Skip to main content
SpringerLink
Log in

Regularity and Recurrence of Switching Diffusions

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This work is concerned with switching diffusion processes, also known as regime-switching diffusions. Our attention focuses on regularity, recurrence, and positive recurrence of the underlying stochastic processes. The main effort is devoted to obtaining easily verifiable conditions for the aforementioned properties. Continuous-state-dependent jump processes are considered. First general criteria on regularity and recurrence using Liapunov functions are obtained. Then we focus on a class of problems, in which both the drift and the diffusion coefficients are “linearizable” with respect to the continuous state, and suppose that the generator of the jump part of the process can be approximated by a generator of an ergodic Markov chain. Sufficient conditions for regularity, recurrence, and positive recurrence are derived, which are linear combination of the averaged coefficients (averaged with respect to the stationary measure of the Markov chain).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. X.Y. Zhou G. Yin (2003) ArticleTitleMarkowitz mean-variance portfolio selection with regime switching: A continuous-time model SIAM J. Control Optim. 42 1466–1482 Occurrence Handle10.1137/S0363012902405583

    Article  Google Scholar 

  2. G. Yin S. Dey (2003) ArticleTitleWeak convergence of hybrid filtering problems involving nearly completely decomposable hidden Markov chains SIAM J. Control Optim. 41 1820–1842 Occurrence Handle10.1137/S0363012901388464

    Article  Google Scholar 

  3. W.M. Wonham (1996) ArticleTitleLiapunov criteria for weak stochastic stability J. Differential Eqs. 2 195–207 Occurrence Handle10.1016/0022-0396(66)90043-X

    Article  Google Scholar 

  4. G. Barone-Adesi R. Whaley (1987) ArticleTitleEfficient analytic approximation of American option values J. Finance 42 301–320 Occurrence Handle10.2307/2328254

    Article  Google Scholar 

  5. X. Mao (1999) ArticleTitleStability of stochastic differential equations with Markovian switching Stochastic Process. Appl. 79 45–67 Occurrence Handle10.1016/S0304-4149(98)00070-2

    Article  Google Scholar 

  6. M.K. Ghosh A. Arapostathis S.I. Marcus (1997) ArticleTitleErgodic control of switching diffusions SIAM J. Control Optim. 35 1952–1988 Occurrence Handle10.1137/S0363012996299302

    Article  Google Scholar 

  7. G. Yin V. Krishnamurthy C. Ion (2004) ArticleTitleRegime switching stochastic approximation algorithms with application to adaptive discrete stochastic optimization SIAM J. Optim. 14 1187–1215 Occurrence Handle10.1137/S1052623403423709

    Article  Google Scholar 

  8. R. Z. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen aan den Rijn, Netherlands, 1980.

  9. C. Zhu and G. Yin, Asymptotic properties of hybrid diffusion systems, preprint, 2006.

  10. R. Z. Khasminskii, C. Zhu, and G. Yin, Stability of regime-switching diffusions, Stochastic Process Appl., to appear.

  11. R. Z. Khasminskii, C. Zhu, and G. Yin, Asymptotic properties of parabolic systems for null-recurrent switching diffusions, Acta Math. Appl. Sinica, to appear.

  12. A. V. Skorohod, Asymptotic Methods in the Theory of Stochastic Differential Equations, Amer. Math. Soc., Providence, RI, 1989.

  13. R.Z. Khasminskii G. Yin (2004) ArticleTitleOn averaging principles: An asymptotic expansion approach SIAM J. Math. Anal. 35 1534–1560 Occurrence Handle10.1137/S0036141002403973

    Article  Google Scholar 

  14. G. C. Papanicolaou, D. Stroock, and S. R. S. Varadhan, Martingale approach to some limit theorems, in Proc. 1976 Duke Univ. Conf. on Turbulence, Durham, NC, 1976.

  15. G. Yin Q. Zhang (1998) Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach Springer-Verleg New York

    Google Scholar 

  16. G. Yin Q. Zhang (2005) Discrete-Time Markov Chains: Two-Time-Scale Methods and Applications Springer New York

    Google Scholar 

  17. G. Yin and C. Zhu, On the notion of weak stability and related issues of hybrid diffusion systems, Nonlinear Anal., Theory, Methods Appl., to appear.

  18. D.R. Cox H.D. Miller (1965) The Theory of Stochastic Processes J. Wiley New York, NY

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, Michigan, 48202, U.S.A.

    George Yin & Chao Zhu

Authors
  1. George Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chao Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to George Yin.

Additional information

This research is supported in part by the National Science Foundation under DMS-0624849, in part by the National Security Agency under MSPF-068-029, and in part by the National Natural Science Foundation of China under Grant No. 60574069.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yin, G., Zhu, C. Regularity and Recurrence of Switching Diffusions. Jrl Syst Sci & Complex 20, 273–283 (2007). https://doi.org/10.1007/s11424-007-9024-3

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-007-9024-3

Keywords

Search

Navigation