The existence of the density for the solution of stochastic differential equation driven by fractional Brownian motion with Markovian switching

This paper studies the solution of stochastic differential equation driven by fractional Brownian motion with Markovian switching. We get the existence and uniqueness for the solution. The conditional Malliavin calculus is defined. And the Malliavin regularity of the solution is obtained under the nondegenerate condition of the diffusion coefficient. This yields that the law of the solution is absolute continuity with respect to the Lebesgue measure.