# Processus de L{\'e}vy avec changements de rythmes

## TL;DR

This paper introduces Switching Processes inspired by PDMPs, incorporating Levy and Itô-Levy processes, and derives their Kolmogorov equations, extending the framework to diffusions and semi-martingales.

## Contribution

It presents a novel class of Switching Processes based on Levy and Itô-Levy processes, with explicit Kolmogorov equations, expanding the theoretical understanding of such stochastic models.

## Key findings

- Derived Kolmogorov equations for Levy-based Switching Processes
- Extended results to Itô-Levy and diffusion processes
- Provided a new framework for processes with regime changes

## Abstract

This paper introduces Switching Processes, called SP. Their constructions are inspired by the PDMP's ones (PDMP stands for Piecewise Deterministic Markov Process). A Markov process, called the intrinsic process, replaces the PDMP's flow. Jumps are added ; they occur randomly as their locations ; their distributions depend on the process's trajectory between them. When the intrinsic process is a Levy process, thanks to its L{\'e}vy-It{\^o} decomposition as a semi-martingale, we obtain the expected Kolmogorov equations for the SP. The results are extended to It{\^o}-L{\'e}vy processes, in particular to diffusion processes.