# Semimartingales on Duals of Nuclear Spaces

## TL;DR

This paper investigates the properties and representations of semimartingales in the dual space of nuclear spaces, establishing conditions for their existence, path regularity, and Lévy process characterization.

## Contribution

It provides new conditions for cylindrical semimartingales to have càdlàg versions and introduces a canonical representation for semimartingales in dual nuclear spaces.

## Key findings

- Conditions for cylindrical semimartingales to have càdlàg versions.
- A canonical representation for semimartingales in dual nuclear spaces.
- Necessary and sufficient conditions for Lévy process characterization.

## Abstract

This work is devoted to the study of semimartingales on the dual of a general nuclear space. We start by establishing conditions for a cylindrical semimartingale in the strong dual $\Phi'$ of a nuclear space $\Phi$ to have a $\Phi'$-valued semimartingale version whose paths are right-continuous with left limits. Results of similar nature but for more specific classes of cylindrical semimartingales and examples are also provided. Later, we will show that under some general conditions every semimartingale taking values in the dual of a nuclear space has a canonical representation. The concept of predictable characteristics is introduced and is used to establish necessary and sufficient conditions for a $\Phi'$-valued semimartingale to be a $\Phi'$-valued L\'{e}vy process.