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

Abstract

Let $\Phi$ be a nuclear space and let $\Phi'_{\beta}$ denote its strong dual. In this paper we introduce sufficient conditions for a cylindrical process in $\Phi'$ to have a version that is a $\Phi'_{\beta}$-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 $\Phi'_{\beta}$. Finally, we apply our results to the study of properties of cylindrical martingales in $\Phi'$.