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

08/06/2020

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by   Nikolaos Halidias, et al.

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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-convergence of the truncated SD method and showed that it can be arbitrarily close to 1/2, see 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.