随机延迟微分方程SK-ROCK方法的延迟相关稳定性分析
Delay Dependent Stability Analysis of SK-ROCK Methods for Stochastic Delay Differential Equations
DOI: 10.12677/aam.2025.144190, PDF,    国家自然科学基金支持
作者: 邢 娟*：青岛大学数学与统计学院，山东 青岛；丁洁玉#：青岛大学计算机科学技术学院，山东 青岛；Alexey S. Eremin：圣彼得堡国立大学信息系统系，俄罗斯 圣彼得堡
关键词: 随机延迟微分方程；显式稳定方法；渐近均方稳定性；延迟相关稳定性；SK-ROCK方法；Stochastic Delay Differential Equation； Explicit Stabilized Methods； Asymptotic Mean Square Stability； Delay Dependent Stability； SK-ROCK Methods
摘要: 本文调整了显式稳定方法，即随机第二类正交Runge-Kutta-Chebyshev (SK-ROCK)方法，用于求解Itô随机延迟微分方程(SDDEs)。SK-ROCK方法实现了沿负实轴的扩展均方稳定区域，解决了现有显式方法在处理刚度和延迟相互作用方面的局限性。通过采用根定位技术，我们严格推导出了所提方法依赖于延迟的渐近均方稳定性条件。数值实验验证了SK-ROCK的收敛阶次和增强稳定性能，证明了它在实际应用中的有效性。
Abstract: This paper adapts the explicit stabilized method, termed the Stochastic Second kind Orthogonal Runge-Kutta-Chebyshev (SK-ROCK) method, for solving Itô stochastic delay differential equations (SDDEs). The SK-ROCK method achieves an extended mean-square stability region along the negative real axis, addressing the limitations of existing explicit methods in handling stiffness and delayed interactions. By employing root locus techniques, we rigorously derive the delay-dependent asymptotic mean-square stability conditions of the proposed method. Numerical experiments validate both the convergence order and enhanced stability performance of SK-ROCK, demonstrating its efficacy in practical applications.
文章引用：邢娟, 丁洁玉, Alexey S. Eremin. 随机延迟微分方程SK-ROCK方法的延迟相关稳定性分析[J]. 应用数学进展, 2025, 14(4): 601-613. https://doi.org/10.12677/aam.2025.144190

