Rooted tree analysis of the order conditions of row-type scheme for stochastic differential equations

  • Yoshio Komori
  • Taketomo Mitsui
  • Hiroshi Sugiura
Article

Abstract

Numerical schemes for initial value problems of stochastic differential equations (SDEs) are considered so as to derive the order conditions of ROW-type schemes in the weak sense. Rooted tree analysis, the well-known useful technique for the counterpart of the ordinary differential equation case, is extended to be applicable to the SDE case. In our analysis, the roots are bi-colored corresponding to the ordinary and stochastic differential terms, whereas the vertices have four kinds of label corresponding to the terms derived from the ROW-schemes. The analysis brings a transparent way for the weak order conditions of the scheme. An example is given for illustration.

AMS subject classification

65U05 60H10 65L06 65L05 

Key words

Stochastic initial value problem time-discrete approximation weak scheme order conditions Stratonovich-Taylor expansion 

References

  1. 1.
    S. S. Artem'ev,The stability of numerical methods for solving stochastic differential equations, Bulletin of the Novosibirsk Computing Center, 2 (1993).Google Scholar
  2. 2.
    S. S. Artem'ev and I. O. Shukurko,Numerical analysis of dynamics of oscillatory stochastic systems, Soviet J. Numer. Anal. Math. Modelling, 6 (1991), pp. 277–298.MathSciNetCrossRefGoogle Scholar
  3. 3.
    T. A. Averina and S. S. Artem'ev,A new family of numerical methods for solving stochastic differential equations, Soviet Math. Dokl., 33 (1986), pp. 736–738.MATHGoogle Scholar
  4. 4.
    J. C. Butcher,The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods, John Wiley & Sons, Chichester, 1987.MATHGoogle Scholar
  5. 5.
    E. Hairer and G. Wanner,Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Systems, Springer-Verlag, Berlin, 1991.Google Scholar
  6. 6.
    N. Hofmann and E. Platen,Stability of weak numerical schemes for stochastic differential equations, Computers Math. Applic., 28 (1994), pp. 45–57.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    P. Kaps and P. Rentrop,Generalized Runge-Kutta methods of order four with stepsize control for stiff ordinary differential equations, Numer. Math., 33 (1979), pp. 55–68.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    P. Kaps and G. Wanner,A study of Rosenbrock-type methods of high order, Numer. Math., 38 (1981), pp. 279–298.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    J. R. Klauder and W. P. Petersen,Numerical integration of multiplicative-noise stochastic differential equations, SIAM J. Numer. Anal., 22 (1985), pp. 1153–1166.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    P. E. Kloeden and E. Platen,Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.MATHGoogle Scholar
  11. 11.
    G. N. Milstein and E. Platen,The integration of stiff stochastic differential equations with stable second moments, Technical Report SRR 014-94, The Australian National University, Canberra 1994.Google Scholar
  12. 12.
    E. Platen.Zur zeitdiskreten Approximation von Itoprozessen, Diss. B, PhD thesis, IMath, Akad. der Wiss. der DDR, 1984.Google Scholar
  13. 13.
    E. Platen,Higher-order weak approximation of Ito diffusions by Markov chains, Probability in the Engineering and Informational Sciences, 6, (1992), pp. 391–408.CrossRefMATHGoogle Scholar
  14. 14.
    E. Platen,On weak implicit and predictor-corrector methods, Mathematics and Computers in Simulation, 38 (1995), pp. 69–76.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© BIT Foundation 1997

Authors and Affiliations

  • Yoshio Komori
    • 1
  • Taketomo Mitsui
    • 2
  • Hiroshi Sugiura
    • 1
  1. 1.Department of Information EngineeringSchool of Engineering Nagoya UniversityNagoyaJapan
  2. 2.Graduate School of Human InformaticsNagoya UniversityNagoyaJapan

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