The It\^{o} integral with respect to an infinite dimensional L\'{e}vy process: A series approach

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


Introduction
The Itô integral with respect to an infinite dimensional Wiener process has been developed in [4,17,10], and for the more general case of an infinite dimensional square-integrable martingale it has been defined in [13,16]. 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, e.g., in [11,7,3]. Moreover, in the article [14], 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 [15] this theory has been extended to general integrands with values in UMD Banach spaces.
Best to 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, e.g., [1,12,18], 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 [8], 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 [8] is that stochastic integrals are considered as L 2 -curves. The connection to the usual Itô integral for a finite dimensional Lévy process has been established in [23], see also Appendix B in [6]. Furthermore, we point out the articles [21] and [9], 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 [2] for cylindrical Lévy processes.
The construction of the Itô integral, which we present in this paper, is divided into the following steps: • For a H-valued process X (with H denoting a separable Hilbert space) and a real-valued square-integrable martingale M we define the Itô integral 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. • Based on the just defined integral, for a 2 (H)-valued process X and a sequence (M j ) j∈N of standard Lévy processes we define the Itô integral as For this, we will ensure convergence of the series. • In the next step, let L denote a 2 λ -valued Lévy process, where 2 λ is a weighted space of sequences (cf. [5]). From the Lévy process L we can construct a sequence (M j ) j∈N of standard Lévy processes, and for a 2 (H)valued process X we define the Itô integral • 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 L 0 2 (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 L 0 2 (H)-valued process X as where Φ : U → 2 λ and Ψ : L 0 2 (H) → 2 (H) are isometric isomorphisms such that Φ(L) is a 2 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 above. In Section 3 we define the Itô integral for Hvalued processes with respect to a real-valued square-integrable martingale, and in Section 4 we define the Itô integral for 2 (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 2 (H)-valued processes with respect to a 2 λ -valued Lévy process, and in Section 7 we define the Itô integral in the general case, where the integrand is a L 0 2 (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 [16].

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.
2.1. Definition. Let p ≥ 1 be arbitrary.   Note that we have the inclusions . The following auxiliary result shows that these inclusions are closed.
2.3. Lemma. Let p ≥ 1 be arbitrary. Then, the following statements are true: . Furthermore, let τ ≤ T be a bounded stopping time. Then we have showing that M τ ∈ L p (Ω, F τ , P; E). Furthermore, we have converges unconditionally in M 2 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 and, by the Cauchy-Schwarz inequality, Therefore, by the Itô isometry for the real-valued Itô integral and Lebesgue's dominated convergence theorem together with (3.1) we obtain Now, Proposition 3.1 gives rise to the following definition: where (f k ) k∈N denotes an orthonormal basis of H.
Proof. Let (f k ) k∈N be an orthonormal basis of H. According to (3.3) we have finishing the proof.
3.5. Proposition. Let X be a H-valued simple process of the form 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 Thus, by the definition of the real-valued Itô integral for simple processes we obtain finishing the proof.
3.6. Lemma. Let X be a H-valued, predictable process satisfying (3.1). Then, for every orthonormal basis where the convergence takes place in A 1 T (R). Proof. We define the integral process and the sequence (I n ) n∈N of partial sums by Furthermore, by Lebesgue's dominated convergence theorem we have which concludes the proof.
3.7. Remark. As a consequence of the Doob-Meyer decomposition theorem, for two square-integrable martingales X, Y ∈ M 2 T (H) there exists a (up to indistinguishability) unique real-valued, predictable process X, Y with finite variation paths and 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 By Proposition 3.1 we have Defining the integral process I by (3.6) and the sequence (I n ) n∈N of partial sums by (3.7), using Lemma 3.6 we have Furthermore, we define the process M ∈ A 1 T (R) and the sequence ( For every k ∈ N the quadratic variation of the real-valued process X, f k H • M is given by see, e.g. [12,Thm. I.4.40.d], which shows that M n is a martingale. Since M n ∈ A 1 T (R), we deduce that M n ∈ M 1 T (R).
Next, we prove that M n → M in A 1 T (R). Indeed, since by the Cauchy-Schwarz inequality and (3.8) we obtain Therefore, together with (3.9) we get , which concludes the proof.
3.9. Theorem. Let N ∈ M 2 T (R) be another square-integrable martingale, and let X, Y be two H-valued, predictable processes satisfying (3.1) and completing the proof.

