Skip to main content
SpringerLink
Log in

Optimal \(\vartheta \)-Methods for Mean-Square Dissipative Stochastic Differential Equations

  • Conference paper
  • First Online:
Computational Science and Its Applications – ICCSA 2021 (ICCSA 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12949))

Included in the following conference series:

Abstract

In this paper, we address our investigation to the numerical integration of nonlinear stochastic differential equations exhibiting a mean-square contractive character along the exact dynamics. We specifically focus on the conservation of this qualitative feature along the discretized dynamics originated by applying stochastic \(\vartheta \)-methods. Retaining the mean-square contractivity under time discretization is translated into a proper stepsize restriction. Here we analyze the choice of the optimal parameter \(\vartheta \) making this restriction less demanding and, at the same time, maximizing the stability interval. A numerical evidence is provided to confirm our theoretical results.

This work is supported by GNCS-INDAM project and by PRIN2017-MIUR project 2017JYCLSF “Structure preserving approximation of evolutionary problems”.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Buckwar, E., D’Ambrosio, R.: Exponential mean-square stability properties of stochastic linear multistep methods. Adv. Comput. Math. 47(4), 1–14 (2021)

    Article  MathSciNet  Google Scholar 

  2. Buckwar, E., Riedler, M.G., Kloeden, P.: The numerical stability of stochastic ordinary differential equations with additive noise. Stoch. Dyn. 11, 265–281 (2011)

    Article  MathSciNet  Google Scholar 

  3. Buckwar, E., Sickenberger, T.: A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods. Math. Comput. Simul. 81, 1110–1127 (2011)

    Article  MathSciNet  Google Scholar 

  4. Caraballo, T., Kloeden, P.: The persistence of synchronization under environmental noise. Proc. Roy. Soc. A 46(2059), 2257–2267 (2005)

    Article  MathSciNet  Google Scholar 

  5. Chen, C., Cohen, D., D’Ambrosio, R., Lang, A.: Drift-preserving numerical integrators for stochastic Hamiltonian systems. Adv. Comput. Math. 46, Article Number 27 (2020)

    Google Scholar 

  6. Citro, V., D’Ambrosio, R.: Long-term analysis of stochastic \(\vartheta \)-methods for damped stochastic oscillators. Appl. Numer. Math. 51, 89–99 (2004)

    Article  MathSciNet  Google Scholar 

  7. Cohen, D.: On the numerical discretization of stochastic oscillators. Math. Comput. Simul. 82, 1478–95 (2012)

    Article  Google Scholar 

  8. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Financial Mathematics Series. Champan & All/CRC (2004)

    Google Scholar 

  9. Conte, D., D’Ambrosio, R., Paternoster, B.: Improved theta-methods for stochastic Volterra integral equations. Comm. Nonlin. Sci. Numer. Simul. 93, Article Number 105528 (2021)

    Google Scholar 

  10. Conte, D., D’Ambrosio, R., Paternoster, B.: On the stability of theta-methods for stochastic Volterra integral equations. Discr. Cont. Dyn. Syst. B 23(7), 2695–2708 (2018)

    MATH  Google Scholar 

  11. Dahlquist, G.: Error analysis for a class of methods for stiff nonlinear initial value problems. Lecture Notes Math. 150, 18–26 (2020)

    Google Scholar 

  12. D’Ambrosio, R., Di Giovacchino, S.: Mean-square contractivity of stochastic \(\vartheta \)-methods. Commun. Nonlinear Sci. Numer. Simul. 96, 105671 (2021)

    Article  MathSciNet  Google Scholar 

  13. D’Ambrosio, R., Di Giovacchino, S.: Nonlinear stability issues for stochastic Runge-Kutta methods. Commun. Nonlinear Sci. Numer. Simul. 94, 105549 (2021)

    Article  MathSciNet  Google Scholar 

  14. D’Ambrosio, R., Scalone, C.: Filon quadrature for stochastic oscillators driven by time-varying forces. Appl. Numer. Math. (to appear)

    Google Scholar 

  15. D’Ambrosio, R., Scalone, C.: Two-step Runge-Kutta methods for stochastic differential equations. Appl. Math. Comput. 403, Article Number 125930 (2021)

    Google Scholar 

  16. D’Ambrosio, R., Scalone, C.: On the numerical structure preservation of nonlinear damped stochastic oscillators. Numer. Algorithms 86(3), 933–952 (2020). https://doi.org/10.1007/s11075-020-00918-5

    Article  MathSciNet  MATH  Google Scholar 

  17. Ginzburg, V.L.: On the theory of superconductivity. Il Nuovo Cimento (1955-1965) 2(6), 1234–1250 (1955). https://doi.org/10.1007/BF02731579

    Article  MATH  Google Scholar 

  18. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics, 2nd edn. Springer, Heidelberg (1996). https://doi.org/10.1007/978-3-642-05221-7

