  In this article, we construct and analyse explicit numerical splitting
methods 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 methods are proved to be
mean-square convergent of order 1 and to preserve important structural
properties of the SDE. In particular, first, they are hypoelliptic in every
iteration step. Second, they are geometrically ergodic and have asymptotically
bounded second moments. Third, they preserve 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 methods to
preserve the aforementioned properties makes them 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.
