Skip to main content
Log in

Effective Filtering for Multiscale Stochastic Dynamical Systems Driven by Lévy Processes*

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

The work is about multiscale stochastic dynamical systems driven by Lévy processes. We prove that such a system can be approximated by a low-dimensional system on a random invariant manifold, and the original filter can be also approximated by the reduced low-dimensional filter. Finally, we investigate the reduction for \(\varepsilon =0\) and obtain that these reduced systems does not approximate these multiscale stochastic dynamical systems.

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

Similar content being viewed by others

References

  1. Applebaum, D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press, Cambridge (2009)

    Book  Google Scholar 

  2. Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)

    Book  Google Scholar 

  3. Bain, A., Crisan, D.: Fundamentals of Stochastic Filtering. Springer, Berlin (2009)

    Book  Google Scholar 

  4. Caraballo, T., Chueshov, I., Langa, J.: Existence of invariant manifolds for coupled parabolic and hyperbolic stochastic partial differential equations. Nonlinearity 18, 747–767 (2005)

    Article  MathSciNet  Google Scholar 

  5. Evensen, G.: Data Assimilation: The Ensemble Kalman Filter. Springer, Berlin (2009)

    Book  Google Scholar 

  6. Fu, H., Liu, X., Duan, J.: Slow manifolds for multi-time-scale stochastic evolutionary systems. Commun. Math. Sci. 11, 141–162 (2013)

    Article  MathSciNet  Google Scholar 

  7. Imkeller, P., Namachchivaya, N., Perkowski, N., Yeong, H.: Dimensional reduction in nonlinear filtering: a homogenization approach. Ann. Appl. Probab. 23, 2290–2326 (2013)

    Article  MathSciNet  Google Scholar 

  8. Ikeda, N., Watanabe, S.: Stochastic Differential Equations And Diffusion Processes, 2nd edn. North-Holland/Kodanska, Amsterdam/Tokyo (1989)

    MATH  Google Scholar 

  9. Khasminskii, R., Yin, G.: On transition densities of singularly perturbed diffusions with fast and slow components. SIAM J. Appl. Math. 56, 1794–1819 (1996)

    Article  MathSciNet  Google Scholar 

  10. Mitchell, L., Gottwald, G.: Data assimilation in slow-fast systems using homogenized climate models. J. Atmos. Sci. 69, 1359–1377 (2012)

    Article  Google Scholar 

  11. Park, J., Namachchivaya, N., Yeong, H.: Particle filters in a multiscale environment: Homogenized hybrid particle filter. J. Appl. Mech. 78, 1–10 (2011)

    Google Scholar 

  12. Park, J., Sowers, R., Namachchivaya, N.: Dimensional reductionin nonlinear filtering. Nonlinearity 23, 305–324 (2010)

    Article  MathSciNet  Google Scholar 

  13. Park, J., Rozovskii, B., Sowers, R.: Efficient nonlinear filtering of a singularly perturbed stochastic hybrid system. LMS J. Comput. Math. 14, 254–270 (2011)

    Article  MathSciNet  Google Scholar 

  14. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)

    Book  Google Scholar 

  15. Qiao, H.: Stationary solutions for stochastic differential equations driven by Lévy processes. J. Dyn. Diff. Equ. 29, 1195–1213 (2017)

    Article  Google Scholar 

  16. Qiao, H.: Convergence of nonlinear filtering for stochastic dynamical systems with Lévy noises, arXiv:1707.07824

  17. Qiao, H.: Effective filtering for multiscale stochastic dynamical systems in Hilbert spaces. J. Math. Anal. Appl 487, 123979 (2020)

    Article  MathSciNet  Google Scholar 

  18. Qiao, H., Zhang, Y., Duan, J.: Effective filtering on a random slow manifold. Nonlinearity 31, 4649–4666 (2018)

    Article  MathSciNet  Google Scholar 

  19. Rozovskii, B.: Stochastic Evolution System: Linear Theory and Application to Nonlinear Filtering. Springer, New York (1990)

    Book  Google Scholar 

  20. Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  21. Schuss, Z.: Nonlinear Filtering and Optimal Phase Tracking. Springer, New York (2012)

    Book  Google Scholar 

  22. Schmalfuß, B., Schneider, R.: Invariant manifolds for random dynamical systems with slow and fast variables. J. Dyna. Diff. Equ. 20, 133–164 (2008)

    Article  MathSciNet  Google Scholar 

  23. Wu, F., Tian, T., Rawling, J., Yin, G.: Approximate method for stochastic chemical kinetics with two-time scales by chemical Langevin equations. J. Chem. Phys. 144, 174112 (2016)

    Article  Google Scholar 

  24. Wang, W., Roberts, A.: Slow manifold and averaging for slow-fast stochastic differential system. J. Math. Anal. Appl 398, 822–839 (2013)

    Article  MathSciNet  Google Scholar 

  25. Yuan, S., Hu, J., Liu, X., Duan, J.: Slow manifolds for stochastic systems with non-Gaussian stable Lévy noise, arXiv:1702.08213

  26. Zhang, Y., Cheng, Z., Zhang, X., Chen, X., Duan, J., Li, X.: Data assimilation and parameter estimation for a multiscale stochastic system with \(\alpha \)-stable Lévy noise. J. Stat. Mech. Theor. Exp. 2017, 113401 (2017)

    Article  Google Scholar 

  27. Zhang, Y., Qiao, H., Duan, J.: Effective filtering analysis for non-Gaussian dynamic systems. Appl. Math. Opt. 83, 437–459 (2021)

Download references

Acknowledgements

The author would like to thank the anonymous referee for valuable comments and suggestions which led to a big improvement of the presentation of the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Huijie Qiao.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by NSF of China (No. 11001051, 11371352, 12071071) and China Scholarship Council under Grant No. 201906095034.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qiao, H. Effective Filtering for Multiscale Stochastic Dynamical Systems Driven by Lévy Processes*. J Dyn Diff Equat 34, 2491–2509 (2022). https://doi.org/10.1007/s10884-021-09981-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-021-09981-5

Keywords

Mathematics Subject Classification

Navigation