    Book  MATH  Google Scholar 

  19. Higham, D.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43, 525–546 (2001)

    Article  MathSciNet  Google Scholar 

  20. Higham, D.: Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal. 38, 753–769 (2000)

    Article  MathSciNet  Google Scholar 

  21. Higham, D., Kloeden, P.: An Introduction to the Numerical Simulation of Stochastic Differential Equations. SIAM (2021)

    Google Scholar 

  22. Higham, D., Kloeden, P.: Numerical methods for nonlinear stochastic differential equations with jumps. Numer. Math. 101, 101–119 (2005)

    Article  MathSciNet  Google Scholar 

  23. Hutzenthaler, M., Jentzen, A.: Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients. Memoirs of the American Mathematical Society, vol. 236, no. 1112 (2015). https://doi.org/10.1090/memo/1112

  24. Koleden, P., Lorenz, T.: Mean-square random dynamical systems. J. Differ. Equ. 253, 1422–1438 (2012)

    Article  MathSciNet  Google Scholar 

  25. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Stochastic Modelling and Applied Probability, vol. 23. Springer, Berlin (1992). https://doi.org/10.1007/978-3-662-12616-5

    Book  MATH  Google Scholar 

  26. Liu, X., Duan, J., Liu, J., Kloeden, P.: Synchronization of dissipative dynamical systems driven by non-Gaussian Levy noises. Int. J. Stoch. Anal. 502803 (2010)

    Google Scholar 

  27. Ma, Q., Ding, D., Ding, X.: Mean-square dissipativity of several numerical methods for stochastic differential equations with jumps. Appl. Numer. Math. 82, 44–50 (2014)

    Article  MathSciNet  Google Scholar 

  28. Majka, M.B.: A note on existence of global solutions and invariant measures for jump SDE with locally one-sided Lipschitz drift. Probab. Math. Stat. 40, 37–57 (2020)

    Article  MathSciNet  Google Scholar 

  29. Melbo, A.H.S., Higham, D.J.: Numerical simulation of a linear stochastic oscillator with additive noise. Appl. Numer. Math. 51, 89–99 (2004)

    Article  MathSciNet  Google Scholar 

  30. Shen, G., Xiao, R., Yin, X., Zhang, J.: Stabilization for hybrid stochastic systems by aperiodically intermittent control. Nonlinear Anal. Hybrid Syst. 29, 100990 (2021)

    Article  MathSciNet  Google Scholar 

  31. Sobczyk, K.: Stochastic Differential Equations with Applications to Physics and Engineering. Mathematics and its Applications, vol. 40. Springer, Dordrecht (1991). https://doi.org/10.1007/978-94-011-3712-6

    Book  MATH  Google Scholar 

  32. Stuart, A.M., Humphries, A.R.: Dynamical Systems and Numerical Analysis. Part of Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  33. Tocino, A.: On preserving long-time features of a linear stochastic oscillators. BIT Numer. Math. 47, 189–196 (2007)

    Article  MathSciNet  Google Scholar 

  34. Wood, G., Zhang, B.: Estimation of the Lipschitz constant of a function. J. Glob. Opt. 8, 91–103 (1996)

    Article  MathSciNet  Google Scholar 

  35. Yao, J., Gan, S.: Stability of the drift-implicit and double-implicit Milstein schemes for nonlinear SDEs. Appl. Math. Comput. 339, 294–301 (2018)

    MathSciNet  MATH  Google Scholar 

  36. Zhao, H., Niu, Y.: Finite-time sliding mode control of switched systems with one-sided Lipschitz nonlinearity. J. Franklin Inst. 357, 11171–11188 (2020)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Information Engineering and Computer Science and Mathematics, University of L’Aquila, L’Aquila, Italy

    Raffaele D’Ambrosio & Stefano Di Giovacchino

Authors
  1. Raffaele D’Ambrosio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Stefano Di Giovacchino
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefano Di Giovacchino .

Editor information

Editors and Affiliations

  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Potenza, Italy

    Beniamino Murgante

  3. Covenant University, Ota, Nigeria

    Sanjay Misra

  4. University of Cagliari, Cagliari, Italy

    Chiara Garau

  5. University of Cagliari, Cagliari, Italy

    Ivan Blečić

  6. Monash University, Clayton, VIC, Australia

    David Taniar

  7. Kyushu Sangyo University, Fukuoka, Japan

    Bernady O. Apduhan

  8. University of Minho, Braga, Portugal

    Ana Maria A. C. Rocha

  9. Polytechnic University of Bari, Bari, Italy

    Eufemia Tarantino

  10. Polytechnic University of Bari, Bari, Italy

    Carmelo Maria Torre

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

D’Ambrosio, R., Di Giovacchino, S. (2021). Optimal \(\vartheta \)-Methods for Mean-Square Dissipative Stochastic Differential Equations. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2021. ICCSA 2021. Lecture Notes in Computer Science(), vol 12949. Springer, Cham. https://doi.org/10.1007/978-3-030-86653-2_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-86653-2_9

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-86652-5

  • Online ISBN: 978-3-030-86653-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation