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.

The Itô integral with respect to an infinite dimensional Wiener process has been developed in [

For stochastic integrals with respect to a Wiener process, series expansions of the Itô integral have been considered, for example, in [

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, [

In [

The idea to use series expansions for the definition of the stochastic integral has also been utilized in the context of cylindrical processes; see [

The construction of the Itô integral, which we present in this paper, is divided into the following steps.

For an

Based on the just defined integral, for an

In the next step, let

Finally, let

The remainder of this text is organized as follows. In Section

In this section, we provide the required preliminary results and some basic notation. Throughout this text, let

Let

We define the Lebesgue space

We denote by

We denote by

We define the factor spaces

Let us emphasize the following.

Since the Skorokhod space

By the completeness of the filtration

The definition of

Note that we have the inclusions

The following auxiliary result shows that these inclusions are closed.

Let

Let

Now, let

Note that, by Doob's martingale inequality [

Let

The series

The series

One has

If the previous conditions are satisfied, then one has

This follows from [

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

Let

Let

Now, let

Now, Proposition

For every

According to Proposition

As the proof of Proposition

For every

Let

Let

Let

Let

We define the integral process

As a consequence of the Doob-Meyer decomposition theorem, for two square-integrable martingales

For every

Let

Next, we prove that

Let

Using Proposition

Let

Using Remark

In this section, we introduce the Itô integral for

A sequence

For the rest of this section, let

For every

For

Therefore, for a

As the proof of Proposition

For each

Using (

Let

For each

In this section, we provide the required results about Lévy processes in Hilbert spaces. Let

A

We have

We have

A

Note that any square-integrable Lévy martingale

Let

The process

Let

Now, let

Let

For each

In this section, we introduce the Itô integral for

Let

For every

Note that

Now, let

is a sequence of standard Lévy processes.

Let

Since

From a geometric point of view, Theorem

In this section, we define the Itô integral with respect to a general Lévy process, which is based on the Itô integral (

Let

The following statements are true.

The process

One has

By Lemma

Now, our idea is to the define the Itô integral for an

In order to show that the Itô integral (

For all

By (

The following statements are true.

For every

The first two statements follow from Lemma

For every

By virtue of Proposition

Now, we will the prove the announced series representation of the Itô integral. According to Proposition

For every

Since

By Remark

For every

By the Itô isometry (Proposition

We shall now prove that the stochastic integral, which we have defined so far, coincides with the Itô integral developed in [

Let

The process

Therefore, and since the space of simple processes is dense in the space of all predictable processes satisfying (

By a standard localization argument, we can extend the definition of the Itô integral to all predictable processes

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