International Journal of Stochastic AnalysisVolume 2013 (2013), Article ID 703769, 14 pageshttp://dx.doi.org/10.1155/2013/703769

Research Article

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

Stefan Tappe

Institut für Mathematische Stochastik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany

Received 6 November 2012; Accepted 20 February 2013

Academic Editor: Josefa Linares-Perez

Copyright © 2013 Stefan Tappe. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 -curves. The connection to the usual Itô integral for a finite dimensional Lévy process has been established in [15]; see also Appendix 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 -valued process (with denoting a separable Hilbert space) and a real-valued square-integrable martingale , we define the Itô integral where denotes an orthonormal basis of , and 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 -valued process and a sequence of standard Lévy processes, we define the Itô integral as For this, we will ensure convergence of the series.(iii)In the next step, let denote an -valued Lévy process, where is a weighted space of sequences (cf. [21]). From the Lévy process , we can construct a sequence of standard Lévy processes, and for a -valued process , we define the Itô integral (iv)Finally, let be a general Lévy process on some separable Hilbert space with covariance operator . Then, there exist sequences of eigenvalues and eigenvectors, which diagonalize the operator . Denoting by an appropriate space of Hilbert Schmidt operators from to , our idea is to utilize the integral from the previous step and to define the Itô integral for a -valued process as where and are isometric isomorphisms such that is an -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 -valued processes with respect to a real-valued square-integrable martingale, and in Section 4, we define the Itô integral for -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 -valued processes with respect to an -valued Lévy process, and in Section 7, we define the Itô integral in the general case, where the integrand is an -valued process and the integrator a general Lévy process on some separable Hilbert space . 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 be a filtered probability space satisfying the usual conditions. For the upcoming results, let be a separable Banach space, and let be a finite time horizon.

Definition 1. Let be arbitrary. (1)We define the Lebesgue space where denotes the Skorokhod space consisting of all càdlàg functions from to , equipped with the supremum norm.(2)We denote by the space of all -valued adapted processes .(3)We denote by the space of all -valued martingales .(4)We define the factor spaces where denotes the subspace consisting of all with up to indistinguishability.

Remark 2. Let us emphasize the following. (1)Since the Skorokhod space equipped with the supremum norm is a Banach space, the Lebesgue space equipped with the standard norm is a Banach space too.(2)By the completeness of the filtration , adaptedness of an element does not depend on the choice of the representative. This ensures that the factor space of adapted processes is well defined.(3)The definition of -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

The following auxiliary result shows that these inclusions are closed.

Lemma 3. Let be arbitrary. Then, the following statements are true: (1) is closed in ;(2) is closed in .

Proof. Let be a sequence, and let be such that in . Furthermore, let be a bounded stopping time. Then, we have showing that . Furthermore, we have By Doob’s optional stopping theorem (which also holds true for -valued martingales; see [2, Remark 2.2.5]), it follows that Using Doob’s optional stopping theorem again, we conclude that , proving the first statement.Now, let be a sequence, and let be such that in . Then, for each , we have and, hence, -almost surely for some subsequence , showing that is -measurable. This proves that , providing the second statement.

Note that, by Doob's martingale inequality [2, Theorem 2.2.7], for , an equivalent norm on is given by Furthermore, if is a separable Hilbert space, then is a separable Hilbert space equipped with the inner product Finally, we recall the following result about series of pairwise orthogonal vectors in Hilbert spaces.

Lemma 4. Let be a separable Hilbert space, and let be a sequence with for . Then, the following statements are equivalent. (1)The series converges in .(2)The series converges unconditionally in .(3)One has . If the previous conditions are satisfied, then one has

Proof. This follows from [22, ] and [23, ].

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 be a separable Hilbert space, and let be a finite time horizon. Furthermore, let be a square-integrable martingale. Recall that the quadratic variation is the (up to indistinguishability) unique real-valued, nondecreasing, predictable process with such that is a martingale.

Proposition 5. Let be an -valued, predictable process with Then, for every orthonormal basis of , the series converges unconditionally in , and its value does not depend on the choice of the orthonormal basis .

Proof. Let be an orthonormal basis of . For with , we have Moreover, by the Itô isometry for the real-valued Itô integral and the monotone convergence theorem, we obtain Therefore, by (17) and Lemma 4, the series (18) converges unconditionally in .Now, let be another orthonormal basis of . We define by Let be arbitrary. Then, we have and the identity For all , 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 (17), we obtain Analogously, we prove that Therefore, denoting by representatives of , , we obtain By separability of , we deduce that Consequently, we have Implying that . 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 -valued, predictable process satisfying (17), we define the Itô integral as where denotes an orthonormal basis of .

According to Proposition 5, definition (31) of the Itô integral is independent of the choice of the orthonormal basis , and the integral process belongs to .

Remark 7. As the proof of Proposition 5 shows, the components of the Itô integral are pairwise orthogonal elements of the Hilbert space .

