- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 350407, 19 pages
The Convergence and MS Stability of Exponential Euler Method for Semilinear Stochastic Differential Equations
Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
Received 23 May 2012; Accepted 4 June 2012
Academic Editor: Yeong-Cheng Liou
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.
- 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.
- P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, Germany, 1992.
- G. N. Milstein and M. Tretyakov, Stochastic Numerics for Mathematical Physics, Springer, Berlin, Germany, 2004.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- D. J. Higham, “An algorithmic introduction to numerical simulation of stochastic differential equations,” SIAM Review, vol. 43, no. 3, pp. 525–546, 2001.
- X. R. Mao, Stochastic Differential Equations and Application, Horwood, New York, NY, USA, 1997.