Stochastic Integration and Stochastic PDEs Driven by Jumps on the Dual of a Nuclear Space

We develop a novel theory of weak and strong stochastic integration for cylindrical martingale-valued measures taking values in the dual of a nuclear space. This is applied to develop a theory of SPDEs with rather general coefficients. In particular, we can then study SPDEs driven by general L'{e}vy processes in this context.