Linear diffusion with stationary switching regime
ESAIM: Probability and Statistics, Tome 8 (2004), pp. 25-35.

Let Y be a Ornstein-Uhlenbeck diffusion governed by a stationary and ergodic process X:dY t =a(X t )Y t dt+σ(X t )dW t ,Y 0 =y 0 . We establish that under the condition α=E μ (a(X 0 ))<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 s0. 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.

DOI : 10.1051/ps:2003017
Classification : 60J60, 60J75
Mots clés : Ornstein-Uhlenbeck diffusion, Markov switching, jump process, random difference equations, ergodicity, existence of moments
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Guyon, Xavier; Iovleff, Serge; Yao, Jian-Feng. Linear diffusion with stationary switching regime. ESAIM: Probability and Statistics, Tome 8 (2004), pp. 25-35. doi : 10.1051/ps:2003017. http://www.numdam.org/articles/10.1051/ps:2003017/

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