Two Implicit Runge-Kutta Methods for Stochastic Differential Equation

DOI: 10.4236/am.2012.310162   PDF   HTML     4,783 Downloads   7,735 Views   Citations

Abstract

In this paper, the Ito-Taylor expansion of stochastic differential equation is briefly introduced. The colored rooted tree theory is applied to derive strong order 1.0 implicit stochastic Runge-Kutta method(SRK). Two fully implicit schemes are presented and their stability qualities are discussed. And the numerical report illustrates the better numerical behavior.

Share and Cite:

F. Lu and Z. Wang, "Two Implicit Runge-Kutta Methods for Stochastic Differential Equation," Applied Mathematics, Vol. 3 No. 10, 2012, pp. 1103-1108. doi: 10.4236/am.2012.310162.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] K. Burrage and P. M. Burrage, “High Strong Order Explicit Runge-Kutta Methods for Stochastic Ordinary Differential Equations,” Applied Numerical Mathematics, Vol. 22, 1996, pp. 81-101. HUdoi:10.1016/S0168-9274(96)00027-XU
[2] P. M. Burrage, “Runge-Kutta Methods for Stochastic Differential Equations,” Ph.D. Thesis, The University of Queensland, Queensland, 1999.
[3] K. Burrage and P. M. Burrage, “Order Condition of Stochastic Runge-Kutta Methods by B-Series,” SIAM Journal on Numerical Analysis, Vol. 38, No. 5, 2000, pp. 1626-1646. HUdoi:10.1137/S0036142999363206U
[4] T. H. Tian, “Implicit Numerical Methods for Stiff Stochastic Differential Equations and Numerical Simulations of Stochasic Models,” Ph.D. Thesis, The University of Queensland, Queensland, 2001.
[5] T. H. Tian and K. Burrage, “Two Stage Runge-Kutta Methods for Stochastic Differential Equations,” BIT, Vol. 42, No. 3, 2002, pp. 625-643. HUdoi:10.1023/A:1021963316988U
[6] P. Wang, “Three-Stage Stochastic Runge-Kutta Methods for Stochastic Differential Equaitons,” Journal of Computational and Applied Mathematics, Vol. 222, No. 2, 2008, pp. 324-332. HUdoi:10.1016/j.cam.2007.11.001U
[7] Z. Y. Wang, “The Stable Study of Stochastic Functional Differential Equation,” Ph.D. Theis, Huazhong University of Science and Technology, Wuhan, 2008
[8] P. E. Kloeden and E. Platen, “Numerical Solution of Stochastic Differential Equations,” Springer-Verlag, Belin, 1992.
[9] Y. Saito and T. Mitsui, “Stability Analysis of Numerical Schemes for Stochastic Differential Equations,” SIAM Journal on Numerical Analysis, Vol. 33, No. 6, 1996, pp. 2254-2267. HUdoi:10.1137/S0036142992228409U

  
comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.