BIT Numerical Mathematics

, Volume 52, Issue 1, pp 109–140 | Cite as

Characterization of bistability for stochastic multistep methods

Article

Abstract

The focus of this article lies on the bistability of multistep methods applied to stochastic ordinary differential equations. Here bistability is understood in the sense of F. Stummel and leads to two-sided estimates of the strong error of convergence. It is shown that bistability can be characterized by Dahlquist’s strong root condition. The main ingredient of the stability analysis is a stochastic version of Spijker’s norm.

We use our results to discuss the maximum order of convergence for higher order schemes. In particular, we are concerned with the stochastic theta method, BDF2-Maruyama and higher order Itô-Taylor schemes.

Keywords

Bistability SODE Itô-Taylor schemes BDF2-Maruyama Stochastic multistep method Stochastic theta method Two-sided error estimate Stochastic Spijker norm 

Mathematics Subject Classification (2000)

65C20 65C30 65L06 65L20 65L70 

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References

  1. 1.
    Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley, New York (1974). Translated from the German MATHGoogle Scholar
  2. 2.
    Baker, C.T.H., Buckwar, E.: Numerical analysis of explicit one-step methods for stochastic delay differential equations. LMS J. Comput. Math. 3, 315–335 (2000) (electronic) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Beyn, W.J., Kruse, R.: Two-sided error estimates for the stochastic theta method. Discrete Contin. Dyn. Syst. Ser. B 14(2), 389–407 (2010) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Buckwar, E., Horváth-Bokor, R., Winkler, R.: Asymptotic mean-square stability of two-step methods for stochastic ordinary differential equations. BIT Numer. Math. 46(2), 261–282 (2006) MATHCrossRefGoogle Scholar
  5. 5.
    Buckwar, E., Kelly, C.: Towards a systematic linear stability analysis of numerical methods for systems of stochastic differential equations. SIAM J. Numer. Anal. 48(1), 298–321 (2010) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Buckwar, E., Winkler, R.: Multistep methods for SDEs and their application to problems with small noise. SIAM J. Numer. Anal. 44(2), 779–803 (2006) (electronic) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Buckwar, E., Winkler, R.: Multi-step Maruyama methods for stochastic delay differential equations. Stoch. Anal. Appl. 25(5), 933–959 (2007) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Clark, J.M.C., Cameron, R.J.: The maximum rate of convergence of discrete approximations for stochastic differential equations. In: Stochastic Differential Systems. Proc. IFIP-WG 7/1 Working Conf., Vilnius, 1978. Lecture Notes in Control and Information Sci., vol. 25, pp. 162–171. Springer, Berlin (1980) CrossRefGoogle Scholar
  9. 9.
    Dahlquist, G.G.: A special stability problem for linear multistep methods. BIT Numer. Math. 3, 27–43 (1963) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Giles, M.B.: Improved multilevel Monte Carlo convergence using the Milstein scheme. In: Monte Carlo and Quasi-Monte Carlo Methods 2006, pp. 343–358. Springer, Berlin (2008) CrossRefGoogle Scholar
  11. 11.
    Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56(3), 607–617 (2008) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Grigorieff, R.D.: Numerik gewöhnlicher Differentialgleichungen, vol. 2. B. G. Teubner, Stuttgart (1977). Mehrschrittverfahren, Unter Mitwirkung von Hans Joachim Pfeiffer, Teubner Studienbücher: Mathematik MATHGoogle Scholar
  13. 13.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations, I, 2nd edn. Springer Series in Computational Mathematics, vol. 8. Springer, Berlin (1993). Nonstiff problems MATHGoogle Scholar
  14. 14.
    Hernández, D.B., Spigler, R.: A-stability of Runge-Kutta methods for systems with additive noise. BIT Numer. Math. 32(4), 620–633 (1992) MATHCrossRefGoogle Scholar
  15. 15.
