The Itô integral with respect to an infinite dimensional Lévy process: A series approach
Stefan Tappe
TL;DR
The paper addresses constructing the Itô integral with respect to infinite-dimensional Lévy processes in a Hilbert space by a series-based approach that reduces the problem to real-valued Itô integrals. It introduces a layered construction: first with a real-valued martingale, then a sequence of standard Lévy processes, then a weighted $\ell_\lambda^2$-valued Lévy process, and finally arbitrary $U$-valued Lévy processes via isometric representations; it proves basis-independence, Itô isometry, and equivalence to the classical Itô integral, and shows consistency across the different spectral decompositions. The method unifies finite- and infinite-dimensional Itô calculus and provides a practical framework for deriving stochastic integrals with complex Hilbert-space-valued Lévy drivers. This series-approach facilitates teaching and leverages finite-dimensional results to address infinite-dimensional stochastic integration problems.
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.