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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 350407, 19 pages
http://dx.doi.org/10.1155/2012/350407
Review Article

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.

Abstract

The numerical approximation of exponential Euler method is constructed for semilinear stochastic differential equations (SDEs). The convergence and mean-square (MS) stability of exponential Euler method are investigated. It is proved that the exponential Euler method is convergent with the strong order 1 / 2 for semilinear SDEs. A mean-square linear stability analysis shows that the stability region of exponential Euler method contains that of EM method and stochastic Theta method ( 0 𝜃 < 1 ) and also contains that of the scale linear SDE, that is, exponential Euler method is analogue mean-square A-stable. Then the exponential stability of the exponential Euler method for scalar semi-linear SDEs is considered. Under the conditions that guarantee the analytic solution is exponentially stable in mean-square sense, the exponential Euler method can reproduce the mean-square exponential stability for any nonzero stepsize. Numerical experiments are given to verify the conclusions.