Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps
This paper is concerned with the stability of analytical and numerical solutions for nonlinear stochastic delay differential equations (SDDEs) with jumps. A sufficient condition for mean-square exponential stability of the exact solution is derived. Then, mean-square stability of the numerical solut...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/831082 |
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| _version_ | 1832565056612597760 |
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| author | Qiyong Li Siqing Gan |
| author_facet | Qiyong Li Siqing Gan |
| author_sort | Qiyong Li |
| collection | DOAJ |
| description | This paper is concerned with the stability of analytical and numerical solutions for nonlinear stochastic delay differential equations (SDDEs) with jumps. A sufficient condition for mean-square exponential stability of the exact solution is derived. Then, mean-square stability of the numerical solution is investigated. It is shown that the compensated stochastic θ methods inherit stability property of the exact solution. More precisely, the methods are mean-square stable for any stepsize Δt=τ/m when 1/2≤θ≤1, and they are exponentially mean-square stable if the stepsize Δt∈(0,Δt0) when 0≤θ<1. Finally, some numerical experiments are given to illustrate the theoretical results. |
| format | Article |
| id | doaj-art-c44887d8c4d94a36adeeab0f3740cc25 |
| 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-c44887d8c4d94a36adeeab0f3740cc252025-02-03T01:09:33ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/831082831082Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with JumpsQiyong Li0Siqing 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, ChinaThis paper is concerned with the stability of analytical and numerical solutions for nonlinear stochastic delay differential equations (SDDEs) with jumps. A sufficient condition for mean-square exponential stability of the exact solution is derived. Then, mean-square stability of the numerical solution is investigated. It is shown that the compensated stochastic θ methods inherit stability property of the exact solution. More precisely, the methods are mean-square stable for any stepsize Δt=τ/m when 1/2≤θ≤1, and they are exponentially mean-square stable if the stepsize Δt∈(0,Δt0) when 0≤θ<1. Finally, some numerical experiments are given to illustrate the theoretical results.http://dx.doi.org/10.1155/2012/831082 |
| spellingShingle | Qiyong Li Siqing Gan Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps Abstract and Applied Analysis |
| title | Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps |
| title_full | Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps |
| title_fullStr | Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps |
| title_full_unstemmed | Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps |
| title_short | Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps |
| title_sort | stability of analytical and numerical solutions for nonlinear stochastic delay differential equations with jumps |
| url | http://dx.doi.org/10.1155/2012/831082 |
| work_keys_str_mv | AT qiyongli stabilityofanalyticalandnumericalsolutionsfornonlinearstochasticdelaydifferentialequationswithjumps AT siqinggan stabilityofanalyticalandnumericalsolutionsfornonlinearstochasticdelaydifferentialequationswithjumps |