¹(𝑤)𝑒_{𝑗𝑘∈ℕ}=^∞_𝑗=1𝑓_𝑘,𝑒_𝑗𝑈0𝑤^𝑗_𝑘∈ℕ=^∞_𝑗=1𝑒_𝑗^(𝜆),𝑓_𝑘^(𝜇)_𝑈𝑤^𝑗_𝑘∈ℕ. (120) Therefore, for all ℎ∈𝐻 and 𝑤∈ℓ²(𝐻), we obtain ⟨ℎ,Ψ(𝑤)⟩_𝐻=^∞_𝑗=1𝑒_𝑗^(𝜆),𝑓_𝑘^(𝜇)_𝑈ℎ,𝑤^𝑗_{𝐻𝑘∈ℕ}=Φ⟨ℎ,𝑤⟩_𝐻, (121) finishing the proof.

Proposition 30. The following statements are true. (1)Φ_𝜆(𝐿) is an ℓ²_𝜆-valued Lévy process with covariance operator 𝑄_Φ𝜆, and one has 𝑄_Φ𝜆𝑔_𝑗^(𝜆)=𝜆_𝑗𝑔_𝑗^(𝜆),∀𝑗∈ℕ. (122)(2)Φ_𝜇(𝐿) is an ℓ²_𝜇-valued Lévy process with covariance operator 𝑄_Φ𝜇, and one has 𝑄_Φ𝜇𝑔_𝑘^(𝜇)=𝜇_𝑘𝑔_𝑘^(𝜇)∀k∈ℕ. (123)(3)For every 𝐿⁰₂(𝐻)-valued, predictable process 𝑋 with (109), one has 𝔼^𝑇₀Ψ_𝜆(𝑋_𝑠)²_ℓ2(𝐻)𝔼𝑑𝑠<∞,_𝑇0Ψ_𝜇(𝑋_𝑠)²_ℓ2(𝐻)𝑑𝑠<∞ (124) and the identity (116).

Proof. The first two statements follow from Lemma 28. Since Ψ_𝜆 and Ψ_𝜇 are isometries, we obtain 𝔼^𝑇₀Ψ_𝜆𝑋_𝑠²_ℓ2(𝐻)𝑑𝑠=𝔼^𝑇₀𝑋_𝑠²_𝐿02(𝐻)𝑑𝑠=𝔼^𝑇₀Ψ_𝜇(𝑋_𝑠)²_ℓ2(𝐻)𝑑𝑠, (125) which, together with (109), yields (124). Now, Theorem 26 applies by virtue of Lemma 29 and yields Ψ_𝜆(𝑋)⋅Φ_𝜆(𝐿)=ΨΨ_𝜆(𝑋)⋅ΦΦ_𝜆(𝐿)=Ψ_𝜇(𝑋)⋅Φ_𝜇(𝐿), (126) proving (116).

Definition 31. For every 𝐿⁰₂(𝐻)-valued process 𝑋 satisfying (109), we define the Itô-Integral 𝑋⋅𝐿=(∫^𝑡₀𝑋_𝑠𝑑𝐿_𝑠)_{𝑡∈[0,𝑇]} by (110).

By virtue of Proposition 30, Definition (110) of the Itô integral neither depends on the choice of the eigenvalues (𝜆_𝑗)_𝑗∈ℕ nor on the eigenvectors (𝑒_𝑗^(𝜆))_𝑗∈ℕ.

Now, we will the prove the announced series representation of the Itô integral. According to Proposition 23, the sequences (𝑀^𝑗)_𝑗∈ℕ and (𝑁^𝑗)_𝑗∈ℕ of real-valued processes given by 𝑀^𝑗1∶=𝜆_𝑗𝐿,𝑒_𝑗^(𝜆)_𝑈,𝑁^𝑗1∶=𝜆_𝑗Φ_𝜆(𝐿),𝑔_𝑗^(𝜆)_ℓ2𝜆 (127) are sequences of standard Lévy processes.

Proposition 32. For every 𝐿⁰₂(𝐻)-valued, predictable process 𝑋 satisfying (109), the process (𝜉^𝑗)_𝑗∈ℕ given by 𝜉^𝑗∶=𝑋𝑒_𝑗,𝑗∈ℕ, (128) is a ℓ²(𝐻)-valued, predictable process, and, one has 𝑋⋅𝐿=_𝑗∈ℕ𝜉^𝑗⋅𝑀^𝑗, (129) where the right-hand side of (129) converges unconditionally in 𝑀²_𝑇(𝐻).

Proof. Since Φ_𝜆 is an isometry, for each 𝑗∈ℕ, we obtain 𝑀^𝑗=1𝜆_𝑗𝐿,𝑒_𝑗^(𝜆)_𝑈=1𝜆_𝑗Φ_𝜆(𝐿),Φ_𝜆𝑒_𝑗^(𝜆)_ℓ2𝜆=1𝜆_𝑗Φ_𝜆(𝐿),𝑔_𝑗^(𝜆)_ℓ2𝜆=𝑁^𝑗. (130) Thus, by Definitions 31 and 24, we obtain 𝑋⋅𝐿=Ψ_𝜆(𝑋)⋅Φ_𝜆(𝐿)=_𝑗∈ℕΨ_𝜆(𝑋)^𝑗⋅𝑁^𝑗=_𝑗∈ℕ𝜉^𝑗⋅𝑀^𝑗, (131) and, by Proposition 16, the series converges unconditionally in 𝑀²_𝑇(𝐻).

