A Comparison of Two Settings for Stochastic Integration with Respect to Lévy Processes in Infinite Dimensions
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We review two settings for stochastic integration with respect to infinite dimensional Lévy processes. We relate notions of stochastic integration with respect to square-integrable Lévy martingales, compound Poisson processes, Poisson random measures and compensated Poisson random measures. We use the Lévy-Khinchin decomposition to decompose stochastic integrals with respect to general, non-square-integrable Lévy processes into a Riemann integral and stochastic integrals with respect to a Wiener process, Poisson random measure and compensated Poisson random measure. Besides its intrinsic interest this review article is also meant as a step toward new studies in stochastic partial differential equations with Lévy noise.
This work was partially supported by the National Science Foundation (NSF) under the grant DMS-1510249 and by the Research Fund of Indiana University. The authors gratefully acknowledge the use of the book  during a semester-long seminar, and some private exchanges of emails with the authors.
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