Advertisement

Science China Mathematics

, Volume 61, Issue 4, pp 659–676 | Cite as

Variational formula for the stability of regime-switching diffusion processes

  • Jinghai Shao
  • Lingdi Wang
Articles
  • 37 Downloads

Abstract

The asymptotical stability in probability is studied for diffusion processes and regime-switching diffusion processes in this work. For diffusion processes, some criteria based on the integrability of the functionals of the coeffcients are given, which yield a useful comparison theorem on stability with respect to some nonlinear systems. For regime-switching diffusion processes, some criteria based on the idea of a variational formula are given. Both state-independent and state-dependent regime-switching diffusion processes are investigated in this work. These conditions are easily verified and are shown to be sharp by examples.

Keywords

stability in probability regime-switching diffusions state-dependent M-matrix 

MSC(2010)

60J27 93E15 60A10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11301030, 11401169 and 11431014), and Key Scientific Research Projects of Henan Province (Grant No. 16A110010).

References

  1. 1.
    Bao J, Shao J. Permanence and extinction of regime-switching predator-prey models. SIAM J Math Anal, 2016, 48: 725–739MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bardet J, Guérin H, Malrieu F. Long time behavior of diffusions with Markov switching. ALEA Lat Am J Probab Math Stat, 2010, 7: 151–170MathSciNetzbMATHGoogle Scholar
  3. 3.
    Berman A, Plemmons R J. Nonnegative Matrices in the Mathematical Sciences. Philadelphia: SIAM, 1994CrossRefzbMATHGoogle Scholar
  4. 4.
    Bhattacharya R N. Criteria for recurrence and existence of invariant measures for multidimensional diffusions. Ann Probab, 1978, 6: 541–553MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chen M F, Wang F Y. Estimation of spectral gap for elliptic operators. Trans Amer Math Soc, 1997, 349: 1239–1267MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cloez B, Hairer M. Exponential ergodicity for Markov processes with random switching. Bernoulli, 2015, 21: 505–536MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    de Saporta B, Yao J F. Tail of linear diffusion with Markov switching. Ann Appl Probab, 2005, 15: 992–1018MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ghosh M, Arapostathis A, Marcus S. Optimal control of switching diffusions with application to exible manufacturing systems. SIAM J Control Optim, 1992, 30: 1–23MathSciNetCrossRefGoogle Scholar
  9. 9.
    Friedman A. Wanding to infinity of diffusion processes. Trans Amer Math Soc, 1973, 184: 185–203MathSciNetGoogle Scholar
  10. 10.
    Friedman A. Stochastic Differential Equations and Applications, 2nd ed. Now York: Cover Publications, 2006zbMATHGoogle Scholar
  11. 11.
    Khasminskii R Z. Stochastic Stability of Differential Equations, 2nd ed. Berlin-Heidelberg: Springer-Verlag, 2012CrossRefzbMATHGoogle Scholar
  12. 12.
    Luo Q, Mao X R. Stochastic population dynamics under regime-switching II. J Math Anal Appl, 2009, 355: 577–593MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mao X. Stochastic Differential Equations and Applications. Chichester: Horwood Publishing Limited, 1997zbMATHGoogle Scholar
  14. 14.
    Mao X, Yuan C. Stochastic Differential Equations with Markovian Switching. London: Imperial College Press, 2006CrossRefzbMATHGoogle Scholar
  15. 15.
    Shao J. Ergodicity of one-dimensional regime-switching diffusion processes. Sci China Math, 2014, 57: 2407–2414MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Shao J. Ergodicity of regime-switching diffusions inWasserstein distances. Stochastic Process Appl, 2015, 125: 739–758MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Shao J. Criteria for transience and recurrence of regime-switching diffusion processes. Electron J Probab, 2015, 20: 1–15MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Shao J. Strong solutions and strong Feller properties for regime-switching diffusion processes in an infinite state space. SIAM J Control Optim, 2015, 53: 2462–2479MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Shao J, Xi F B. Strong ergodicity of the regime-switching diffusion processes. Stochastic Process Appl, 2013, 123: 3903–3918MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shao J, Xi F B. Stability and recurrence of regime-switching diffusion processes. SIAM J Control Optim, 2014, 52: 3496–3516MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Skorokhod A V. Asymptotic Methods in the Theory of Stochastic Differential Equations. Providence: Amer Math Soc, 1989zbMATHGoogle Scholar
  22. 22.
    Wu Z, Shi P, Karimi H R. Stability of stochastic nonlinear systems with state-dependent switching. IEEE Trans Automat Control, 2013, 58: 1904–1918MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Xi F B, Yin G. Almost sure stability and instability for switching-jump-diffusion systems with state-dependent switch-ing. J Math Anal Appl, 2013, 400: 460–474MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Yin G, Xi F B. Stability of regime-switching jump diffusions. SIAM J Control Optim, 2010, 48: 4525–4549MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Yin G, Zhu C. Hybrid Switching Diffusions: Properties and Applications. New York: Springer, 2010CrossRefzbMATHGoogle Scholar
  26. 26.
    Yuan C, Mao X. Asymptotic stability in distribution of SDEs with Markovian switching. Stochastic Process Appl, 2003, 103: 277–291MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zhu C, Yin G. On strong Feller recurrence and weak stabilization of regime-switching diffusions. SIAM J Control Optim, 2009, 48: 2003–2031MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Center for Applied MathematicsTianjin UniversityTianjinChina
  2. 2.School of Mathematics and StatisticsHenan UniversityKaifengChina

Personalised recommendations