Volume 8, August 2004
|Page(s)||25 - 35|
|Published online||15 September 2004|
Linear diffusion with stationary switching regime
SAMOS, Université Paris 1, France; Xavier.Guyon@univ-paris1.fr.
2 LMA, Université de Lille 1, France; email@example.com.
3 IRMAR, Université de Rennes 1, France; Jian-Feng.Yao@univ-rennes1.fr.
Revised: 17 March 2003
Let Y be a Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic process X : dYt = a(Xt)Ytdt + σ(Xt)dWt,Y0 = y0. We establish that under the condition α = Eµ(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.
Mathematics Subject Classification: 60J60 / 60J75
Key words: Ornstein–Uhlenbeck diffusion / Markov switching / jump process / random difference equations / ergodicity / existence of moments.
© EDP Sciences, SMAI, 2004
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