Smoothness of density for stochastic differential equations with Markovian switching

TL;DR

This paper investigates the smoothness of the probability density for solutions to stochastic differential equations with Markovian switching, using Malliavin calculus and establishing the strong Feller property under Hörmander conditions.

Contribution

It applies Malliavin calculus to SDEs with Markovian switching to prove density smoothness and derives a Bismut type formula for the strong Feller property.

Findings

Density of solutions is smooth under Hörmander condition

Established a Bismut type formula for these SDEs

Proved the strong Feller property for the process

Abstract

This paper is concerned with a class of stochastic differential equations with Markovian switching. The Malliavin calculus is used to study the smoothness of the density of the solution under a H\"{o}rmander type condition. Furthermore, we obtain a Bismut type formula which is used to establish the strong Feller property.