Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

On the exponential stability of switching-diffusion processes with jumps


Authors: Chenggui Yuan and Jianhai Bao
Journal: Quart. Appl. Math. 71 (2013), 311-329
MSC (2010): Primary 60H15; Secondary 60J28, 60J60
DOI: https://doi.org/10.1090/S0033-569X-2012-01292-8
Published electronically: October 18, 2012
MathSciNet review: 3087425
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we focus on the pathwise stability of mild solutions for a class of stochastic partial differential equations which are driven by switching-diffusion processes with jumps. In comparison to the existing literature, we show that: (i) the criterion to guarantee pathwise stability does not rely on the moment stability of the system; (ii) the sample Lyapunov exponent obtained is generally smaller than that of the counterpart driven by a Wiener process; (iii) due to the Markovian switching the overall system can become pathwise exponentially stable although some subsystems are not stable.


References [Enhancements On Off] (What's this?)

  • 1. M. J. Anabtawi and G. S. Ladde, Convergence and stability analysis of partial differential equations under Markovian structural perturbations-I and II: Vector Lyapunov-like functionals, Stoch. Anal. Appl. 18 (2000), no. 4, 493-524. MR1763938 (2001k:35136); 18 (2000), no. 5, 671-696. MR 1780165 (2001k:35137)
  • 2. M. J. Anabtawi and S. Sathananthan, Quantitative analysis of hybrid parabolic systems with Markovian regime switching via practical stability, Nonlinear Anal. Hybrid Syst. 2 (2008), no. 3, 980-992. MR 2431729 (2009g:35127)
  • 3. D. Applebaum, Lévy Processes and Stochastic Calculus, $ 2^{nd}$ Edition, Cambridge University Press, Cambridge, 2009. MR 2512800 (2010m:60002)
  • 4. D. Applebaum and M. Siakalli, Asymptotic stability of stochastic differential equations driven by Lévy noise, J. Appl. Probab. 46 (2009), no. 4, 1116-1129. MR 2582710 (2011g:60098)
  • 5. D. Applebaum and M. Siakalli, Stochastic stabilization of dynamical systems using Lévy noise, Stoch. Dyn. 10 (2010), no. 4, 509-527. MR 2740700 (2011j:93124)
  • 6. J. Bao, A. Truman, and C. Yuan, Stability in distribution of mild solutions to stochastic partial differential delay equations with jumps, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009), no. 2107, 2111-C2134. MR 2515633 (2010i:60192)
  • 7. Z. Brzeźniak, E. Hausenblas, and J. Zhu, Maximal inequality of stochastic convolution driven by compensated random measures in Banach spaces, arXiv:1005.1600.
  • 8. T. Caraballo, K. Liu, and X. Mao, On stabilization of partial differential equations by noise, Nagoya Math. J. 161 (2001), 155-170. MR 1820216 (2002b:60110)
  • 9. P. L. Chow, Stochastic Partial Differential Equations, Chapman & Hall/CRC, Boca Raton, 2007. MR 2295103 (2008d:35243)
  • 10. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992. MR 1207136 (95g:60073)
  • 11. U. G. Haussmann, Asymptotic Stability of the Linear Itô Equation in Infinite Dimensions, J. Math. Anal. Appl. 65 (1978), no. 1, 219-235. MR 0501750 (80b:60082)
  • 12. Z. Hou, J. Bao, and C. Yuan, Exponential stability of energy solutions to stochastic partial differential equations with variable delays and jumps, J. Math. Anal. Appl. 366 (2010), no. 1, 44-54. MR 2593632 (2011e:60144)
  • 13. A. Ichikawa, Stability of semilinear stochastic evolution equations, J. Math. Anal. Appl. 90 (1982), no. 1, 12-44. MR 0680861 (84g:60091)
  • 14. R. Li, P. Leung, and W. Pang, Convergence of numerical solutions to stochastic age-dependent population equations with Markovian switching, J. Comput. Appl. Math. 233 (2009), no. 4, 1046-1055. MR 2557294 (2010k:92100)
  • 15. R. Lipster, A strong law of large numbers for local martingales, Stochastics 3 (1980), no. 3, 217-228. MR 0573205 (83d:60057)
  • 16. K. Liu, Stability of Infinite Dimensional Stochastic Differential Equations with Applications, Chapman and Hall/CRC, Boca Raton, FL, 2006. MR 2165651 (2006f:60060)
  • 17. J. Luo and K. Liu, Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps, Stochastic Process. Appl. 118 (2008), no. 5, 864-895. MR 2411525 (2009c:60155)
  • 18. X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College, London, 2006. MR 2256095 (2008f:60002)
  • 19. C. Marinelli, C. Prevôt, and M. Röckner, Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise, J. Funct. Anal. 258 (2010), no. 2, 616-649. MR 2557949 (2011a:60230)
  • 20. J. R. Norris, Markov Chains, Cambridge University Press, 1998. MR 1600720 (99c:60144)
  • 21. B. Øksendal and A. Sulem, Applied stochastic control of jump diffusions, $ 2^{nd}$ Edition, Springer, Berlin, 2007. MR 2322248 (2008b:93003)
  • 22. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. MR 0710486 (85g:47061)
  • 23. W. Woyczyński, Lévy Processes in the physical sciences, Birkhäuser, Boston, MA, 2001. MR 1833700 (2002d:82029)
  • 24. S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations with Lévy Noise: An Evolution Equation Approach, Cambridge University Press, 2007. MR 2356959 (2009b:60200)
  • 25. G. Yin and C. Zhu, Hybrid Switching Diffusion: Properties and Applications, Springer, New York, 2010. MR 2559912 (2010i:60226)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 60H15, 60J28, 60J60

Retrieve articles in all journals with MSC (2010): 60H15, 60J28, 60J60


Additional Information

Chenggui Yuan
Affiliation: Department of Mathematics, Swansea University, Swansea SA2 8PP, UK
Email: C.Yuan@swansea.ac.uk

Jianhai Bao
Affiliation: Department of Mathematics, Swansea University, Swansea SA2 8PP, UK
Email: majb@swansea.ac.uk

DOI: https://doi.org/10.1090/S0033-569X-2012-01292-8
Keywords: Lévy noise, maximal inequality, exponential martingale inequality with jumps, sample Lyapunov exponent
Received by editor(s): July 6, 2011
Published electronically: October 18, 2012
Article copyright: © Copyright 2012 Brown University
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society