Hostname: page-component-74656f9884-54qdv Total loading time: 0 Render date: 2026-11-10T01:06:42.548Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear diffusion with stationary switching regime

Published online by Cambridge University Press:  15 September 2004

Xavier Guyon ,
Serge Iovleff  and
Jian-Feng Yao
Xavier Guyon
Affiliation:
SAMOS, Université Paris 1, France; Xavier.Guyon@univ-paris1.fr.
Serge Iovleff
Affiliation:
LMA, Université de Lille 1, France; serge.iovleff@univ-lille1.fr.
Jian-Feng Yao
Affiliation:
IRMAR, Université de Rennes 1, France; Jian-Feng.Yao@univ-rennes1.fr.
Get access

Abstract

Let Y be a Ornstein–Uhlenbeck diffusion governed by astationary and ergodic process X : dYt = a(Xt)Yt dt + 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 theinvariant law of Y when X is a Markov jump process having a finite number of states.Using results on random difference equationson one hand and the fact that conditionally toX, 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.

Keywords

Ornstein–Uhlenbeck diffusionMarkov switchingjump processrandom difference equationsergodicityexistence of moments.

Information

Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 8 , August 2004 , pp. 25 - 35
Copyright
© EDP Sciences, SMAI, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Basak, G.K., Bisi, A. and Ghosh, M.K., Stability of random diffusion with linear drift. J. Math. Anal. Appl. 202 (1996) 604-622. CrossRef
Bougerol, P. and Picard, N., Strict stationarity of generalized autoregressive processe. Ann. Probab. 20 (1992) 1714-1730. CrossRef
Brandt, A., The stochastic equation Yn+1 = AnYn + Bn with stationnary coefficients. Adv. Appl. Probab. 18 (1986) 211-220.
C. Cocozza–Thivent, Processus stochastiques et fiabilité des systèmes. Springer (1997).
W. Feller,  An Introduction to Probability Theory, Vol. II. Wiley (1966).
Francq, C. and Roussignol, M., Ergodicity of autoregressive processes with Markov switching and consistency of maximum-likelihood estimator. Statistics 32 (1998) 151-173. CrossRef
Hamilton, J.D., A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57 (1989) 151-173. CrossRef
Hamilton, J.D., Analysis of time series subject to changes in regime. J. Econometrics 45 (1990) 39-70. CrossRef
Hamilton, J.D., Specification testing in Markov-switching time series models. J. Econometrics 70 (1996) 127-157. CrossRef
Hansen, B., The likelihood ratio test under nonstandard conditions: Testing the Markov switching model of GNP. J. Appl. Econometrics 7 (1996) 61-82. CrossRef
R.A. Horn and C.R. Johnson, Matrix Analysis. Cambridge University Press (1985).
Ji, Y. and Chizeck, H.J., Controllability, stabilizability and continuous-time Markovian jump linear quadratic control. IEEE Trans. Automat. Control 35 (1990) 777-788. CrossRef
I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic calculus, 2nd Ed. Springer, New York (1991).
M. Mariton,  Jump linear systems in Automatic Control. Dekker (1990).
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.
B. Øksendal, Stochastic Differential Equations, 5th Ed. Springer-Verlag, Berlin (1998).
Yao, J.F. and Attali, J.G., On stability of nonlinear AR processes with Markov switching. Adv. Appl. Probab. 32 (2000) 394-407. CrossRef