Copyright © 2012 Chunmei Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

1. K. Burrage, P. M. Burrage, and T. Tian, “Numerical methods for strong solutions of stochastic differential equations: an overview,” Proceedings of The Royal Society of London A, vol. 460, no. 2041, pp. 373–402, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
2. P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, Germany, 1992.
3. G. N. Milstein and M. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer, Berlin, Germany, 2004.
4. D. J. Higham, X. Mao, and C. Yuan, “Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations,” SIAM Journal on Numerical Analysis, vol. 41, no. 2, pp. 592–609, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
5. S. Pang, F. Deng, and X. R. Mao, “Almost sure and moment exponential stability of Euler-Maruyama discretizations for hybrid stochastic differential equations,” Journal of Computational and Applied Mathematics, vol. 213, no. 1, pp. 127–141, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
6. W. R. Cao, M. Z. Liu, and Z. C. Fan, “MS-stability of the Euler-Maruyama method for stochastic differential delay equations,” Applied Mathematics and Computation, vol. 159, no. 1, pp. 127–135, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
7. Y. Saito and T. Mitsui, “Mean-square stability of numerical schemes for stochastic differential systems,” Vietnam Journal of Mathematics, vol. 30, pp. 551–560, 2002. View at Zentralblatt MATH
8. E. Buckwar and C. Kelly, “Towards a systematic linear stability analysis of numerical methods for systems of stochastic differential equations,” SIAM Journal on Numerical Analysis, vol. 48, no. 1, pp. 298–321, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
9. D. J. Higham, “Mean-square and asymptotic stability of the stochastic theta method,” SIAM Journal on Numerical Analysis, vol. 38, no. 3, pp. 753–769, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
10. B. V. Minchev and W. M. Wright, “A review of exponential integrators for first order semi-linear problems,” Tech. Rep. 2/05, Department of Mathematics, NTNU, 2005.
11. J. D. Lawson, “Generalized Runge-Kutta processes for stable systems with large Lipschitz constants,” SIAM Journal on Numerical Analysis, vol. 4, pp. 372–380, 1967. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
12. A. Ostermann, M. Thalhammer, and W. Wright, “A class of explicit exponential general linear methods,” BIT Numerical Mathematics, vol. 46, no. 2, pp. 409–431, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
13. M. Hochbruck and A. Ostermann, “Explicit exponential Runge-Kutta methods for semilinear parabolic problems,” SIAM Journal on Numerical Analysis, vol. 43, no. 3, pp. 1069–1090, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
14. S. Maset and M. Zennaro, “Unconditional stability of explicit exponential Runge-Kutta methods for semi-linear ordinary differential equations,” Mathematics of Computation, vol. 78, no. 266, pp. 957–967, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
15. X. R. Mao, Y. Shen, and A. Gray, “Almost sure exponential of backward Euler-Maruyama discretization for differential equations,” Journal of Computational and Applied Mathematics, vol. 235, pp. 1213–1226, 2010.
16. Z. Y. Wang and C. J. Zhang, “An analysis of stability of Milstein method for stochastic differential equations with delay,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1445–1452, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
17. M. Kunze and J. Neerven, “Approximating the coefficients in semilinear stochastic partial differential equations,” Journal of Evolution Equations, vol. 11, no. 3, pp. 577–604, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
18. Y. Saito and T. Mitsui, “Stability analysis of numerical schemes for stochastic differential equations,” SIAM Journal on Numerical Analysis, vol. 33, no. 6, pp. 2254–2267, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
19. D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Review, vol. 43, no. 3, pp. 525–546, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
20. X. R. Mao, Stochastic Differential Equations and Application, Horwood, New York, NY, USA, 1997.