Runge-Kutta methods for Itô stochastic differential equations with scalar noise

BIT Numerical Mathematics

Volumn 46, Issue 1, 2006, Pages 97-110

  Rößler, Andreas ^a  

^a DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Colored rooted tree analysis; Numerical method; Stochastic differential equation; Stochastic Runge Kutta method; Weak approximation

Indexed keywords

EID: 33645029424     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10543-005-0039-7     Document Type: Article

References (17)

1. M. I. Abukhaled and E. J. Allen, A class of second-order Runge-Kutta methods for numerical solution of stochastic differential equations, Stochastic Anal. Appl., 16 (1998), pp. 977-991.
2. K. Burrage and P. M. Burrage, General order conditions for stochastic Runge-Kutta methods for both commuting and non-commuting stochastic ordinary differential equation systems, Appl. Numer. Math., 28 (1998), pp. 161-177.
3. E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I, Springer, Berlin, 1993.
4. I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, New-York, Berlin, 1999.
5. P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1999.
6. Y. Komori and T. Mitsui, Stable ROW-type weak scheme for stochastic differential equations, Monte Carlo Methods Appl., 1 (1995), pp. 279-300.
7. J. Lehn, A. Rößler and O. Schein, Adaptive schemes for the numerical solution of SDEs - a comparison, J. Comput. Appl. Math., 138 (2002), pp. 297-308.
8. G. N. Milstein, Numerical Integration of Stochastic Differential Equations, Kluwer Academic Publishers, Dordrecht, 1995.
9. N. J. Newton, Asymptotically efficient Runge-Kutta methods for a class of Itô and Stratonovich equations, SIAM J. Appl. Math., 51 (1991), pp. 542-567.
10. A. Rößler, Coefficients of Runge-Kutta Schemes for Itô Stochastic Differential Equations, Proc. Appl. Math. Mech., 3 (2003), pp. 571-572.
11. A. Rößler, Rooted tree analysis for order conditions of stochastic Runge-Kutta methods for the weak approximation of stochastic differential equations, Stochastic Anal. Appl., 24 (2006), pp. 97-134.
12. A. Rößler, Runge-Kutta methods for Stratonovich stochastic differential equation systems with commutative noise, J. Comput. Appl. Math., 164-165 (2004), pp. 613-627.
13. A. Rößler, Runge-Kutta Methods for the Numerical Solution of Stochastic Differential Equations, Ph.D. thesis, Darmstadt University of Technology, Shaker, Aachen, 2003.
14. A. Rößler, Stochastic Taylor expansions for the expectation of functionals of diffusion processes, Stochastic Anal. Appl., 22 (2004), pp. 1553-1576.
15. W. Rümelin, Numerical treatment of stochastic differential equations, SIAM J. Numer. Anal., 19 (1982), pp. 604-613.
16. D. Talay, Efficient numerical schemes for the approximation of expectations of functionals of the solution of an S.D.E., and applications, Filtering and Control of Random Processes, Lecture Notes in Control and Information Sciences, vol. 61, pp. 294-313, 1984.
17. A. Tocino and J. Vigo-Aguiar, Weak second order conditions for stochastic Runge-Kutta methods, SIAM J. Sci. Comput., 24 (2002), pp. 507-523.