New S-ROCK methods for stochastic differential equations with commutative noise

Document Type : Research Article

Author

• A. Haghighi

Razi University, Kermanshah, Iran.

10.22067/ijnao.v9i1.69454

Abstract

The class of strong stochastic Runge–Kutta (SRK) methods for stochas tic differential equations with a commutative noise proposed by R¨ oßler (2010) is considered. Motivated by Komori and Burrage (2013), we design a class of explicit stochastic orthogonal Runge–Kutta Chebyshev (SROCKC2) meth ods of strong order one for the approximation of the solution of Itˆo SDEs with an m-dimensional commutative noise.The mean-square and asymptotic stability analysis of the newly proposed methods applied to a scalar linear test equation with a multiplicative noise is presented. Finally, some numer ical experiments for stochastic models arising in applications are given that confirm the theoretical discussion.

Keywords

• Stochastic differential equations
• Runge-Kutta methods
• Stochastic mean square stability
• Stiff equations
• Commutative noise

20.1001.1.24236977.2019.9.1.6.9

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References

1. Abdulle, A. and Cirilli, S. S-ROCK: Chebyshev methods for stiff stochastic differential equations, SIAM J. Sci. Comput. 30(2) (2008), 997–1014.

2. Abdulle, A. and Li, T. S-ROCK methods for stiff Itˆo SDEs, Commun. Math. Sci. 6(4) (2008), 845–868.

3. Abdulle, A. and Medovikov, A. Second order Chebyshev methods based on orthogonal polynomials, Numer. Math. 90 (2001), 1–18.

4. Burrage, K., Burrage, P. and Tian, T. Numerical methods for strong solutions of stochastic differential equations: an overview, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460(2041) (2004), 373–402.

5. Burrage, K. and 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 (1998), 161–177.

6. Falsone, G. Stochastic differential calculus for Gaussian and non-Gaussian noises: A critical review, Commum. Nonlinear. Sci. 56 (2018), 198–216.

7. Haghighi, A. and Hosseini, S.M. A class of split-step balanced methods for stiff stochastic differential equations, Numer. Algorithms. 61 (2012), 141–162.

8. Haghighi, A., Hosseini, S.M. and R¨ oßler, A. Diagonally drift-implicit Runge–Kutta methods of strong order one for stiff stochastic differential systems, J. Comput. Appl. Math. 293 (2016), 82–93.

9. Higham, D. Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal. 38(3) (2000), 753–769.

10. Holden, H., Øksendal, B., Ubøe, J., and Zhang, T. Stochastic partial differential equations, In Stochastic Partial Differential Equations, pp. 141–191. Birkh¨auser Boston, 1996.

11. Ilie, S., Jackson, K.R. and Enright, W.H. Adaptive time-stepping for the strong numerical solution of stochastic differential equations, Numer Algor. 68 (2015), 791–812.

12. Kloeden, P. and Platen, E. Numerical solution of stochastic differential equations, Vol. 23, Springer-Verlag, Berlin, 1999.

13. Komori, Y. and Burrage, K. A bound on the maximum strong order of stochastic Runge–Kutta methods for stochastic ordinary differential equations, BIT 37(4) (1997), 771–780.

14. Komori, Y. and Burrage, K. Strong first order S-ROCK methods for stochastic differential equations, J. Comput. Appl. Math. 242 (2013), 261–274.

15. Komori, Y., Mitsui, T. and Sugiura, H. Rooted tree analysis of the order conditions of row-type scheme for stochastic differential equations, BIT 37(1) (1997), 43–66.

16. Milstein, G.N. and Tretyakov, M.V. Stochastic numerics for mathematical physics, Scientific Computing, Springer-Verlag, Berlin and New York, 2004.

17. Namjoo, M. Strong approximation for Itˆo stochastic differential equations, Iranian Journal of Numerical Analysis and Optimization 5(1) (2015), 1–12.

18. Riha, W. Optimal stability polynomials, Computing 9 (1972), 37–43.

19. R¨ oßler, A. Second order Runge–Kutta methods for Itˆo stochastic differential equations, SIAM J. Numer. Anal. 47(3) (2009), 1713–1738.

20. R¨ oßler, A. Runge–Kutta methods for the strong approximation of solutions of stochastic differential equations, SIAM J. Numer. Anal. 48(3) (2010), 922–952.

21. Saito, Y. and Mitsui, T. T-stability of numerical scheme for stochastic differential equations, World Sci. Ser. Appl. Anal. 2 (1993), 333–343.

22. Sobczyk, K. Stochastic differential equations: with applications to physics and engineering, Vol. 40. Springer Science and Business Media, 2013.

23. Soheili, A.R. and Soleymani, F. Iterative methods for nonlinear systems associated with finite difference approach in stochastic differential equations, Numer Algor 71(1) (2016), 89–102.

24. Soleymani, F. and Soheili, A.R. A revisit of stochastic theta method with some improvements, Filomat. 31(3) (2017), 585–596.