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

We present an alternative construction of the infinite dimensional It？ 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 It？ integral is given by a series of It？ integrals with respect to standard Lévy processes. We also prove that this stochastic integral coincides with the It？ integral that has been developed in the literature. 1. Introduction The It？ 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 It？ integral for elementary processes and then extends it via the It？ 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 It？ integral have been considered, for example, in [6–8]. Moreover, in [9], series expansions have been used in order to define the It？ 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 It？ 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 It？ integral; see, for example, [11–13], and where the It？ integral is given by a series of It？ 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 It？ 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 It？ 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