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

TL;DR

This paper provides conditions under which cylindrical processes in the dual of a nuclear space have versions that are continuous or càdlàg in the strong dual space, with applications to cylindrical martingales.

Contribution

It introduces new sufficient conditions for the existence of continuous or càdlàg versions of cylindrical processes in the dual of a nuclear space, including moment conditions in Hilbert spaces.

Findings

Established conditions for continuous versions of cylindrical processes.

Derived criteria for càdlàg versions with finite moments.

Applied results to cylindrical martingales in dual nuclear spaces.

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 $\Phi'$.