Stochastic integration with respect to cylindrical semimartingales

In this work we introduce a theory of stochastic integration with respect to general cylindrical semimartingales defined on locally convex space $\Phi$. Our construction the integral is based tensor products topological vector spaces and property good integrators real-valued semimartingales. This further developed in case where $\Phi$ complete, barrelled, nuclear space, obtain complete description class integrands as $\Phi$-valued bounded weakly predictable processes. Several other properties are proven, including Riemann representation, by parts formula Fubini theorem. then applied provide sufficient necessary conditions for existence uniqueness solutions linear evolution equations driven semimartingale noise taking values strong dual $\Phi'$ last part article apply our define integrals sequence