Proposition 8. For every -valued, predictable process satisfying (17), one has the Itô isometry

Proof. Let be an orthonormal basis of . According to (19), we have Thus, by Lemma 4 and (20), we obtain finishing the proof.

Proposition 9. Let be a -valued simple process of the form with and -measurable random variables for . Then, one has

Proof. Let be an orthonormal basis of . Then, for each , the process 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.

Lemma 10. Let be a -valued, predictable process satisfying (17). Then, for every orthonormal basis of , one has where the convergence takes place in .

Proof. We define the integral process and the sequence of partial sums by By (17) we have and . Furthermore, by Lebesgue's dominated convergence theorem, we have which concludes the proof.

Remark 11. As a consequence of the Doob-Meyer decomposition theorem, for two square-integrable martingales , there exists (up to indistinguishability) a unique real-valued, predictable process with finite variation paths and such that is a martingale.

Proposition 12. For every -valued, predictable process satisfying (17), one has

Proof. Let be an orthonormal basis of . We define the process and the sequence of partial sums by By Proposition 5, we have Defining the integral process by (40) and the sequence of partial sums by (41), using Lemma 10, we have Furthermore, we define the process and the sequence as Then, we have . Indeed, for each , we have For every , the quadratic variation of the real-valued process is given by see, for example, [12, ], which shows that is a martingale. Since , we deduce that .Next, we prove that in . Indeed, since by the Cauchy-Schwarz inequality and (45) we obtain Therefore, together with (46), we get showing that in . Now, Lemma 3 yields that , which concludes the proof.

Theorem 13. Let be another square-integrable martingale, and let be two -valued, predictable processes satisfying (17) and Then, one has

Proof . Using Proposition 12 and the identities identity (54) follows from a straightforward calculation.

Proposition 14. Let be another square-integrable martingale such that , and let be two -valued, predictable processes satisfying (17) and (53). Then, one has

Proof. Using Remark 11, Theorem 13, and the hypothesis , we obtain 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 -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 which, equipped with the inner product is a separable Hilbert space.

Definition 15. A sequence of real-valued Lévy processes is called a sequence of standard Lévy processes if it consists of square-integrable martingales with for all . Here, denotes the Kronecker delta For the rest of this section, let be a sequence of standard Lévy processes.

Proposition 16. For every -valued, predictable process with the series converges unconditionally in .

Proof. For with , we have , and, hence, by Proposition 14, we obtain Moreover, by the Itô isometry (Proposition 8) and the monotone convergence theorem, we have Thus, by (61) and Lemma 4, the series (62) converges unconditionally in .

Therefore, for a -valued, predictable process 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 are pairwise orthogonal elements of the Hilbert space .

Proposition 18. For each -valued, predictable process satisfying (61), one has the Itô isometry

Proof . Using (63), Lemma 4, and identity (64), we obtain completing the proof.

Proposition 19. Let be a -valued simple process of the form with and -measurable random variables for . Then, one has

Proof . For each , the process is a -valued simple process having the representation Hence, by Proposition 9, we obtain 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 be a separable Hilbert space.

Definition 20. A -valued càdlàg, adapted process is called a Lévy process if the following conditions are satisfied. (1)We have .(2) is independent of for all .(3)We have for all .

Definition 21. A -valued Lévy process with and for all is called a square-integrable Lévy martingale.

Note that any square-integrable Lévy martingale is indeed a martingale; that is, see [5, ]. According to [5, ], for each square-integrable Lévy martingale , there exists a unique self-adjoint, nonnegative definite trace class operator , called the covariance operator of , such that for all and , we have Moreover, for all , the angle bracket process is given by see [5, ].

Lemma 22. Let be a -valued square-integrable Lévy martingale with covariance operator , let be another separable Hilbert space, and let be an isometric isomorphism. Then, the process is a -valued square-integrable Lévy martingale with covariance operator .

Proof . The process is a -valued càdlàg, adapted process with . Let be arbitrary. Then, the random variable is independent of , and we have Moreover, for each , we have showing that is a -valued square-integrable Lévy martingale.Let and , be arbitrary, and set ,  . Then, we have showing that the Lévy martingale has the covariance operator .

Now, let be a self-adjoint, positive definite trace class operator. Then, there exists a sequence with and an orthonormal basis of such that We define the sequence of pairwise orthogonal vectors as

Proposition 23. Let be a -valued square-integrable Lévy martingale with covariance operator . Then, the sequence given by is a sequence of standard Lévy processes.

Proof. For each , the process is a real-valued square-integrable Lévy martingale. By (73), for all , we obtain showing that is a sequence of standard Lévy processes.

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

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

Let be a sequence with and denote by the weighted space of sequences which, equipped with the inner product is a separable Hilbert space. Note that we have the strict inclusion , where denotes the space of sequences We denote by the standard orthonormal basis of , which is given by Then, the system defined as is an orthonormal basis of . Let be a linear operator such that Then, is a nuclear, self-adjoint, positive definite operator. Let be an -valued, square-integrable Lévy martingale with covariance operator . According to Proposition 23, the sequence given by is a sequence of standard Lévy processes.

