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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 ℒ 2 {\mathcal{L}^{2}} -convergence of the truncated SD method and showed that it can be arbitrarily close to 1 2 {\frac{1}{2}} ; see [I. S. Stamatiou and N. Halidias, Convergence rates of the semi-discrete method for stochastic differential equations, Theory Stoch. Process. 24 2019, 2, 89–100]. 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. Numerical simulations support our theoretical findings. [ABSTRACT FROM AUTHOR]

We study the two-factor CIR model and using the main idea of two of our previous papers, we propose explicit numerical schemes that preserves positivity and converges strongly in the mean square sense to the true solution. Our results apply for the multi-factor and the one-factor cases as well. [ABSTRACT FROM AUTHOR]