A Study of Stochastic Differential Equations Driven by Cylindrical Lévy Processes

This thesis develops a rigorous framework for stochastic integration in Banach spaces, with a particular focus on integration with respect to cylindrical Lévy processes – an infinite-dimensional generalization of Lévy processes that contains jumps and discontinuities. Motivated by the need to work in non-Hilbertian spaces, the thesis makes original contributions, including a generalization of the Itô integral for Banach space-valued processes, a stochastic Fubini theorem for generalised cylindrical Lévy processes, existence and uniqueness solution for Lévy driven stochastic differential equations in Banach spaces, as well as a key extension of the Brze´zniak and Hausenblas [8] framework to cylindrical Lévy settings using p-summing operators. These results are important in the theory of stochastic partial differential equations and infinite-dimensional stochastic systems. Future work will involve two research articles: one based on Sections 5–7, and another based on the advanced results of Section 8.