Methodology and Computing in Applied Probability

, Volume 8, Issue 4, pp 431–447 | Cite as

Estimating Stochastic Dynamical Systems Driven by a Continuous-Time Jump Markov Process

Article
First Online:
Received:
Revised:
Accepted:

Abstract

We discuss the use of a continuous-time jump Markov process as the driving process in stochastic differential systems. Results are given on the estimation of the infinitesimal generator of the jump Markov process, when considering sample paths on random time intervals. These results are then applied within the framework of stochastic dynamical systems modeling and estimation. Numerical examples are given to illustrate both consistency and asymptotic normality of the estimator of the infinitesimal generator of the driving process. We apply these results to fatigue crack growth modeling as an example of a complex dynamical system, with applications to reliability analysis.

Keywords

Stochastic dynamical system Markov process Estimation Fatigue crack growth 

AMS 2000 Subject Classification

60H10 60K40 62M05 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Albert, “Estimating the infinitesimal generator of a continuous time, finite state Markov process,” Annals of Mathematical Statistics vol. 38 pp. 727–753, 1962.MathSciNetGoogle Scholar
  2. P. Billingsley, Statistical Inference for Markov Processes, University of Chicago Press: Chicago, 1961.MATHGoogle Scholar
  3. J. Chiquet, N. Limnios, and M. Eid, “Modeling and estimating the reliability of stochastic dynamical systems with markovian switching.” In ESREL 2006 - Safety and Reliability for managing Risks, 2006.Google Scholar
  4. R. Dautray (ed.), Méthodes probabilistes pour les équations de la physique, Synthèse. Eyrolles: Paris, France, 1989.Google Scholar
  5. M. H. A. Davis, Markov Models and Optimization, Monographs On statistics and Applied Probability 49, Chapman & Hall, UK, 1993.Google Scholar
  6. S. Goldstein, “On diffusion by discontinuous movements, and on the telegraph equation,” Quarterly Journal of Mechanics and Applied Mathematics vol. 4 pp. 129–156, 1951.MathSciNetMATHGoogle Scholar
  7. W. Härdle, Applied Nonparametric Regression, Econometric Society Monographs, Cambridge University Press, Cambridge, UK, 1990.Google Scholar
  8. A. Jankunas and R. Z. Khasminskii, “Estimation of parameters of linear homogeneous stochastic differential equation,” Stochastic Processes and their Applications vol. 72 pp. 205–219, 1997.MathSciNetCrossRefGoogle Scholar
  9. V. S. Korolyuk and N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific: Singapore, 2005.Google Scholar
  10. Y. A. Kutoyants, Statistical Inference for Ergodic Diffusion Processes, Springer: Berlin Heidelberg New York, 2004.MATHGoogle Scholar
  11. B. Lapeyre and E. Pardoux, Méthodes de Monte-Carlo pour les équations de transport et de diffusion, Springer: Berlin Heidelberg New York, 1998.MATHGoogle Scholar
  12. Y. K. Lin and J. N. Yang, “A stochastic theory of fatigue crack propagation,” AIAA Journal vol. 23 pp. 117–124, 1985.MATHGoogle Scholar
  13. B. Øksendal, Stochastic Differential Equations, 6th edn., Springer: Berlin Heidelberg New York, 2003.MATHGoogle Scholar
  14. A. Sadek and N. Limnios, “Nonparametric estimation of reliability and survival function for continuous time finite Markov process,” Journal of Statistical Planning and Inference vol. 133 pp. 1–21, 2005.MathSciNetCrossRefMATHGoogle Scholar
  15. K. Sobczyk, Stochastic Differential Equations with Applications to Physics and Engineering, Kluwer: Dordrecht, The Netherlands, 1991.MATHGoogle Scholar
  16. K. Sobczyk (ed.), Stochastic Approach to Fatigue, Springer: Berlin Heidelberg New York, 1993.MATHGoogle Scholar
  17. B. F. Spencer, J. Tang, and M. E. Artley, “A stochastic approach to modeling fatigue crack growth,” AIAA Journal vol. 114 pp. 1628–1635, 1989.CrossRefGoogle Scholar
  18. H. Tanaka, “Importance sampling simulation for a stochastic fatigue crack growth model,” In R. E. Melchers and M. G. Stewart (eds.), Proceeding of ICASP8 vol. 2, pp. 907–914, 1999.Google Scholar
  19. A. Tsurui and H. Ishikawa, “Application of the Fokker-Plank equation to a stochastic fatigue crack growth model,” Structural Safety vol. 4 pp. 15–29, 1986.CrossRefGoogle Scholar
  20. D. A. Virkler, B. M. Hillberry, and P. K. Goel, “The statistical nature of fatigue crack propagation,” Journal of Engineering Material Technology ASME vol. 101 pp. 148–153, 1979.CrossRefGoogle Scholar
  21. J. N. Yang and S. D. Manning, “A simple second order approximation for stochastic crack growth analysis,” Engineering Fracture Mechanics vol. 53 pp. 677–686, 1996.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  1. 1.Université de Technologie de Compiègne, Centre de Recherche de Royallieu, LMACCompiègne cedexFrance
  2. 2.Commissariat à l’Énergie Atomique, Centre de Recherche de Saclay, DM2S/SERMA/LCAGif sur Yvette cedexFrance

Personalised recommendations