Existence of Continuous and Càdlàg Versions for Cylindrical Processes in the Dual of a Nuclear Space

Abstract

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

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