An elementary proof that the first hitting time of an open set by a jump process is a stopping time

TL;DR

This paper provides a simple and elementary proof demonstrating that the first hitting time of an open set by a jump process of a cadlag adapted process is a stopping time.

Contribution

It introduces a straightforward proof confirming the stopping time property for the first hitting time of open sets by jump processes.

Findings

First hitting time is a stopping time for jump processes.

Elementary proof simplifies existing arguments.

Applicable to cadlag adapted processes.

Abstract

We give a short and elementary proof that the first hitting time of an open set by the jump process of a c\`adl\`ag adapted process is a stopping time.