Abstract
First, we provide a comprehensive theory of stochastic integration with respect to cylindrical Lévy processes in Hilbert space. In fact, we go further than simply introducing the stochastic integral, and give a complete analytic characterisation of the largest set of predictable Hilbert-Schmidt operator-valued processes integrable with respect to a cylindrical Lévy process. We demonstrate the strength of the developed integration theory by establishing a stochastic dominated convergence result.Second, we prove an Itô formula for Itô processes driven by cylindrical α-stable noise. It turns out that in the case of standard symmetric α-stable cylindrical Lévy processes, our integration theory simplifies significantly and it is possible to identify the largest space of predictable Hilbert-Schmidt operator-valued integrands with the collection of all predictable processes that have paths in the Bochner space La. As an application of our developed integration theory, we carry out an in-depth analysis of the jump structure of stochastic integral processes driven by standard symmetric α-stable cylindrical Lévy processes, which allows us to establish an Itô formula in this setting.
Finally, we consider stochastic evolution equations driven by α-stable noise and prove the existence of a mild solution, establish long-term regularity of the solutions via a Lyapunov functional approach, and prove an Itô formula for mild solutions to the evolution equations under consideration. The main tool for establishing these results is a Yoshida approximation of the solution, which we combine with the crucial observation that these approximations converge in the space C([0, T];Lp(Ω, H)) of p-th mean continuous Hilbert space-valued processes for any p < α.
| Date of Award | 1 Oct 2023 |
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| Original language | English |
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| Supervisor | Markus Riedle (Supervisor) |