# Local Skorokhod topology on the space of cadlag processes

## TL;DR

This paper introduces a localized version of the Skorokhod topology for cadlag paths to include explosions, studies tightness of probability measures, and compares local and global topologies, aiding analysis of Lévy-type processes.

## Contribution

It develops a local Skorokhod topology on cadlag paths, characterizes tightness of measures, and compares it with the global topology, facilitating the study of Lévy processes with unbounded coefficients.

## Key findings

- Established tightness criteria for probability measures under the local topology.
- Provided a comparison between local and global Skorokhod topologies using time change.
- Applied results to Lévy-type processes with unbounded coefficients.

## Abstract

We modify the global Skorokhod topology, on the space of cadlag paths, by localising with respect to space variable, in order to include the eventual explosions. The tightness of families of probability measures on the paths space endowed with this local Skorokhod topology is studied and a characterization of Aldous type is obtained. The local and global Skorokhod topologies are compared by using a time change transformation. A number of results in the paper should play an important role when studying L\'evy-type processes with unbounded coefficients by martingale problem approach.