Abstract and Applied Analysis

The Convergence and MS Stability of Exponential Euler Method for Semilinear Stochastic Differential Equations

Chunmei Shi, Yu Xiao, and Chiping Zhang

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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.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 350407, 19 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364475813

Digital Object Identifier
doi:10.1155/2012/350407

Mathematical Reviews number (MathSciNet)
MR2965478

Zentralblatt MATH identifier
1253.65009

Citation

Shi, Chunmei; Xiao, Yu; Zhang, Chiping. The Convergence and MS Stability of Exponential Euler Method for Semilinear Stochastic Differential Equations. Abstr. Appl. Anal. 2012 (2012), Article ID 350407, 19 pages. doi:10.1155/2012/350407. https://projecteuclid.org/euclid.aaa/1364475813


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