Long time behavior of diffusions with Markov switching

Jean-Baptiste Bardet (1) , Hélène Guérin (2) , Florent Malrieu (2)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem
2 IRMAR - Institut de Recherche Mathématique de Rennes

Let $Y$ be an Ornstein-Uhlenbeck diffusion governed by an ergodic finite state Markov process $X$: $dY_t=-\lambda(X_t)Y_tdt+\sigma(X_t)dB_t$, $Y_0$ given. Under ergodicity condition, we get quantitative estimates for the long time behavior of $Y$. We also establish a trichotomy for the tail of the stationary distribution of $Y$: it can be heavy (only some moments are finite), exponential-like (only some exponential moments are finite) or Gaussian-like (its Laplace transform is bounded below and above by Gaussian ones). The critical moments are characterized by the parameters of the model.

Mots clés: Ornstein–Uhlenbeck diffusion; Markov switching; jump process; random difference equation; light tail; heavy tail; Laplace transform

Domaines: Probabilités [math.PR]

Jean-Baptiste Bardet, Hélène Guérin, Florent Malrieu. Long time behavior of diffusions with Markov switching. ALEA : Latin American Journal of Probability and Mathematical Statistics, 2010, 7, pp.151-170. ⟨hal-00441645v2⟩