The Convergence and MS Stability of Exponential Euler Method for Semilinear Stochastic Differential Equations
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 or...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/350407 |
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| author | Chunmei Shi Yu Xiao Chiping Zhang |
| author_facet | Chunmei Shi Yu Xiao Chiping Zhang |
| author_sort | Chunmei Shi |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-bd84f988ef5143c49967df6cdc1229b3 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-bd84f988ef5143c49967df6cdc1229b32025-02-03T07:25:54ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/350407350407The Convergence and MS Stability of Exponential Euler Method for Semilinear Stochastic Differential EquationsChunmei Shi0Yu Xiao1Chiping Zhang2Department of Mathematics, Harbin Institute of Technology, Harbin 150001, ChinaDepartment of Mathematics, Harbin Institute of Technology, Harbin 150001, ChinaDepartment of Mathematics, Harbin Institute of Technology, Harbin 150001, ChinaThe 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.http://dx.doi.org/10.1155/2012/350407 |
| spellingShingle | Chunmei Shi Yu Xiao Chiping Zhang The Convergence and MS Stability of Exponential Euler Method for Semilinear Stochastic Differential Equations Abstract and Applied Analysis |
| title | The Convergence and MS Stability of Exponential Euler Method for Semilinear Stochastic Differential Equations |
| title_full | The Convergence and MS Stability of Exponential Euler Method for Semilinear Stochastic Differential Equations |
| title_fullStr | The Convergence and MS Stability of Exponential Euler Method for Semilinear Stochastic Differential Equations |
| title_full_unstemmed | The Convergence and MS Stability of Exponential Euler Method for Semilinear Stochastic Differential Equations |
| title_short | The Convergence and MS Stability of Exponential Euler Method for Semilinear Stochastic Differential Equations |
| title_sort | convergence and ms stability of exponential euler method for semilinear stochastic differential equations |
| url | http://dx.doi.org/10.1155/2012/350407 |
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