On Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations
A split-step theta (SST) method is introduced and used to solve the nonlinear neutral stochastic delay differential equations (NSDDEs). The mean square asymptotic stability of the split-step theta (SST) method for nonlinear neutral stochastic delay differential equations is studied. It is proved tha...
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| Format: | Article |
| Language: | English |
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Wiley
2016-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2016/7397941 |
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| author | Haiyan Yuan Jihong Shen Cheng Song |
| author_facet | Haiyan Yuan Jihong Shen Cheng Song |
| author_sort | Haiyan Yuan |
| collection | DOAJ |
| description | A split-step theta (SST) method is introduced and used to solve the nonlinear neutral stochastic delay differential equations (NSDDEs). The mean square asymptotic stability of the split-step theta (SST) method for nonlinear neutral stochastic delay differential equations is studied. It is proved that under the one-sided Lipschitz condition and the linear growth condition, the split-step theta method with θ∈(1/2,1] is asymptotically mean square stable for all positive step sizes, and the split-step theta method with θ∈[0,1/2] is asymptotically mean square stable for some step sizes. It is also proved in this paper that the split-step theta (SST) method possesses a bounded absorbing set which is independent of initial data, and the mean square dissipativity of this method is also proved. |
| format | Article |
| id | doaj-art-f6cf40579bb74b638c7fa64acab8a1e6 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-f6cf40579bb74b638c7fa64acab8a1e62025-02-03T01:10:32ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2016-01-01201610.1155/2016/73979417397941On Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential EquationsHaiyan Yuan0Jihong Shen1Cheng Song2College of Automation, Harbin Engineering University, Harbin 150001, ChinaCollege of Automation, Harbin Engineering University, Harbin 150001, ChinaSchool of Management, Harbin Institute of Technology, Harbin 150001, ChinaA split-step theta (SST) method is introduced and used to solve the nonlinear neutral stochastic delay differential equations (NSDDEs). The mean square asymptotic stability of the split-step theta (SST) method for nonlinear neutral stochastic delay differential equations is studied. It is proved that under the one-sided Lipschitz condition and the linear growth condition, the split-step theta method with θ∈(1/2,1] is asymptotically mean square stable for all positive step sizes, and the split-step theta method with θ∈[0,1/2] is asymptotically mean square stable for some step sizes. It is also proved in this paper that the split-step theta (SST) method possesses a bounded absorbing set which is independent of initial data, and the mean square dissipativity of this method is also proved.http://dx.doi.org/10.1155/2016/7397941 |
| spellingShingle | Haiyan Yuan Jihong Shen Cheng Song On Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations Discrete Dynamics in Nature and Society |
| title | On Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations |
| title_full | On Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations |
| title_fullStr | On Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations |
| title_full_unstemmed | On Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations |
| title_short | On Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations |
| title_sort | on mean square stability and dissipativity of split step theta method for nonlinear neutral stochastic delay differential equations |
| url | http://dx.doi.org/10.1155/2016/7397941 |
| work_keys_str_mv | AT haiyanyuan onmeansquarestabilityanddissipativityofsplitstepthetamethodfornonlinearneutralstochasticdelaydifferentialequations AT jihongshen onmeansquarestabilityanddissipativityofsplitstepthetamethodfornonlinearneutralstochasticdelaydifferentialequations AT chengsong onmeansquarestabilityanddissipativityofsplitstepthetamethodfornonlinearneutralstochasticdelaydifferentialequations |