Linear diffusion with stationary switching regime

¹ SAMOS, Université Paris 1, France; Xavier.Guyon@univ-paris1.fr.
² LMA, Université de Lille 1, France; serge.iovleff@univ-lille1.fr.
³ IRMAR, Université de Rennes 1, France; Jian-Feng.Yao@univ-rennes1.fr.

Received: 4 February 2003
Revised: 17 March 2003

Abstract

Let Y be a Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic process X : dY_t = a(X_t)Y_tdt + σ(X_t)dW_t,Y₀ = y₀. We establish that under the condition α = E_µ(a(X₀)) < 0 with μ the stationary distribution of the regime process X, the diffusion Y is ergodic. We also consider conditions for the existence of moments for the invariant law of Y when X is a Markov jump process having a finite number of states. Using results on random difference equations on one hand and the fact that conditionally to X, Y is Gaussian on the other hand, we give such a condition for the existence of the moment of order s ≥ 0. Actually we recover in this case a result that Basak et al. [J. Math. Anal. Appl. 202 (1996) 604–622] have established using the theory of stochastic control of linear systems.