Research ArticleOpen AccessGoogle Scholar indexed
Two Implicit Runge-Kutta Methods for Stochastic Differential Equation
Department of Mathematics, University of electronic Science and Technology of China, Chengdu Sichuan, China
Department of Mathematics, University of electronic Science and Technology of China, Chengdu Sichuan, China
- 1 Department of Mathematics, University of electronic Science and Technology of China, Chengdu Sichuan, China
- 2 Department of Mathematics, University of electronic Science and Technology of China, Chengdu Sichuan, China
Applied Mathematics·Volume 03 (2012)·Pages 1103–1108·Published 11 October 2012·DOI10.4236/am.2012.310162
Copy link · social · email
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.
KeywordsStochastic Differential EquationImplicit Stochastic Runge-Kutta MethodOrder Condition
- 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
- P. M. Burrage, “Runge-Kutta Methods for Stochastic Differential Equations,” Ph.D. Thesis, The University of Queensland, Queensland, 1999.
- 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
- 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.
- 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
- 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
- Z. Y. Wang, “The Stable Study of Stochastic Functional Differential Equation,” Ph.D. Theis, Huazhong University of Science and Technology, Wuhan, 2008
- P. E. Kloeden and E. Platen, “Numerical Solution of Stochastic Differential Equations,” Springer-Verlag, Belin, 1992.
- 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