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Komori, Yoshio

Weak second-order stochastic Runge-Kutta methods for non-commutative stochastic differential equations. (English) Zbl 1117.65010

J. Comput. Appl. Math. 206, No. 1, 158-173 (2007).
Summary: A new explicit stochastic Runge-Kutta scheme of weak order 2 is proposed for non-commutative stochastic differential equations (SDEs), which is derivative-free and which attains order 4 for ordinary differential equations. The scheme is directly applicable to Stratonovich SDEs and uses \(2m-1\) random variables for one step in the \(m\)-dimensional Wiener process case. It is compared with other derivative-free and weak second-order schemes in numerical experiments.

Cited in 1 Review
Cited in 21 Documents

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations

Keywords:

multi-dimensional Wiener process; derivative-free; multiplicative noise; multi-colored rooted tree; explicit stochastic Runge-Kutta scheme; (SDEs); Stratonovich SDEs; numerical experiments

Software:

RODAS
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Full Text: DOI

References:

[1] Burrage, K.; Burrage, P. M., General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems, Appl. Numer. Math., 28, 161-177 (1998) · Zbl 0924.65146
[2] Burrage, K.; Burrage, P. M., Order conditions of stochastic Runge-Kutta methods by B-series, SIAM J. Numer. Anal., 38, 5, 1626-1646 (2000) · Zbl 0983.65006
[3] Burrage, K.; Burrage, P. M.; Tian, T., Numerical methods for strong solutions of stochastic differential equations: an overview, Proc. Roy. Soc. London A, 460, 373-402 (2004) · Zbl 1048.65004
[4] Butcher, J. C., Numerical Methods for Ordinary Differential Equations (2003), Wiley: Wiley Chichester · Zbl 1032.65512
[5] Eichenauer-Herrmann, J., Inversive congruential pseudorandom numbers: a tutorial, Internat. Statist. Rev., 60, 2, 167-176 (1992) · Zbl 0766.65002
[6] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Systems (1996), Springer: Springer Berlin · Zbl 0859.65067
[7] P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Corrected 3rd printing, Springer, New York, 1999.; P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Corrected 3rd printing, Springer, New York, 1999.
[8] Y. Komori, Multi-colored rooted tree analysis of the weak order conditions of a stochastic Runge-Kutta family, Appl. Numer. Math., in press, doi: 10.1016/j.apnum.2006.02.002; Y. Komori, Multi-colored rooted tree analysis of the weak order conditions of a stochastic Runge-Kutta family, Appl. Numer. Math., in press, doi: 10.1016/j.apnum.2006.02.002 · Zbl 1135.65003
[9] Y. Komori, Weak order stochastic Runge-Kutta methods for commutative stochastic differential equations, J. Comput. Appl. Math., in press, doi: 10.1016/j.cam.2006.03.010; Y. Komori, Weak order stochastic Runge-Kutta methods for commutative stochastic differential equations, J. Comput. Appl. Math., in press, doi: 10.1016/j.cam.2006.03.010 · Zbl 1112.65005
[10] Komori, Y.; Mitsui, T., Stable ROW-type weak scheme for stochastic differential equations, Monte Carlo Methods Appl., 1, 4, 279-300 (1995) · Zbl 0841.65146
[11] Komori, Y.; Saito, Y.; Mitsui, T., Some issue in discrete approximate solution for stochastic differential equations, Comput. Math. Appl., 28, 10-12, 269-278 (1994) · Zbl 0810.65145
[12] Platen, E., Higher-order weak approximation of Ito diffusions by Markov chains, Probab. Eng. Inform. Sci., 6, 391-408 (1992) · Zbl 1134.60357
[13] A. Rößler, Runge-Kutta methods for the numerical solution of stochastic differential equations, Ph.D. Thesis, Darmstadt University of Technology, Germany, 2003.; A. Rößler, Runge-Kutta methods for the numerical solution of stochastic differential equations, Ph.D. Thesis, Darmstadt University of Technology, Germany, 2003. · Zbl 1017.65002
[14] Rößler, A., Runge-Kutta methods for Stratonovich stochastic differential equation systems with commutative noise, J. Comput. Appl. Math., 164-165, 613-627 (2004) · Zbl 1038.65003
[15] Rößler, A., Rooted tree analysis for order conditions of stochastic Runge-Kutta methods for the weak approximation of stochastic differential equations, Stochastic Anal. Appl., 24, 1, 97-134 (2006) · Zbl 1094.65008
[16] D. Talay, Simulation of stochastic differential systems, Probab. Methods Appl. Phys. (1995) 54-96.; D. Talay, Simulation of stochastic differential systems, Probab. Methods Appl. Phys. (1995) 54-96. · Zbl 0837.65150
[17] Talay, D.; Tubaro, L., Expansion of the global error for numerical schemes solving stochastic differential equations, Stochastic Anal. Appl., 8, 4, 483-509 (1990) · Zbl 0718.60058
[18] Tocino, A.; Vigo-Aguiar, J., Weak second order conditions for stochastic Runge-Kutta methods, SIAM J. Sci. Comput., 24, 2, 507-523 (2002) · Zbl 1025.65011
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