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Symmetry
Volume 14
Issue 11
10.3390/sym14112413
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Article

Convergence and Stability of a Split-Step Exponential Scheme Based on the Milstein Methods

by
Leila Torkzadeh
1,*,
Hassan Ranjbar
1,
Sanda Micula
2 and
Kazem Nouri
1
1
Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan P.O. Box 35195-363, Iran
2
Department of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania
*
Author to whom correspondence should be addressed.
Symmetry 2022, 14(11), 2413; https://doi.org/10.3390/sym14112413
Submission received: 18 October 2022 / Revised: 8 November 2022 / Accepted: 11 November 2022 / Published: 15 November 2022
(This article belongs to the Special Issue Recent Progress in Studies of Stability of Numerical Schemes)
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Abstract

We introduce two approaches by modifying split-step exponential schemes to study stochastic differential equations. Under the Lipschitz condition and linear-growth bounds, it is shown that our explicit schemes converge to the solution of the corresponding stochastic differential equations with the order 1.0 in the mean-square sense. The mean-square stability of our methods is investigated through some linear stochastic test systems. Additionally, asymptotic mean-square stability is analyzed for the two-dimensional system with symmetric and asymmetric coefficients and driven by two commutative noise terms. In particular, we prove that our methods are mean-square stable for any step-size. Finally, some numerical experiments are carried out to confirm the theoretical results.
Keywords: mean-square stability; stochastic differential equations; strong convergence; ODE solver; Milstein method; split-step schemes mean-square stability; stochastic differential equations; strong convergence; ODE solver; Milstein method; split-step schemes

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MDPI and ACS Style

Torkzadeh, L.; Ranjbar, H.; Micula, S.; Nouri, K. Convergence and Stability of a Split-Step Exponential Scheme Based on the Milstein Methods. Symmetry 2022, 14, 2413. https://doi.org/10.3390/sym14112413

AMA Style

Torkzadeh L, Ranjbar H, Micula S, Nouri K. Convergence and Stability of a Split-Step Exponential Scheme Based on the Milstein Methods. Symmetry. 2022; 14(11):2413. https://doi.org/10.3390/sym14112413

Chicago/Turabian Style

Torkzadeh, Leila, Hassan Ranjbar, Sanda Micula, and Kazem Nouri. 2022. "Convergence and Stability of a Split-Step Exponential Scheme Based on the Milstein Methods" Symmetry 14, no. 11: 2413. https://doi.org/10.3390/sym14112413

APA Style

Torkzadeh, L., Ranjbar, H., Micula, S., & Nouri, K. (2022). Convergence and Stability of a Split-Step Exponential Scheme Based on the Milstein Methods. Symmetry, 14(11), 2413. https://doi.org/10.3390/sym14112413

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