Equivalent preservation of asymptotic mean square stability and instability of balanced midpoint Milstein methods (BMMMs) applied to stochastic differential equations (SDEs) driven by standard Wiener processes is shown whenever the underlying SDE has an asymptotically mean square stable equilibrium or not in \(\mathbb {C}^1\) , respectively (called mean square E-stability by this article). These are certain numerical methods built up by the class of implicit Milstein methods combined with midpoint drift-implicitness and additional balanced terms. The paper verifies that it is indeed possible to construct such higher order numerical methods for SDEs, which are asymptotically mean square stable for all possible step sizes \(h>0\) if and only if the standard test class of bi-linear SDEs (the stochastic 1D test equation in \(\mathbb {C}^1\) in the sense of Dahlquist) with multiplicative noise has an asymptotically mean square stable trivial solution. This investigation goes far beyond the common requirement of mean square A-stability of stochastic-numerical methods. Previously, it has been shown that this is the case for the drift-implicit midpoint method, which represents a lower mean square order stochastic Theta method for SDEs with parameter \(\theta =0.5\) .
