Tail of a linear diffusion with Markov switching

Let Y be an Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic Markov jump process X: dYt=a(Xt)Yt dt+σ(Xt) dWt, Y0=y0. Ergodicity conditions for Y have been obtained. Here we investigate the tail propriety of the stationary distribution of this model. A characterization of either heavy or light tail case is established. The method is based on a renewal theorem for systems of equations with distributions on ℝ.