Skip to main content
Log in

Invariant density, Lapunov exponent, and almost sure stability of Markovian-regime-switching linear systems

Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This paper is concerned with stability of a class of randomly switched systems of ordinary differential equations. The system under consideration can be viewed as a two-component process (X(t), α(t)), where the system is linear in X(t) and α(t) is a continuous-time Markov chain with a finite state space. Conditions for almost surely exponential stability and instability are obtained. The conditions are based on the Lyapunov exponent, which in turn, depends on the associate invariant density. Concentrating on the case that the continuous component is two dimensional, using transformation techniques, differential equations satisfied by the invariant density associated with the Lyapunov exponent are derived. Conditions for existence and uniqueness of solutions are derived. Then numerical solutions are developed to solve the associated differential equations.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

References

  1. M. M. Benderskii and L. A. Pastur, The spectrum of the one-dimensional Schrodinger equation with random potential, Mat. Sb., 1972, 82: 273–284.

    MathSciNet  Google Scholar 

  2. M. M. Benderskii and L. A. Pastur, Asymptotic behavior of the solutions of a second order equation with random coefficients, Teor. Funkciz̆ Funkcional. Anal. i Priložen, 1973, 22: 3–14.

    Google Scholar 

  3. K. A. Loparo and G. L. Blankenship, Almost sure instability of a class of linear stochastic systems with jump process coefficients, Lyapunov Exponents, 160–190, Lecture Notes in Math., 1186, Springer, Berlin, 1986.

    Chapter  Google Scholar 

  4. G. Barone-Adesi and R. Whaley, Efficient analytic approximation of American option values, J. Finance, 1987, 42: 301–320.

    Article  Google Scholar 

  5. M. H. A. Davis, Markov Models and Optimization, Chapman & Hall, London, UK, 1993.

    MATH  Google Scholar 

  6. G. B. Di Masi, Y. M. Kabanov, and W. J. Runggaldier, Mean variance hedging of options on stocks with Markov volatility, Theory of Probab. Appl., 1994, 39: 173–181.

    Google Scholar 

  7. G. Yin, V. Krishnamurthy, and C. Ion, Regime switching stochastic approximation algorithms with application to adaptive discrete stochastic optimization, SIAM J. Optim., 2004, 14: 1187–1215.

    Article  MATH  MathSciNet  Google Scholar 

  8. G. Yin and C. Zhu, Hybrid Switching Diffusions: Properties and Applications, Springer, New York, 2010.

    Book  MATH  Google Scholar 

  9. C. Zhu, G. Yin, and Q. S. Song, Stability of random-switching systems of differential equations, Quarterly Appl. Math., 2009, 67: 201–220.

    MATH  MathSciNet  Google Scholar 

  10. X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Impreial College Press, London, 2006.

    MATH  Google Scholar 

  11. R. Z. Khasminskii, C. Zhu, and G. Yin, Stability of regime-switching diffusions, Stochastic Process. Appl., 2007, 117: 1037–1051.

    Article  MATH  MathSciNet  Google Scholar 

  12. I. Ia. Kac and N. N. Krasovskii, On the stability of systems with random parameters, J. Appl. Math. Mech., 1960, 24: 1225–1246.

    Article  Google Scholar 

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

    Google Scholar 

  14. X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl., 1999, 79: 45–67.

    Article  MATH  MathSciNet  Google Scholar 

  15. X. Mao, G. Yin, and C. Yuan, Stabilization and destabilization of hybrid systems of stochastic differential equations, Automatica, 2007, 43: 264–273.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Qi He.

Additional information

This research was supported in part by the National Science Foundation under Grant No. DMS-0907753, in part by the Air Force Office of Scientific Research under Grant No. FA9550-10-1-0210, and in part by the National Natural Science Foundation of China under Grant No. 70871055.

This paper was recommended for publication by Editor Jifeng ZHANG

Rights and permissions

Reprints and permissions

About this article

Cite this article

He, Q., Yin, G.G. Invariant density, Lapunov exponent, and almost sure stability of Markovian-regime-switching linear systems. J Syst Sci Complex 24, 79–92 (2011). https://doi.org/10.1007/s11424-011-9018-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-011-9018-z

Key words

Navigation