Definition 24. For every -valued, predictable process satisfying (61), we define the Itô integral as

Remark 25. Note that , where denotes the space of Hilbert-Schmidt operators from to . In [21], the Itô integral for -valued processes with respect to an -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 be another sequence with , and let be an isometric isomorphism such that By Lemma 22, the process is a -valued, square integrable Lévy martingale with covariance operator , and by Proposition 23, the sequence given by

is a sequence of standard Lévy processes.

Theorem 26. Let be an isometric isomorphism such that Then, for every -valued, predictable process satisfying (61), one has and the identity

Proof. Since is an isometry, by (61), we have showing (92). Moreover, by (89), we have and, hence, we get By (86) and (96), the vectors and are eigenvectors of with corresponding eigenvalues and . Therefore, and since is an isometry, for with , we obtain Let be arbitrary. Then, we have Since and are eigenvalues of , for each , there are only finitely many such that . Therefore, by (97), and since is an orthonormal basis of , we obtain Thus, taking into account (91) gives us Since was arbitrary, using the separability of 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 be a separable Hilbert space, and let be a nuclear, self-adjoint, positive definite linear operator. Then, there exist a sequence with and an orthonormal basis of   such that namely, are the eigenvalues of , and each is an eigenvector corresponding to . The space , equipped with the inner product is another separable Hilbert space, and the sequence given by is an orthonormal basis of . We denote by the space of Hilbert-Schmidt operators from into , which, endowed with the Hilbert-Schmidt norm itself is a separable Hilbert space. We define the isometric isomorphisms Recall that denotes the orthonormal basis of , which we have defined in (85). Let be a -valued square-integrable Lévy martingale with covariance operator .

Lemma 28. The following statements are true. (1)The process is an -valued square-integrable Lévy martingale with covariance operator .(2)One has

Proof. By Lemma 22, the process is an -valued square-integrable Lévy martingale with covariance operator . Furthermore, by (105) and (101), for all , we obtain showing (107).

Now, our idea is to the define the Itô integral for an -valued, predictable process with by setting 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 and eigenvectors . In order to prove that this is not the case, let be another sequence with , and let be another orthonormal basis of   such that Then, the sequence given by is an orthonormal basis of . Analogous to (105) and (106), we define the isometric isomorphisms Furthermore, we define the isometric isomorphisms The following diagram illustrates the situation: xy(115)

In order to show that the Itô integral (110) is well defined, we have to show that For this, we prepare the following auxiliary result.

Lemma 29. For all and , one has

Proof. By (101) and (111), the vectors and are eigenvectors of with corresponding eigenvalues and . Therefore, for with , we have . For each , we obtain Let be arbitrary. By (106), we have and, hence, Therefore, for all and , we obtain finishing the proof.

Proposition 30. The following statements are true. (1) is an -valued Lévy process with covariance operator , and one has (2) is an -valued Lévy process with covariance operator , and one has (3)For every -valued, predictable process with (109), one has and the identity (116).

Proof. The first two statements follow from Lemma 28. Since and are isometries, we obtain which, together with (109), yields (124). Now, Theorem 26 applies by virtue of Lemma 29 and yields proving (116).

Definition 31. For every -valued process satisfying (109), we define the Itô-Integral by (110).

By virtue of Proposition 30, Definition (110) of the Itô integral neither depends on the choice of the eigenvalues nor on the eigenvectors .

Now, we will the prove the announced series representation of the Itô integral. According to Proposition 23, the sequences and of real-valued processes given by are sequences of standard Lévy processes.

Proposition 32. For every -valued, predictable process satisfying (109), the process given by is a -valued, predictable process, and, one has where the right-hand side of (129) converges unconditionally in .

Proof. Since is an isometry, for each , we obtain Thus, by Definitions 31 and 24, we obtain and, by Proposition 16, the series converges unconditionally in .

Remark 33. By Remark 17 and the proof of Proposition 32, the components of the Itô integral are pairwise orthogonal elements of the Hilbert space .

Proposition 34. For every -valued process satisfying (109), one has the Itô isometry

Proof. By the Itô isometry (Proposition 18), and since is an isometry, we obtain 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 , the restriction belongs to , because

Proposition 35. Let be a -valued simple process of the form with and -measurable random variables for . Then, one has

Proof. The process is an -valued simple process having the representation Thus, by Proposition 19, and since 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 (109); see, for example, [5, ], the Itô integral (110) coincides with that in [5] for every -valued, predictable process 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 satisfying Since the respective spaces of predictable and adapted, measurable processes are isomorphic (see [24]), proceeding as in [24, ], we can further extend the definition of the Itô integral to all adapted, measurable processes satisfying (139).

Acknowledgment

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

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