Linear diffusion with stationary switching regime

Guyon, Xavier; Iovleff, Serge; Yao, Jian-Feng

ESAIM: Probability and Statistics, Tome 8 (2004), pp. 25-35

Let $Y$ be a Ornstein-Uhlenbeck diffusion governed by a stationary and ergodic process $X:\mathrm{d}{Y}_t=a\left(X_t\right)Y_t\mathrm{d}t+\sigma \left(X_t\right)\mathrm{d}{W}_t,Y_0=y_0$. We establish that under the condition $\alpha =E_{\mu }\left(a\left(X_0\right)\right)<0$ with $\mu$ 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 $s\ge 0$. 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.

Classification : 60J60, 60J75
Keywords: Ornstein-Uhlenbeck diffusion, Markov switching, jump process, random difference equations, ergodicity, existence of moments

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