  We study the asymptotic stability of the semi-discrete (SD) numerical method
for the approximation of stochastic differential equations. Recently, we
examined the order of $\mathcal L^2$-convergence of the truncated SD method and
showed that it can be arbitrarily close to $1/2,$ see \textit{Stamatiou,
Halidias (2019), Convergence rates of the Semi-Discrete method for stochastic
differential equations, Theory of Stochastic Processes, 24(40)}. We show that
the truncated SD method is able to preserve the asymptotic stability of the
underlying SDE. Motivated by a numerical example, we also propose a different
SD scheme, using the Lamperti transformation to the original SDE, which we call
Lamperti semi-discrete (LSD). Numerical simulations support our theoretical
findings.
