Second order Runge-Kutta methods for stratonovich stochastic differential equations

BIT Numerical Mathematics

Volumn 47, Issue 3, 2007, Pages 657-680

  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: 34548664171     PISSN: 00063835     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10543-007-0130-3     Document Type: Article

References (19)

1. K. Burrage and P. M. Burrage, High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Appl. Numer. Math., 22(1-3) (1996), pp. 81-101.
2. K. Burrage and P. M. Burrage, Order conditions of stochastic Runge-Kutta methods by B-series, SIAM J. Numer. Anal., 38(5) (2000), pp. 1626-1646.
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. P. E. Kloeden, E. Platen, and N. Hofmann, Extrapolation methods for the weak approximation of Itô diffusions, SIAM J. Numer. Anal., 32(5) (1995), pp. 1519-1534.
7. Y. Komori, Weak second-order stochastic Runge-Kutta methods for non-commutative stochastic differential equations, J. Comput. Appl. Math., in press, doi:10.1016/j.cam.2006.06.006.
8. Y. Komori, T. Mitsui, and H. Sugiura, Rooted tree analysis of the order conditions of ROW-type scheme for stochastic differential equations, BIT, 37(1) (1997), pp. 43-66.
9. G. N. Milstein and M. V. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer, Berlin, 2004.
10. K.-S. Moon, A. Szepessy, R. Tempone, and G. E. Zouraris, Convergence rates for adaptive weak approximation of stochastic differential equations, Stochastic Anal. Appl., 23 (2005), pp. 511-558.
11. N. J. Newton, Asymptotically efficient Runge-Kutta methods for a class of Itô and Stratonovich equations, SIAM J. Appl. Math., 51(2) (1991), pp. 542-567.
12. 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(1) (2006), pp. 97-134.
13. A. Rößler, Runge-Kutta methods for Itô stochastic differential equations with scalar noise, BIT, 46(1) (2006), pp. 97-110.
14. A. Rößler, Runge-Kutta methods for Stratonovich stochastic differential equation systems with commutative noise, J. Comput. Appl. Math., 164-165 (2004), pp. 613-627.
15. A. Rößler, Stochastic Taylor Expansions for the Expectation of Functionals of Diffusion Processes, Stochastic Anal. Appl., 22(6) (2004), pp. 1553-1576.
16. W. Rümelin, Numerical treatment of stochastic differential equations, SIAM J. Numer. Anal., 19(3) (1982), pp. 604-613.
17. D. Talay, Second-order discretization schemes of stochastic differential systems for the computation of the invariant law, Stochastics Stochastics Rep., 29(1) (1990), pp. 13-36.
18. D. Talay and L. Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations, Stochastic Anal. Appl., 8(4) (1990), pp. 483-509.
19. A. Tocino and J. Vigo-Aguiar, Weak second order conditions for stochastic Runge-Kutta methods, SIAM J. Sci. Comput., 24(2) (2002), pp. 507-523.