Abstract
In this paper, a split-step theta (SST) method is introduced and analyzed for nonlinear neutral stochastic differential delay equations (NSDDEs). The asymptotic mean square stability of the split-step theta (SST) method is considered for nonlinear neutral stochastic differential equations. It is proved that, under the one-sided Lipschitz condition and the linear growth condition, for all positive stepsizes, the split-step theta method with \( \theta \in (1/2,1] \) is asymptotically mean square stable. The stability for the method with \( \theta \in [0,1/2] \) is also obtained under a stronger assumption. It further studies the mean square dissipativity of the split-step theta method with \( \theta \in (1/2,1] \) and proves that the method possesses a bounded absorbing set in mean square independent of initial data.
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Acknowledgements
This work was supported by the Natural Science Foundation of Heilongjiang Province (A201418) and the Creative Talent Project Foundation of Heilongjiang Province Education Department (UNPYSCT-2015102).
Declare. The authors declare that there is no conflict of interests regarding the publication of this article.
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Yuan, H., Shen, J., Song, C. (2016). Mean Square Stability and Dissipativity of Split-Step Theta Method for Stochastic Delay Differential Equations with Poisson White Noise Excitations. In: Silhavy, R., Senkerik, R., Oplatkova, Z.K., Silhavy, P., Prokopova, Z. (eds) Automation Control Theory Perspectives in Intelligent Systems. CSOC 2016. Advances in Intelligent Systems and Computing, vol 466. Springer, Cham. https://doi.org/10.1007/978-3-319-33389-2_9
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DOI: https://doi.org/10.1007/978-3-319-33389-2_9
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