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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. S. Artem'ev,The stability of numerical methods for solving stochastic differential equations, Bulletin of the Novosibirsk Computing Center, 2 (1993).
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.
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.
J. C. Butcher,The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods, John Wiley & Sons, Chichester, 1987.
E. Hairer and G. Wanner,Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Systems, Springer-Verlag, Berlin, 1991.
N. Hofmann and E. Platen,Stability of weak numerical schemes for stochastic differential equations, Computers Math. Applic., 28 (1994), pp. 45–57.
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.
P. Kaps and G. Wanner,A study of Rosenbrock-type methods of high order, Numer. Math., 38 (1981), pp. 279–298.
J. R. Klauder and W. P. Petersen,Numerical integration of multiplicative-noise stochastic differential equations, SIAM J. Numer. Anal., 22 (1985), pp. 1153–1166.
P. E. Kloeden and E. Platen,Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992.
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.
E. Platen.Zur zeitdiskreten Approximation von Itoprozessen, Diss. B, PhD thesis, IMath, Akad. der Wiss. der DDR, 1984.
E. Platen,Higher-order weak approximation of Ito diffusions by Markov chains, Probability in the Engineering and Informational Sciences, 6, (1992), pp. 391–408.
E. Platen,On weak implicit and predictor-corrector methods, Mathematics and Computers in Simulation, 38 (1995), pp. 69–76.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Komori, Y., Mitsui, T. & Sugiura, H. Rooted tree analysis of the order conditions of row-type scheme for stochastic differential equations. Bit Numer Math 37, 43–66 (1997). https://doi.org/10.1007/BF02510172
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02510172