A note on the asymptotic stability of the Semi-Discrete method for Stochastic Differential Equations

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.