The Itô integral with respect to a sequence of standard Lévy processes
In this section, we introduce the Itô integral for 2 (H)-valued processes with respect to a sequence of standard Lévy processes, which is based on the Itô integral (3.5) from the previous section. We define the space of sequences which, equipped with the inner product is a separable Hilbert space.

Definition.
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 For the rest of this section, let (M j ) j∈N be a sequence of standard Lévy processes.

Proposition. For every 2 (H)-valued, predictable process X with
converges unconditionally in M 2 T (H). Proof. For j, k ∈ N with j = k we have M j , M k = 0, and hence, by Proposition 3.10 we obtain Moreover, by the Itô isometry (Proposition 3.4) and the monotone convergence theorem we have Thus, by (4.1) and Lemma 2.4, the series (4.2) converges unconditionally in M 2 T (H).

4.5.
Proposition. Let X be a 2 (H)-valued simple process of the form with 0 = t 1 < . . . < t n+1 = T and F ti -measurable random variables X i : Ω → 2 (H) for i = 0, . . . , n. Then we have Proof. For each j ∈ N the process X j is a H-valued simple process having the representation Hence, by Proposition 3.5 we obtain which finishes the proof.

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.

5.1.
Definition. An U -valued càdlàg, adapted process L is called a Lévy process if the following conditions are satisfied: Note that any square-integrable Lévy martingale L is indeed a martingale, that is see [16,Prop. 3.25]. According to [16,Thm. 4.44], 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 u 1 , Moreover, for all u 1 , u 2 ∈ U the angle bracket process is given by see [16,Thm. 4.49].

Lemma.
Let L be an 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(L 0 ) = Φ(0) = 0. Let s ≤ t be arbitrary. Then the random variable Φ Let t, s ∈ R + and v i ∈ V , i = 1, 2 be arbitrary, and set u i := Φ −1 v i ∈ U , i = 1, 2. Then we have 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 exist a sequence (λ j ) j∈N ⊂ (0, ∞) with ∞ j=1 λ j < ∞ and an orthonormal basis (e (λ) j ) j∈N of U and such that We define the sequence of pairwise orthogonal vectors (e j ) j∈N as e j := λ j e (λ) j , j ∈ N. 5.4. Proposition. Let L be an U -valued square-integrable Lévy martingale with covariance operator Q. Then the sequence (M j ) j∈N given by 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 (5.1), for all j, k ∈ N we obtain showing that (M j ) j∈N is a sequence of standard Lévy processes.
6. The Itô integral with respect to a 2 λ -valued Lévy process In this section, we introduce the Itô integral for 2 (H)-valued processes with respect to a 2 λ -valued Lévy process, which is based on the Itô integral (4.2) from Section 4.

