Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion
A class of drift-implicit one-step schemes are proposed for the neutral stochastic delay differential equations (NSDDEs) driven by Poisson processes. A general framework for mean-square convergence of the methods is provided. It is shown that under certain conditions global error estimates for a met...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2011/917892 |
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| _version_ | 1832548503583195136 |
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| author | Lin Hu Siqing Gan |
| author_facet | Lin Hu Siqing Gan |
| author_sort | Lin Hu |
| collection | DOAJ |
| description | A class of drift-implicit one-step schemes are proposed for the neutral stochastic delay differential equations (NSDDEs) driven by Poisson processes. A general framework for mean-square convergence of the methods is provided. It is shown that under certain conditions global error estimates for a method can be inferred from estimates on its local error. The applicability of the mean-square convergence theory is illustrated by the stochastic θ-methods and the balanced implicit methods. It is derived from Theorem 3.1 that the order of the mean-square convergence of both of them for NSDDEs with jumps is 1/2. Numerical experiments illustrate the theoretical results. It is worth noting that the results of mean-square convergence of the stochastic θ-methods and the balanced implicit methods are also new. |
| format | Article |
| id | doaj-art-ea4b9af4286d4ebfb4d08d63c4220eb2 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-ea4b9af4286d4ebfb4d08d63c4220eb22025-02-03T06:13:59ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2011-01-01201110.1155/2011/917892917892Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump DiffusionLin Hu0Siqing Gan1School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410075, ChinaSchool of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410075, ChinaA class of drift-implicit one-step schemes are proposed for the neutral stochastic delay differential equations (NSDDEs) driven by Poisson processes. A general framework for mean-square convergence of the methods is provided. It is shown that under certain conditions global error estimates for a method can be inferred from estimates on its local error. The applicability of the mean-square convergence theory is illustrated by the stochastic θ-methods and the balanced implicit methods. It is derived from Theorem 3.1 that the order of the mean-square convergence of both of them for NSDDEs with jumps is 1/2. Numerical experiments illustrate the theoretical results. It is worth noting that the results of mean-square convergence of the stochastic θ-methods and the balanced implicit methods are also new.http://dx.doi.org/10.1155/2011/917892 |
| spellingShingle | Lin Hu Siqing Gan Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion Discrete Dynamics in Nature and Society |
| title | Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion |
| title_full | Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion |
| title_fullStr | Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion |
| title_full_unstemmed | Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion |
| title_short | Mean-Square Convergence of Drift-Implicit One-Step Methods for Neutral Stochastic Delay Differential Equations with Jump Diffusion |
| title_sort | mean square convergence of drift implicit one step methods for neutral stochastic delay differential equations with jump diffusion |
| url | http://dx.doi.org/10.1155/2011/917892 |
| work_keys_str_mv | AT linhu meansquareconvergenceofdriftimplicitonestepmethodsforneutralstochasticdelaydifferentialequationswithjumpdiffusion AT siqinggan meansquareconvergenceofdriftimplicitonestepmethodsforneutralstochasticdelaydifferentialequationswithjumpdiffusion |