# Certain Properties Related to Well Posedness of Switching Diffusions

## TL;DR

This paper investigates well-posedness properties of switching diffusions with a continuous component influenced by a discrete switching component, establishing differentiability, continuity, smoothness, and Feller properties, with an application to a Lotka-Volterra model.

## Contribution

It provides new theoretical results on the differentiability, continuity, and Feller property of switching diffusions where the discrete component depends on the continuous one, extending prior work on Markovian switching.

## Key findings

- Differentiability with respect to initial data established.
- Uniform continuity and smoothness of functionals demonstrated.
- Feller property proved under local Lipschitz conditions.

## Abstract

This work is devoted to switching diffusions that have two components (a continuous component and a discrete component). Different from the so-called Markovian switching diffusions, in the setup, the discrete component (the switching) depends on the continuous component (the diffusion process). The objective of this paper is to provide a number of properties related to the well posedness. First, the differentiability with respect to initial data of the continuous component is established. Then, further properties including uniform continuity with respect to initial data, and smoothness of certain functionals are obtained. Moreover, Feller property is obtained under only local Lipschitz continuity. Finally, an example of Lotka-Voterra model under regime switching is provided as an illustration.