Tail of a linear diffusion with Markov switching

Let Y be a Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic Markov jump process X, i.e. $\mathrm{d}{Y}_t=a(X_t)Y_t\mathrm{d}t+\sigma (X_t)\mathrm{d}{W}_t$, $Y_0=y_0$. Ergodicity conditions for Y have been obtained. Here we investigate the tail property of the stationary distribution of this model. A characterization of the only two possible cases is established: light tail or polynomial tail. Our method is based on discretizations and renewal theory. To cite this article: B. de Saporta, J.-F. Yao, C. R. Acad. Sci. Paris, Ser. I 339 (2004).