Existence of Continuous and C\`adl\`ag Versions for Cylindrical Processes in the Dual of a Nuclear Space

Let $\mathrm{\Phi }$ be a nuclear space and let ${\mathrm{\Phi }}_{\beta }^{\mathrm{\prime }}$ denote its strong dual. In this paper we introduce sufficient conditions for a cylindrical process in ${\mathrm{\Phi }}^{\mathrm{\prime }}$ to have a version that is a ${\mathrm{\Phi }}_{\beta }^{\mathrm{\prime }}$-valued continuous or c'{a}dl'{a}g process. We also establish sufficient conditions for the existence of such a version taking values and having finite moments in a Hilbert space continuously embedded in ${\mathrm{\Phi }}_{\beta }^{\mathrm{\prime }}$. Finally, we apply our results to the study of properties of cylindrical martingales in ${\mathrm{\Phi }}^{\mathrm{\prime }}$.