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.

[1] G.K. Basak, A. Bisi and M.K. Ghosh, Stability of random diffusion with linear drift. J. Math. Anal. Appl. 202 (1996) 604-622. | MR | Zbl

[2] P. Bougerol and N. Picard, Strict stationarity of generalized autoregressive processe. Ann. Probab. 20 (1992) 1714-1730. | MR | Zbl

[3] A. Brandt, The stochastic equation $Y_{n+1}=A_n{Y}_n+B_n$ with stationnary coefficients. Adv. Appl. Probab. 18 (1986) 211-220. | MR | Zbl

[4] C. Cocozza-Thivent, Processus stochastiques et fiabilité des systèmes. Springer (1997). | Zbl

[5] W. Feller, An Introduction to Probability Theory, Vol. II. Wiley (1966). | MR | Zbl

[6] C. Francq and M. Roussignol, Ergodicity of autoregressive processes with Markov switching and consistency of maximum-likelihood estimator. Statistics 32 (1998) 151-173. | MR | Zbl

[7] J.D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57 (1989) 151-173. | MR | Zbl

[8] J.D. Hamilton, Analysis of time series subject to changes in regime. J. Econometrics 45 (1990) 39-70. | MR | Zbl

[9] J.D. Hamilton, Specification testing in Markov-switching time series models. J. Econometrics 70 (1996) 127-157. | MR | Zbl

[10] B. Hansen, The likelihood ratio test under nonstandard conditions: Testing the Markov switching model of GNP. J. Appl. Econometrics 7 (1996) 61-82.

[11] R.A. Horn and C.R. Johnson, Matrix Analysis. Cambridge University Press (1985). | MR | Zbl

[12] Y. Ji and H.J. Chizeck, Controllability, stabilizability and continuous-time Markovian jump linear quadratic control. IEEE Trans. Automat. Control 35 (1990) 777-788. | MR | Zbl

[13] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic calculus, 2nd Ed. Springer, New York (1991). | MR | Zbl

[14] M. Mariton, Jump linear systems in Automatic Control. Dekker (1990).

[15] R.E. Mccullogh and R.S. Tsay, Statistical analysis of econometric times series via Markov switching models, J. Time Ser. Anal. 15 (1994) 523-539. | Zbl

[16] B. Øksendal, Stochastic Differential Equations, 5th Ed. Springer-Verlag, Berlin (1998). | MR | Zbl

[17] J.F. Yao and J.G. Attali, On stability of nonlinear AR processes with Markov switching. Adv. Appl. Probab. 32 (2000) 394-407. | MR | Zbl