Stability of the drift-implicit and double-implicit Milstein schemes for nonlinear SDEs

This paper examines the stability of numerical solutions of nonlinear stochastic differential equations (SDEs) with non-global Lipschitz continuous coefficients. Two implicit Milstein schemes, called drift-implicit Milstein scheme and double-implicit Milstein scheme, are considered to simulate the underlying SDEs. It is proved that the schemes can preserve the stability and contractivity in mean square of the underlying systems.