Ornstein-Uhlenbeck Processes Driven by Cylindrical Levy Processes

In this article we introduce a theory of integration for deterministic, operator-valued integrands with respect to cylindrical L,vy processes in separable Banach spaces. Here, a cylindrical L,vy process is understood in the classical framework of cylindrical random variables and cylindrical measures, and thus, it can be considered as a natural generalisation of cylindrical Wiener processes or white noises. Depending on the underlying Banach space, we provide necessary and/or sufficient conditions for a function to be integrable. In the last part, the developed theory is applied to define Ornstein-Uhlenbeck processes driven by cylindrical L,vy processes and several examples are considered.