Remark 33. By Remark 17 and the proof of Proposition 32, the components of the Itô integral ∑_𝑗∈ℕ𝜉^𝑗⋅𝑀^𝑗 are pairwise orthogonal elements of the Hilbert space 𝑀²_𝑇(𝐻).

Proposition 34. For every 𝐿⁰₂(𝐻)-valued process 𝑋 satisfying (109), one has the Itô isometry 𝔼^𝑇₀𝑋_𝑠𝑑𝐿_𝑠²_𝐻=𝔼^𝑇₀𝐿_𝑠²_𝐿02(𝐻)𝑑𝑠. (132)

Proof. By the Itô isometry (Proposition 18), and since Ψ_𝜆 is an isometry, we obtain 𝔼^𝑇₀𝑋_𝑠𝑑𝐿_𝑠²_𝐻=𝔼^𝑇₀Ψ_𝜆(𝑋_𝑠)𝑑Φ_𝜆(𝐿)_𝑠²_𝐻=𝔼^𝑇₀Ψ_𝜆(𝑋_𝑠)²_ℓ2(𝐻)𝑑𝑠=𝔼^𝑇₀𝑋_𝑠²_𝐿02(𝐻)𝑑𝑠, (133) 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 𝐿⁰₂(𝐻), because ^∞_𝑗=1𝑆𝑒_𝑗²_𝐻≤^∞_𝑗=1‖𝑆‖²_{𝐿(𝑈,𝐻)}𝑒_𝑗²_𝑈=‖𝑆‖²_𝐿()^∞_{𝑈,𝐻𝑗=1}𝜆_𝑗𝑒_𝑗^(𝜆)2_𝑈0=‖𝑆‖^2∞_{𝐿(𝑈,𝐻)𝑗=1}𝜆_𝑗<∞. (134)

Proposition 35. Let 𝑋 be a 𝐿(𝑈,𝐻)-valued simple process of the form 𝑋=𝑋₀𝟙_{0}+^𝑛_𝑖=1𝑋_𝑖𝟙_{(𝑡𝑖,𝑡𝑖+1]} (135) with 0=𝑡₁<⋯<𝑡_𝑛+1=𝑇 and ℱ_𝑡𝑖-measurable random variables 𝑋_𝑖∶Ω→𝐿(𝑈,𝐻) for 𝑖=0,…,𝑛. Then, one has 𝑋|_𝑈0⋅𝐿=^𝑛_𝑖=1𝑋_𝑖𝐿^𝑡𝑖+1−𝐿^𝑡𝑖. (136)

Proof. The process Ψ_𝜆(𝑋|_𝑈0) is an ℓ²(𝐻)-valued simple process having the representation Ψ_𝜆𝑋|_𝑈0=Ψ_𝜆𝑋₀|_𝑈0𝟙_{0}+^𝑛_𝑖=1Ψ_𝜆𝑋_𝑖|_𝑈0𝟙_{(𝑡𝑖,𝑡𝑖+1]}. (137) Thus, by Proposition 19, and since Φ_𝜆 is an isometry, we obtain 𝑋|_𝑈0⋅𝐿=Ψ_𝜆𝑋|_𝑈0⋅Φ_𝜆(𝐿)=^𝑛_{𝑖=1𝑗∈ℕ}Ψ_𝜆𝑋_𝑖|_𝑈0^𝑗𝑁^{𝑗𝑡𝑖+1}−𝑁^𝑗𝑡𝑖=^𝑛_{𝑖=1𝑗∈ℕ}𝑋_𝑖𝑒_𝑗Φ_𝜆(𝐿^𝑡𝑖+1−𝐿^𝑡𝑖),𝑔_𝑗^(𝜆)_ℓ2𝜆𝜆_𝑗=^𝑛_{𝑖=1𝑗∈ℕ}𝑋_𝑖𝑒_𝑗^(𝜆)𝐿^𝑡𝑖+1−𝐿^𝑡𝑖,Φ_𝜆⁻¹𝑔_𝑗^(𝜆)_𝑈=^𝑛_{𝑖=1𝑗∈ℕ}𝑋_𝑖𝑒_𝑗^(𝜆)𝐿^𝑡𝑖+1−𝐿^𝑡𝑖,𝑒_𝑗^(𝜆)_𝑈=^𝑛_𝑖=1𝑋_{𝑖𝑗∈ℕ}𝐿^𝑡𝑖+1−𝐿^𝑡𝑖,𝑒_𝑗^(𝜆)_𝑈𝑒_𝑗^(𝜆)=^𝑛_𝑖=1𝑋_𝑖𝐿^𝑡𝑖+1−𝐿^𝑡𝑖, (138) 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 𝐿⁰₂(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 processes 𝑋 satisfying ℙ^𝑇₀𝑋_𝑠²_𝐿02(𝐻)ds<∞=1,∀𝑇>0. (139) 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).

Acknowledgment

The author is grateful to an anonymous referee for valuable comments and suggestions.