Analysis of Stability for the Semi Implicit Scheme for SDEs with Polynomial Growth Condition

摘要

Numerical stability plays an important role in numerical analysis. The author analysis the numerical stability of the stochastic delay differential equations (SDDEs). Traditional stability theory for numerical methods applied to SDDEs requires a global Lipschitz assumption or one-side linear growth condition on the coefficients. In this paper we want to further relax the condition. Under polynomial growth condition, this paper shows that the semi implicit method can reproduce almost sure exponential stability of the exact solutions to the SDDEs. This improves the existing results considerably.