论文部分内容阅读
This talk is concerned with mean square stability of the classical stochastic theta method and the so called split-step theta method for multidimensional stochastic systems.First,we consider linear autonomous systems.Under a sufficient and necessary condition for exponential mean square stability of exact solution,it is proved that the two classes of theta methods with θ>0.5 are exponentially mean square stable for all positive stepsizes and the methods with θ<0.5 are stable for some small stepsizes.Then,we study stability of the methods for nonlinear non-autonomous systems.Under a coupled condition on the drift and diffusion coefficients,it is proved that the split-step theta method with θ> 0.5 still unconditionally preserves the exponential mean square stability of the underlying systems but the stochastic theta method does have this property.Later,we consider stochastic differential equations with jumps and stochastic delay differential equations.Some similar results are derived.Finally,we further investigate the mean square dissipativity of the split-step theta method with θ>0.5 and prove that the method possesses a bounded absorbing set in mean square independent of initial data.