    Higham, D.J.: Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal. 38(3), 753–769 (2000) (electronic) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Higham, D.J., Mao, X., Stuart, A.M.: Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J. Numer. Anal. 40(3), 1041–1063 (2002) (electronic) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Hutzenthaler, M., Jentzen, A., Kloeden, P.E.: Strong and weak divergence in finite time of Euler’s method for SDEs with non-globally Lipschitz continuous coefficients. Proc. R. Soc. Lond. Ser. A 467(2130), 1563–1576 (2011) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Jentzen, A., Kloeden, P.E., Neuenkirch, A.: Pathwise convergence of numerical schemes for random and stochastic differential equations. In: Cucker, F., Pinkus, A., Todd, M. (eds.) Foundation of Computational Mathematics, Hong Kong, 2008, pp. 140–161. Cambridge University Press, Cambridge (2009) Google Scholar
  19. 19.
    Jentzen, A., Kloeden, P.E., Neuenkirch, A.: Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients. Numer. Math. 112(1), 41–64 (2009) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Kloeden, P.E., Neuenkirch, A.: The pathwise convergence of approximation schemes for stochastic differential equations. LMS J. Comput. Math. 10, 235–253 (2007) (electronic) MathSciNetMATHGoogle Scholar
  21. 21.
    Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York), vol. 23. Springer, Berlin (1992) MATHGoogle Scholar
  22. 22.
    Kruse, R.: Discrete approximation of stochastic differential equations. Bol. Soc. Esp. Mat. Apl. 51, 83–91 (2010) MathSciNetGoogle Scholar
  23. 23.
    Lax, P.D., Richtmyer, R.D.: Survey of the stability of linear finite difference equations. Commun. Pure Appl. Math. 9, 267–293 (1956) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Mao, X.: Stochastic Differential Equations and Their Applications. Horwood Publishing Series in Mathematics & Applications. Horwood Publishing Limited, Chichester (1997) MATHGoogle Scholar
  25. 25.
    Milstein, G.N.: Numerical Integration of Stochastic Differential Equations. Mathematics and Its Applications, vol. 313. Kluwer Academic, Dordrecht (1995). Translated and revised from the 1988 Russian original Google Scholar
  26. 26.
    Milstein, G.N., Tretyakov, M.V.: Stochastic Numerics for Mathematical Physics. Scientific Computation. Springer, Berlin (2004) MATHGoogle Scholar
  27. 27.
    Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications, 6th edn. Universitext. Springer, Berlin (2003) Google Scholar
  28. 28.
    Rydén, T., Wiktorsson, M.: On the simulation of iterated Itô integrals. Stoch. Process. Appl. 91(1), 151–168 (2001) MATHCrossRefGoogle Scholar
  29. 29.
    Spijker, M.N.: Stability and convergence of finite-difference methods. Doctoral dissertation, University of Leiden. Rijksuniversiteit te Leiden, Leiden (1968) Google Scholar
  30. 30.
    Spijker, M.N.: On the structure of error estimates for finite-difference methods. Numer. Math. 18, 73–100 (1971/72) MathSciNetCrossRefGoogle Scholar
  31. 31.
    Stetter, H.J.: Analysis of Discretization Methods for Ordinary Differential Equations. Springer Tracts in Natural Philosophy, vol. 23. Springer, New York (1973) MATHCrossRefGoogle Scholar
  32. 32.
    Stummel, F.: Approximation methods in analysis. Matematisk Institut, Aarhus Universitet, Aarhus (1973). Lectures delivered during the Spring Semester, 1973, Lecture Notes Series, No. 35 Google Scholar
  33. 33.
    Wiktorsson, M.: Joint characteristic function and simultaneous simulation of iterated Itô integrals for multiple independent Brownian motions. Ann. Appl. Probab. 11(2), 470–487 (2001) MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Winkler, R.: Stochastic differential algebraic equations of index 1 and applications in circuit simulation. J. Comput. Appl. Math. 157(2), 477–505 (2003) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsBielefeld UniversityBielefeldGermany

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