Asymptotic mean-square stability of two-step Maruyama schemes for stochastic differential equations

The mean-square stability for two-step schemes applied to scalar stochastic differential equations is studied. Necessary and sufficient conditions in terms of the parameters of the schemes guaranteeing their MS-stability are derived. Particular members of the studied family are considered, their stability regions are plotted and compared with the stability region of the linear test equation. It is proved that the stochastic two-step BDF scheme is unconditionally MS-stable. Numerical experiments that confirm the theoretical results are shown.