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

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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)

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.

Suggested Citation

  • Halidias Nikolaos & Stamatiou Ioannis S., 2022. "A note on the asymptotic stability of the semi-discrete method for stochastic differential equations," Monte Carlo Methods and Applications, De Gruyter, vol. 28(1), pages 13-25, March.
    • Handle: RePEc:bpj:mcmeap:v:28:y:2022:i:1:p:13-25:n:5
    DOI: 10.1515/mcma-2022-2102
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    References listed on IDEAS

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    1. Nikolaos Halidias & Ioannis Stamatiou, 2015. "Approximating explicitly the mean reverting CEV process," Papers 1502.03018, arXiv.org, revised May 2015.
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