  In this article, we construct and analyse an explicit numerical splitting
method for a class of semi-linear stochastic differential equations (SDEs) with
additive noise, where the drift is allowed to grow polynomially and satisfies a
global one-sided Lipschitz condition. The method is proved to be mean-square
convergent of order 1 and to preserve important structural properties of the
SDE. First, it is hypoelliptic in every iteration step. Second, it is
geometrically ergodic and has an asymptotically bounded second moment. Third,
it preserves oscillatory dynamics, such as amplitudes, frequencies and phases
of oscillations, even for large time steps. Our results are illustrated on the
stochastic FitzHugh-Nagumo model and compared with known mean-square convergent
tamed/truncated variants of the Euler-Maruyama method. The capability of the
proposed splitting method to preserve the aforementioned properties may make it
applicable within different statistical inference procedures. In contrast,
known Euler-Maruyama type methods commonly fail in preserving such properties,
yielding ill-conditioned likelihood-based estimation tools or computationally
infeasible simulation-based inference algorithms.
