Runge-Kutta methods for the strong approximation of solutions of stochastic differential equations

SIAM Journal on Numerical Analysis

Volumn 48, Issue 3, 2010, Pages 922-952

  Rößler, Andreas ^a  

^a DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Additive noise; Commutative noise; Diagonal noise; Multicolored rooted tree analysis; Numerical method; Stochastic differential equation; Stochastic Runge Kutta method; Strong approximation

Indexed keywords

COMMUTATIVE NOISE; DIAGONAL NOISE; ROOTED TREE ANALYSIS; STOCHASTIC DIFFERENTIAL EQUATIONS; STRONG APPROXIMATION;

ADDITIVE NOISE; COMPUTATIONAL COMPLEXITY; DIFFERENTIAL EQUATIONS; DIFFERENTIATION (CALCULUS); MEASUREMENT THEORY; NUMBER THEORY; NUMERICAL METHODS; STOCHASTIC SYSTEMS; TREES (MATHEMATICS);

RUNGE KUTTA METHODS;

EID: 77956044362     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/09076636X     Document Type: Article

References (30)

1. K. BURRAGE AND P. M. BURRAGE, High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations, Appl. Numer. Math., 22 (1996), pp. 81-101.
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. K. BURRAGE AND P. M. BURRAGE, Order conditions of stochastic Runge-Kutta methods by B-series, SIAM J. Numer. Anal., 38 (2000), pp. 1626-1646.
4. P. M. BURRAGE, Runge-Kutta Methods for Stochastic Differential Equations, Ph.D. thesis, Dept. of Math., University of Queensland, Australia, 1999.
5. J. M. C. CLARK AND R. J. CAMERON, The maximum rate of convergence of discrete approximations for stochastic differential equations, in Stochastic Differential Systems, Proceedings of the IFIP-WG 7/1 Working Conference, Lecture Notes in Control Inform. Sci. 25, Springer, Berlin, 1980, pp. 162-171.
6. K. DEBRABANT AND A. KVÅRNØ, B-series analysis of stochastic Runge-Kutta methods that use an iterative scheme to compute their internal stage values, SIAM J. Numer. Anal., 47 (2008), pp. 181-203.
7. J. G. GAINES AND T. J. LYONS, Random generation of stochastic area integrals, SIAM J. Appl. Math., 54 (1994), pp. 1132-1146.
8. M. B. GILES, Improved multilevel Monte Carlo convergence using the Milstein scheme, in Monte Carlo and Quasi-Monte Carlo Methods 2006, Springer, Berlin, 2008, pp. 343-358.
9. H. GILSING AND T. SHARDLOW, SDELab: A package for solving stochastic differential equations in MATLAB, J. Comput. Appl. Math., 205 (2007), pp. 1002-1018.
10. E. HAIRER, S. P. NØRSETT, AND G. WANNER, Solving Ordinary Differential Equations I, Springer, Berlin, 1993.
11. J. KANEKO, Explicit order 1.5 Runge-Kutta scheme for solutions of Itô stochastic differential equations, in Various Problems in Stochastic Numerical Analysis II (Kyoto), Sūrikaiseki-kenkyūsho Kōkyūroku 932, Kyoto University Research Institute for Mathematical Sciences, Kyoto, Japan, 1995, pp. 46-60.
12. P. E. KLOEDEN AND E. PLATEN, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1999.
13. P. E. KLOEDEN, E. PLATEN, AND I. W. WRIGHT, The approximation of multiple stochastic integrals, Stoch. Anal. Appl., 10 (1992), pp. 431-441.
14. Y. KOMORI, Multi-colored rooted tree analysis of the weak order conditions of a stochastic Runge-Kutta family, Appl. Numer. Math., 57 (2007), pp. 147-165.
15. Y. KOMORI, Weak second-order stochastic Runge-Kutta methods for non-commutative stochastic differential equations, J. Comput. Appl. Math., 206 (2007), pp. 158-173. (Pubitemid 46817751)
16. Y. KOMORI, Weak order stochastic Runge-Kutta methods for commutative stochastic differential equations, J. Comput. Appl. Math., 203 (2007), pp. 57-79. (Pubitemid 46349467)
17. Y. KOMORI, T. MITSUI, AND H. SUGIURA, Rooted tree analysis of the order conditions of ROW-type scheme for stochastic differential equations, BIT, 37 (1997), pp. 43-66.
18. D. KÜPPER, J. LEHN, AND A. RÖSSLER, A step size control algorithm for the weak approximation of stochastic differential equations, Numer. Algorithms, 44 (2007), pp. 335-346.
19. G. N. MILSTEIN, Numerical Integration of Stochastic Differential Equations, Kluwer Academic, Dordrecht, The Netherlands, 1995.
20. G. N. MILSTEIN AND M. V. TRETYAKOV, Stochastic Numerics for Mathematical Physics, Springer, Berlin, 2004.
21. 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.
22. A. RÖSSLER, Stochastic Taylor expansions for the expectation of junctionals of diffusion processes, Stoch. Anal. Appl., 22 (2004), pp. 1553-1576.
23. A. RÖSSLER, Rooted tree analysis for order conditions of stochastic Runge-Kutta methods for the weak approximation of stochastic differential equations, Stoch. Anal. Appl., 24 (2006), pp. 97-134.
24. A. RÖSSLER, Second order Runge-Kutta methods for Stratonovich stochastic differential equations, BIT, 47 (2007), pp. 657-680.
25. A. RÖSSLER, Stochastic Taylor expansions for junctionals of diffusion processes, Stoch. Anal. Appl., 28 (2010), pp. 415-429.
26. A. RÖSSLER, Second order Runge-Kutta methods for Itô stochastic differential equations, SIAM J. Numer. Anal., 47 (2009), pp. 1713-1738.
27. W. RÜMELIN, Numerical treatment of stochastic differential equations, SIAM J. Numer. Anal., 19 (1982), pp. 604-613.
28. D. TALAY, Second-order discretization schemes of stochastic differential systems for the computation of the invariant law, Stoch. Stoch. Rep., 29 (1990), pp. 13-36.
29. A. TOCINO AND J. VIGO-AGUIAR, Weak second order conditions for stochastic Runge-Kutta methods, SIAM J. Sci. Comput., 24 (2002), pp. 507-523.
30. M. WIKTORSSON, Joint characteristic function and simultaneous simulation of iterated Itô integrals for multiple independent Brownian motions, Ann. Appl. Probab., 11 (2001), pp. 470-487.