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

## TL;DR

This paper introduces a new series-based construction of the infinite dimensional Itô integral for Hilbert space valued Lévy processes, connecting it with classical real-valued stochastic integration.

## Contribution

It provides an alternative, series-based approach to defining the infinite dimensional Itô integral, aligning it with existing literature.

## Key findings

- The series approach is equivalent to existing definitions.
- The construction simplifies understanding of infinite dimensional Lévy integrals.
- The method bridges real-valued and infinite dimensional stochastic calculus.

## Abstract

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.