Then the system (g
is an orthonormal basis of 2 λ . Let Q ∈ L( 2 λ ) be a linear operator such that Qg Then Q is a nuclear, self-adjoint, positive definite operator. Let L be a 2 λ -valued, square-integrable Lévy martingale with covariance operator Q. According to Proposition 5.4, the sequence (M j ) j∈N given by is a sequence of standard Lévy processes. 6.2. Remark. Note that L 0 2 (H) ∼ = 2 (H), where L 0 2 (H) denotes the space of Hilbert-Schmidt operators from 2 to H. In [5], the Itô integral for L 0 2 (H)-valued processes with respect to a 2 λ -valued Wiener process has been constructed in the usual fashion (first for elementary and afterwards for general processes), and then the series representation (6.3) has been proven, see [5,Prop. 2.2.1]. Now, let (µ k ) k∈N be another sequence with ∞ k=1 µ k < ∞, and let Φ : 2 λ → 2 µ be an isometric isomorphism such that for all k ∈ N. (6.4) By Lemma 5.3, the process Φ(L) is a 2 µ -valued, square integrable Lévy martingale with covariance operator Q Φ , and by Proposition 5.4, the sequence (N k ) k∈N given by is a sequence of standard Lévy processes. Proof. Since Ψ is an isometry, by (4.1) we have showing (6.6). Moreover, by (6.4) we have and hence, we get k ) for all k ∈ N. k ) 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 Let h ∈ H be arbitrary. Then we have 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 (6.9), and since (Φ(g Thus, taking into account (6.5) gives us Since h ∈ H was arbitrary, using the separability of H as in the proof of Proposition 3.1, we arrive at (6.7).
6.4. Remark. From a geometric point of view, Theorem 6.3 says that the "angle" measured by the Itô integral is preserved under isometries.

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 (6.3) 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=1 λ j < ∞ and an orthonormal basis (e  is an eigenvector corresponding to λ j . The space U 0 := Q 1/2 (U ), equipped with the inner product u, v U0 := Q −1/2 u, Q −1/2 v U , is another separable Hilbert space and the sequence (e j ) j∈N given by e j = λ j e (λ) j , j ∈ N is an orthonormal basis of U 0 . We denote by L 0 2 (H) := L 2 (U 0 , H) the space of Hilbert-Schmidt operators from U 0 into H, which, endowed with the Hilbert-Schmidt norm itself is a separable Hilbert space. We define the isometric isomorphisms j ) j∈N denotes the orthonormal basis of 2 λ , which we have defined in (6.1). Let L be an U -valued square-integrable Lévy martingale with covariance operator Q.
(2) We have Proof. By Lemma 5.3, the process Φ λ (L) is a 2 λ -valued square-integrable Lévy martingale with covariance operator Q Φ λ . Furthermore, by (7.2) and (7.1), for all j ∈ N we obtain j , showing (7.4). Now, our idea is to the define the Itô integral for a L 0 2 (H)-valued, predictable process X with E T 0 X s 2 L 0 2 (H) ds < ∞ (7.5) by setting X • L := Ψ λ (X) • Φ λ (L), (7.6) where the right-hand side of (7.6) denotes the Itô integral (6.3) from Definition 6.1. One might suspect that this definition depends on the choice of the eigenvalues (λ j ) j∈N and eigenvectors (e (λ) j ) j∈N . In order to prove that this is not the case, let (µ k ) k∈N ⊂ (0, ∞) be another sequence with ∞ k=1 µ k < ∞ and let (f Then the sequence (f k ) k∈N given by k , k ∈ N is an orthonormal basis of U 0 . Analogous to (7.2) and (7.3), we define the isometric isomorphisms (1) Φ λ (L) is a 2 λ -valued Lévy process with covariance operator Q Φ λ , and we have (2) Φ µ (L) is a 2 µ -valued Lévy process with covariance operator Q Φµ , and we have Thus, by Proposition 4.5, and since Φ λ is an isometry, we obtain completing the proof.
Therefore, and since the space of simple processes is dense in the space of all predictable processes satisfying (7.5), see, e.g. [16,Cor. 8.17], the Itô integral (7.6) coincides with that in [16] for every L 0 2 (H)-valued, predictable process X satisfying (7.5). In particular, for a driving Wiener process, it coincides with the Itô integral from [4,17,10].
By a standard localization argument, we can extend the definition of the Itô integral to all predictable processes X satisfying P T 0 X s 2 L 0 2 (H) ds < ∞ = 1 for all T > 0. (7.11) Since the respective spaces of predictable and adapted, measurable processes are isomorphic (see [22]), proceeding as in the [22,Sec. 3.2], we can further extend the definition of the Itô integral to all adapted, measurable processes X satisfying (7.11).