Numerical contractivity of split-step backward Milstein-type schemes for commutative SDEs with non-globally Lipschitz continuous coefficients

This work investigates the mean-square contractivity of two types of split-step backward Milstein schemes for commutative stochastic differential equations (SDEs) with non-globally Lipschitz continuous coefficients. Our setting allows the drift coefficient to satisfy a onesided Lipschitz condition and the diffusion coefficient to satisfy a global Lipschitz condition, thereby including well-known examples such as the stochastic Ginzburg-Landau equation and the stochastic Verhulst equation. Our results demonstrate that both of the numerical schemes considered can accurately reproduce the mean-square contractivity of the nonlinear SDEs mentioned. Finally, some numerical experiments are performed to illustrate the validity of the theoretical results.