参考文献

[1] Carletti, M. (2006) Numerical Solution of Stochastic Differential Problems in the Biosciences. Journal of Computational and Applied Mathematics, 185, 422-440. [Google Scholar] [CrossRef] 
[2] Mao, X., Yuan, C. and Zou, J. (2005) Stochastic Differential Delay Equations of Population Dynamics. Journal of Mathematical Analysis and Applications, 304, 296-320. [Google Scholar] [CrossRef] 
[3] Tian, T., Burrage, K., Burrage, P.M. and Carletti, M. (2007) Stochastic Delay Differential Equations for Genetic Regulatory Networks. Journal of Computational and Applied Mathematics, 205, 696-707. [Google Scholar] [CrossRef] 
[4] Baker, C.T.H. and Buckwar, E. (2000) Numerical Analysis of Explicit One-Step Methods for Stochastic Delay Differential Equations. LMS Journal of Computation and Mathematics, 3, 315-335. [Google Scholar] [CrossRef] 
[5] Buckwar, E. (2000) Introduction to the Numerical Analysis of Stochastic Delay Differential Equations. Journal of Computational and Applied Mathematics, 125, 297-307. [Google Scholar] [CrossRef] 
[6] Mao, X. and Sabanis, S. (2003) Numerical Solutions of Stochastic Differential Delay Equations under Local Lipschitz Condition. Journal of Computational and Applied Mathematics, 151, 215-227. [Google Scholar] [CrossRef] 
[7] Hu, Y., Mohammed, S.A. and Yan, F. (2004) Discrete-Time Approximations of Stochastic Delay Equations: The Milstein Scheme. The Annals of Probability, 32, 265-314. [Google Scholar] [CrossRef] 
[8] Huang, C., Gan, S. and Wang, D. (2012) Delay-Dependent Stability Analysis of Numerical Methods for Stochastic Delay Differential Equations. Journal of Computational and Applied Mathematics, 236, 3514-3527. [Google Scholar] [CrossRef] 
[9] Hu, P. and Huang, C. (2021) Delay Dependent Asymptotic Mean Square Stability Analysis of the Stochastic Exponential Euler Method. Journal of Computational and Applied Mathematics, 382, Article ID: 113068. [Google Scholar] [CrossRef] 
[10] Abdulle, A. and Cirilli, S. (2008) S-ROCK: Chebyshev Methods for Stiff Stochastic Differential Equations. SIAM Journal on Scientific Computing, 30, 997-1014. [Google Scholar] [CrossRef] 
[11] Abdulle, A. and Li, T. (2008) S-ROCK Methods for Stiff Ito SDEs. Communications in Mathematical Sciences, 6, 845-868. [Google Scholar] [CrossRef] 
[12] Abdulle, A., Vilmart, G. and Zygalakis, K.C. (2013) Weak Second Order Explicit Stabilized Methods for Stiff Stochastic Differential Equations. SIAM Journal on Scientific Computing, 35, A1792-A1814. [Google Scholar] [CrossRef] 
[13] Komori, Y. and Burrage, K. (2012) Weak Second Order S-ROCK Methods for Stratonovich Stochastic Differential Equations. Journal of Computational and Applied Mathematics, 236, 2895-2908. [Google Scholar] [CrossRef] 
[14] Komori, Y. and Burrage, K. (2013) Strong First Order S-Rock Methods for Stochastic Differential Equations. Journal of Computational and Applied Mathematics, 242, 261-274. [Google Scholar] [CrossRef] 
[15] Guo, Q., Qiu, M. and Mitsui, T. (2016) Asymptotic Mean-Square Stability of Explicit Runge-Kutta Maruyama Methods for Stochastic Delay Differential Equations. Journal of Computational and Applied Mathematics, 296, 427-442. [Google Scholar] [CrossRef] 
[16] Komori, Y., Eremin, A. and Burrage, K. (2019) S-ROCK Methods for Stochastic Delay Differential Equations with One Fixed Delay. Journal of Computational and Applied Mathematics, 353, 345-354. [Google Scholar] [CrossRef] 
[17] Abdulle, A., Almuslimani, I. and Vilmart, G. (2018) Optimal Explicit Stabilized Integrator of Weak Order 1 for Stiff and Ergodic Stochastic Differential Equations. SIAM/ASA Journal on Uncertainty Quantification, 6, 937-964. [Google Scholar] [CrossRef] 
[18] van Der Houwen, P.J. and Sommeijer, B.P. (1980) On the Internal Stability of Explicit, m‐Stage Runge‐Kutta Methods for Large m‐Values. ZAMM—Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 60, 479-485. [Google Scholar] [CrossRef] 
[19] Küchler, U. and Platen, E. (2000) Strong Discrete Time Approximation of Stochastic Differential Equations with Time Delay. Mathematics and Computers in Simulation, 54, 189-205. [Google Scholar] [CrossRef] 
[20] Appleby, J.A.D., Mao, X. and Riedle, M. (2008) Geometric Brownian Motion with Delay: Mean Square Characterisation. Proceedings of the American Mathematical Society, 137, 339-348. [Google Scholar] [CrossRef] 
[21] Zhang, H., Gan, S. and Hu, L. (2009) The Split-Step Backward Euler Method for Linear Stochastic Delay Differential Equations. Journal of Computational and Applied Mathematics, 225, 558-568. [Google Scholar] [CrossRef] 
[22] Eremin, A.S. and Zubakhina, T.S. (2022) Real-Valued Stability Analysis of Runge-Kutta-Chebyshev Methods for Delay Differential Equations. AIP Conference Proceedings, 2425, Article ID: 090006. [Google Scholar] [CrossRef] 
[23] Eremin, A.S. (2024) Stability Analysis of First Order S-ROCK Methods for Stochastic Differential Equations with Delays in the Deterministic Part. AIP Conference Proceedings, 3094, Article ID: 220006. [Google Scholar] [CrossRef] 

为你推荐