Tail of a linear diffusion with Markov switching

Benoîte de Saporta (1, 2, 3, 4) , Jian-Feng Yao (1, 5)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 CQFD - Quality control and dynamic reliability
3 IMB - Institut de Mathématiques de Bordeaux
4 GREThA - Groupe de Recherche en Economie Théorique et Appliquée
5 VISTA - Vision spatio-temporelle et active

Let Y be an Ornstein-Uhlenbeck diffusion governed by a stationary and ergodic Markov jump process X: dY_t=a(X_t)Y_t dt+\sigma(X_t) dW_t, Y_0=y_0. 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.

Mots clés: Ornstein-Uhlenbeck diffusion; Markov switching; random difference equation; light tail; heavy tail; renewal theory; Perron-Frobenius theory; ladder heights

Domaines: Matière Molle [cond-mat.soft]

Benoîte de Saporta, Jian-Feng Yao. Tail of a linear diffusion with Markov switching. The Annals of Applied Probability, 2005, 15, pp.922-1018. ⟨10.1214/105051604000000828⟩. ⟨hal-00111278⟩