The Annals of Applied Probability

Tail of a linear diffusion with Markov switching

Benoîte de Saporta and Jian-Feng Yao

Full-text: Open access


Let Y be an Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic Markov jump process X: dYt=a(Xt)Ytdt+σ(Xt) dWt, Y0=y0. Ergodicity conditions for Y have been obtained. Here we investigate the tail propriety of the stationary distribution of this model. A characterization of either heavy or light tail case is established. The method is based on a renewal theorem for systems of equations with distributions on ℝ.

Article information

Ann. Appl. Probab., Volume 15, Number 1B (2005), 992-1018.

First available in Project Euclid: 1 February 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65] 60J75: Jump processes 60H25: Random operators and equations [See also 47B80]
Secondary: 60K05: Renewal theory 60J15

Ornstein–Uhlenbeck diffusion Markov switching random difference equation light tail heavy tail renewal theory Perron–Frobenius theory ladder heights


de Saporta, Benoîte; Yao, Jian-Feng. Tail of a linear diffusion with Markov switching. Ann. Appl. Probab. 15 (2005), no. 1B, 992--1018. doi:10.1214/105051604000000828.

Export citation


  • Athreya, K. and Rama Murthy, K. (1976). Feller's renewal theorem for systems of renewal equations. J. Indian Inst. Sci. 58 437–459.
  • Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. Roy. Statist. Soc. Ser. B 63 167–241.
  • Basak, G., Bisi, A. and Ghosh, M. K. (1996). Stability of random diffusion with linear drift. J. Math. Anal. Appl. 202 604–622.
  • Brockwell, P. J. (2001). Lévy-driven CARMA processes. Ann. Inst. Statist. Math. 53 113–124.
  • Chow, S. C. and Teicher, H. (1978). Probability Theory. Independence, Interchangeability, Martingales. Springer, New York.
  • Crump, K. (1970). On systems of renewal equations. J. Math. Anal. Appl. 30 425–434.
  • de Saporta, B. (2003). Renewal theorem for a system of renewal equations. Ann. Inst. H. Poincaré Probab. Statist. 39 823–838.
  • de Saporta, B. (2004). Tail of the stationary solution of the stochastic equation $y_{n+1}=a_ny_n+b_n$ with Markovian coefficients. Stochastic Process. Appl. To appear.
  • Feller, W. (1966). Introduction to Probability Theory and Its Applications
  • Wiley, New York.
  • Goldie, C. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 26–166.
  • Grincevičius, A. K. (1980). Products of random affine transformations. Lithuanian Math. J. 20 279–282.
  • Guyon, X., Iovleff, S. and Yao, J.-F. (2004). Linear diffusion with stationary switching regime. ESAIM Probab. Statist. 8 25–35.
  • Hamilton, J. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57 151–173.
  • Hamilton, J. (1990). Analysis of time series subject to changes in regime. J. Econometrics 45 39–70.
  • Horn, R. and Johnson, C. (1985). Matrix Analysis. Cambridge Univ. Press.
  • Karatzas, I. and Shreve, S. (1991). Brownian Motion and Stochastic Calculus. Springer, New York.
  • Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207–248.
  • Le Page, E. (1983). Théorèmes de renouvellement pour les produits de matrices aléatoires. Equations aux différences aléatoires. In Séminaires de Probabilités de Rennes 116. Publ. Sem. Math., Univ. Rennes 1, Rennes.
  • Norris, J. (1998). Markov Chains. Cambridge Univ. Press.