scholarly journals Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Qiyong Li ◽  
Siqing Gan
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Mean Square ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Analytical And Numerical Solutions ◽  
Mean Square Stability ◽  
Delay Differential

This paper is concerned with the stability of analytical and numerical solutions fornonlinearstochastic 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 stepsizeΔt=τ/mwhen1/2≤θ≤1, and they are exponentially mean-square stable if the stepsizeΔt∈(0,Δt0)when0≤θ<1. Finally, some numerical experiments are given to illustrate the theoretical results.

scholarly journals On Mean Square Stability and Dissipativity of Split-Step Theta Method for Nonlinear Neutral Stochastic Delay Differential Equations

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Haiyan Yuan ◽  
Jihong Shen ◽  
Cheng Song
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Mean Square ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Mean Square Stability ◽  
Linear Growth Condition ◽  
The Mean ◽  
Delay Differential ◽  
Theta Method

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 withθ∈(1/2,1]is asymptotically mean square stable for all positive step sizes, and the split-step theta method withθ∈[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.

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