Advertisement

BIT Numerical Mathematics

, Volume 37, Issue 4, pp 771–780 | Cite as

A bound on the maximum strong order of stochastic Runge-Kutta methods for stochastic ordinary differential equations

  • K. Burrage
  • P. M. Burrage
  • J. A. Belward
Article

Abstract

In Burrage and Burrage [1] it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods, the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular, methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present in the formulation. In this present paper, general order results are proven about the maximum attainable strong order of these stochastic Runge-Kutta methods (SRKs) in terms of the order of the Stratonovich integrals appearing in the Runge-Kutta formulation. In particular, it will be shown that if ans-stage SRK contains Stratonovich integrals up to orderp then the strong order of the SRK cannot exceed min{(p+1)/2, (s−1)/2},p≥2,s≥3 or 1 ifp=1.

AMS subject classification

65L06 

Key words

Stochastic ordinary differential equation Runge-Kutta methods strong order 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    K. Burrage and P. M. Burrage,High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Appl. Numer. Math., 20 (1996), pp. 1–21.MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    P. M. Burrage,Numerical methods for stochastic differential equations, Ph.D. thesis, Department of Mathematics, University of Queensland, Australia, 1998.Google Scholar
  3. 3.
    J. C. Butcher,The Numerical Analysis of Ordinary Differential Equations, Wiley, U.K., 1987.MATHGoogle Scholar
  4. 4.
    T. C. Gard,Introduction to Stochastic Differential Equations, Marcel Dekker, New York, 1988.MATHGoogle Scholar
  5. 5.
    P. E. Kloeden and E. Platen,Relations between multiple Itô and Stratonovich integrals, Stochastic Anal. Appl., 9(3) (1991), pp. 311–321.MATHMathSciNetGoogle Scholar
  6. 6.
    P. E. Kloeden and E. Platen,The Numerical Solution of Stochastic Differential Equations, Springer-Verlag, 1992.Google Scholar
  7. 7.
    G. N. Milstein, E. Platen, and H. Schurz,Balanced implicit methods for stiff stochastic systems, Inst. of Advanced Studies, ANU, Canberra, Australia, 1994.Google Scholar
  8. 8.
    W. Rümelin,Numerical treatment of stochastic differential equations, SIAM J. Numer. Anal., 19 (1982), pp. 604–613.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Swets & Zeitlinger 1997

Authors and Affiliations

  • K. Burrage
    • 1
  • P. M. Burrage
    • 1
  • J. A. Belward
    • 1
  1. 1.Department of MathematicsThe University of QueenslandBrisbaneAustralia

Personalised recommendations