A note on the asymptotic stability of the semi-discrete method for stochastic differential equations

Halidias Nikolaos () and Stamatiou Ioannis S. ()
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Halidias Nikolaos: Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Mytilini, Greece
Stamatiou Ioannis S.: Department of Biomedical Sciences, University of West Attica, Athens, Greece

Monte Carlo Methods and Applications, 2022, vol. 28, issue 1, 13-25

Abstract: 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.

Keywords: Explicit numerical scheme; semi-discrete method; non-linear stochastic differential equations; asymptotic stability (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1515/mcma-2022-2102

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