Numerical contractivity preserving implicit balanced Milstein-type schemes for SDEs with non-global Lipschitz coefficients

Abstract

Stability analysis, which was investigated in this paper, is one of the main issues related to numerical analysis for stochastic dynamical systems (SDS) and has the same important significance as the convergence one. To this end, we introduced the concept of $ p $-th moment stability for the $ n $-dimensional nonlinear stochastic differential equations (SDEs). Specifically, if $ p = 2 $ and the $ p $-th moment stability constant $ \bar{K} < 0 $, we speak of strict mean square contractivity. The present paper put the emphasis on systematic analysis of the numerical mean square contractivity of two kinds of implicit balanced Milstein-type schemes, e.g., the drift implicit balanced Milstein (DIBM) scheme and the semi-implicit balanced Milstein (SIBM) scheme (or double-implicit balanced Milstein scheme), for SDEs with non-global Lipschitz coefficients. The requirement in this paper allowed the drift coefficient $ f(x) $ to satisfy a one-sided Lipschitz condition, while the diffusion coefficient $ g(x) $ and the diffusion function $ L^{1}g(x) $ are globally Lipschitz continuous, which includes the well-known stochastic Ginzburg Landau equation as an example. It was proved that both of the mentioned schemes can well preserve the numerical counterpart of the mean square contractivity of the underlying SDEs under appropriate conditions. These outcomes indicate under what conditions initial perturbations are under control and, thus, have no significant impact on numerical dynamic behavior during the numerical integration process. Finally, numerical experiments intuitively illustrated the theoretical results.

Keywords

• stochastic differential equations
• $ p $-th moment stability
• mean square contractivity
• drift-implicit balanced milstein scheme
• semi-implicit balanced milstein scheme
• initial perturbation