Diagonally drift-implicit Runge-Kutta methods of weak order one and two for Itô SDEs and stability analysis

Applied Numerical Mathematics

Volumn 59, Issue 3-4, 2009, Pages 595-607

  Debrabant, Kristian ^a   Rößler, Andreas ^a  

^a DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Asymptotic stability; Implicit method; Mean square stability; Stochastic differential equation; Stochastic Runge Kutta method; Weak approximation

Indexed keywords

ASYMPTOTIC ANALYSIS; ASYMPTOTIC STABILITY; CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIAL EQUATIONS; DISPERSIONS; MEASUREMENT THEORY; NUMERICAL METHODS; POLYNOMIAL APPROXIMATION; STATISTICAL TESTS; STOCHASTIC CONTROL SYSTEMS;

IMPLICIT METHOD; MEAN-SQUARE STABILITY; STOCHASTIC DIFFERENTIAL EQUATION; STOCHASTIC RUNGE-KUTTA METHOD; WEAK APPROXIMATION;

RUNGE KUTTA METHODS;

EID: 58249086593     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.apnum.2008.03.011     Document Type: Article

References (27)

1. Burrage K., Burrage P.M., and Mitsui T. Numerical solutions of stochastic differential equations - implementation and stability issues. J. Comput. Appl. Math. 125 (2000) 171-182
2. Burrage K., and Tian T. Implicit stochastic Runge-Kutta methods for stochastic differential equations. BIT 44 (2004) 21-39
3. Debrabant K., and Kværnø A. B-Series analysis of stochastic Runge-Kutta methods that use an iterative scheme to compute their internal stage values. SIAM J. Numer. Anal. 47 1 (2008) 181-203
4. Debrabant K., and Rößler A. Families of efficient second order Runge-Kutta methods for the weak approximation of Itô stochastic differential equations. Appl. Numer. Math. 59 3-4 (2009) 582-594
5. Hairer E., and Wanner G. Solving Ordinary Differential Equations II (1996), Springer-Verlag, Berlin
6. Has'minskii R.Z. Stochastic Stability of Differential Equations (1980), Sijthoff & Noordhoff, Alphen aan den Rijn, Germantown
7. Hernandez D.B., and Spigler R. A-stability of Runge-Kutta methods for systems with additive noise. BIT 32 4 (1992) 620-633
8. Hernandez D.B., and Spigler R. Convergence and stability of implicit Runge-Kutta methods for systems with multiplicative noise. BIT 33 4 (1993) 654-669
9. Higham D.J. Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal. 38 3 (2000) 753-769
10. Hofmann N., and Platen E. Stability of weak numerical schemes for stochastic differential equations. Comput. Math. Appl. 28 10-12 (1994) 45-57
11. Kloeden P.E., and Platen E. Numerical Solution of Stochastic Differential Equations. Applications of Mathematics vol. 23 (1999), Springer-Verlag, Berlin
12. Y. Komori, Weak first- or second-order implicit Runge-Kutta methods for stochastic differential equations with a scalar Wiener process, J. Comput. Appl. Math. (2007), doi:10.1016/j.cam.2007.06.024
13. Komori Y., and Mitsui T. Stable ROW-type weak scheme for stochastic differential equations. Monte Carlo Methods Appl. 1 4 (1995) 279-300
14. Milstein G.N., Platen E., and Schurz H. Balanced implicit methods for stiff stochastic systems. SIAM J. Numer. Anal. 35 3 (1998) 1010-1019
15. Milstein G.N., and Tretyakov M.V. Stochastic Numerics for Mathematical Physics. Scientific Computation (2004), Springer-Verlag, Berlin
16. Peterson W.P. A general implicit splitting for stabilizing numerical simulations of Itô stochastic differential equations. SIAM J. Numer. Anal. 35 (1998) 1439-1451
17. 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 (2006) 97-134
18. Rößler A. Runge-Kutta methods for Itô stochastic differential equations with scalar noise. BIT 46 1 (2006) 97-110
19. Rößler A. Runge-Kutta methods for Stratonovich stochastic differential equation systems with commutative noise. J. Comput. Appl. Math. 164-165 (2004) 613-627
20. A. Rößler, Second order Runge-Kutta methods for Itô stochastic differential equations, Preprint No. 2479, Technische Universität Darmstadt, 2006
21. Rößler A. Second order Runge-Kutta methods for Stratonovich stochastic differential equations. BIT 47 3 (2007) 657-680
22. Saito Y., and Mitsui T. T-stability of numerical scheme for stochastic differential equations. World Sci. Ser. Appl. Anal. 2 (1993) 333-344
23. Saito Y., and Mitsui T. Stability analysis of numerical schemes for stochastic differential equations. SIAM J. Numer. Anal. 33 6 (1996) 2254-2267
24. Schurz H. Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise. Stochastic Anal. Appl. 14 (1996) 313-354
25. Schurz H. The invariance of asymptotic laws of linear stochastic systems under discretization. Z. Angew. Math. Mech. 79 6 (1999) 375-382
26. Tocino A. Mean-square stability of second-order Runge-Kutta methods for stochastic differential equations. J. Comput. Appl. Math. 175 (2005) 355-367
27. Tocino A., and Vigo-Aguiar J. Weak second order conditions for stochastic Runge-Kutta methods. SIAM J. Sci. Comput. 24 2 (2002) 507-523