Mean Square Stability and Dissipativity of Split-Step Theta Method for Stochastic Delay Differential Equations with Poisson White Noise Excitations
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.
KeywordsSplit-step theta method Nonlinear neutral stochastic differential delay equations Mean square stability Dissipativity
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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