Mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations. (English) Zbl 1291.65020
Summary: We first study the mean square stability of numerical methods for stochastic delay differential equations under a coupled condition on the drift and diffusion coefficients. This condition admits that the diffusion coefficient can be highly nonlinear, i.e., it does not necessarily satisfy a linear growth or global Lipschitz condition. It is proved that, for all positive stepsizes, the classical stochastic theta method with \({\theta} \geq 0.5\) is asymptotically mean square stable and the split-step theta method with \({\theta} > 0.5\) is exponentially mean square stable. Conditional stability results for the methods with \({\theta} < 0.5\) are also obtained under a stronger assumption. Finally, we further investigate the mean square dissipativity of the split-step theta method with \({\theta} > 0.5\) and prove that the method possesses a bounded absorbing set in mean square independent of initial data.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
Keywords:
stochastic delay differential equations; mean square stability; exponential stability; theta method; dissipativitySoftware:
RODASReferences:
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