分段连续型随机微分方程平衡方法的均方收敛性与稳定性
The Mean Square Convergence and Stability of the Balanced Implicit Method of Stochastic Differential Equations with Piecewise Continuous Arguments
摘要: 针对一类分段连续型随机微分方程，证明了其采用全隐式平衡方法的情况下，该方法具有良好的均方收敛性和稳定性，并且证明了平衡方法的强收敛阶为1/2，同时通过数值分析实验验证了此类方程数学分析的正确以及合理性，证明了强平衡隐式方法以及弱平衡隐式方法都是均方稳定的。
Abstract: For a class of piecewise continuous stochastic differential equations, the mean square convergence and stability of the fully implicit method-balanced method are studied by using the balanced method. It is proved that the strong convergence order of the balanced method is 1/2, and the rationality and correctness of the analysis are verified by numerical examples. It is shown that both the strong balanced implicit method and the weak balanced implicit method are mean square stable.
文章引用：蒋泽楷, 胡琳, 李元林. 分段连续型随机微分方程平衡方法的均方收敛性与稳定性[J]. 理论数学, 2025, 15(12): 7-23. https://doi.org/10.12677/pm.2025.1512289

参考文献

[1] Van Kampen, N.G. (1976) Stochastic Differential Equations. Physics Reports, 24, 171-228. [Google Scholar] [CrossRef] 
[2] Kloeden, P.E., Platen, E., Kloeden, P.E., et al. (1992) Stochastic Differential Equations. Springer.
[3] Mao, X., Wei, F. and Wiriyakraikul, T. (2021) Positivity Preserving Truncated Euler-Maruyama Method for Stochastic Lotka-Volterra Competition Model. Journal of Computational and Applied Mathematics, 394, Article 113566. [Google Scholar] [CrossRef] 
[4] Li, X., Liu, W., Luo, Q. and Mao, X. (2022) Stabilisation in Distribution of Hybrid Stochastic Differential Equations by Feedback Control Based on Discrete-Time State Observations. Automatica, 140, Article 110210. [Google Scholar] [CrossRef] 
[5] Li, X., Mao, X., Mukama, D.S. and Yuan, C. (2020) Delay Feedback Control for Switching Diffusion Systems Based on Discrete-Time Observations. SIAM Journal on Control and Optimization, 58, 2900-2926. [Google Scholar] [CrossRef] 
[6] Lu, J., Li, Y., Mao, X. and Pan, J. (2022) Stabilization of Nonlinear Hybrid Stochastic Delay Systems by Feedback Control Based on Discrete-Time State and Mode Observations. Applicable Analysis, 101, 1077-1100. [Google Scholar] [CrossRef] 
[7] Rihan, F.A. (2021) Delay Differential Equations and Applications to Biology. Springer.
[8] Song, M. and Zhang, L. (2012) Numerical Solutions of Stochastic Differential Equations with Piecewise Continuous Arguments under Khasminskii‐Type Conditions. Journal of Applied Mathematics, 2012, Article ID: 696849. [Google Scholar] [CrossRef] 
[9] Chen, C., Hong, J. and Lu, Y. (2023) Stochastic Differential Equation with Piecewise Continuous Arguments: Markov Property, Invariant Measure and Numerical Approximation. Discrete and Continuous Dynamical Systems-B, 28, Article 765. [Google Scholar] [CrossRef] 
[10] Shi, H., Song, M. and Liu, M. (2023) Strong Convergence of Explicit Numerical Schemes for Stochastic Differential Equations with Piecewise Continuous Arguments. Numerical Algorithms, 97, 779-800. [Google Scholar] [CrossRef] 
[11] Geng, Y., Song, M. and Liu, M. (2023) The Convergence of Truncated Euler-Maruyama Method for Stochastic Differential Equations with Piecewise Continuous Arguments under Generalized One-Sided Lipschitz Condition. Journal of Computational Mathematics, 41, 663-682. [Google Scholar] [CrossRef] 
[12] Zhu, Q., Shen, Y., Li, D. and Lin, W. (2022) Neural Piecewise-Constant Delay Differential Equations. Proceedings of the AAAI Conference on Artificial Intelligence, 36, 9242-9250. [Google Scholar] [CrossRef] 
[13] 刘国清, 张玲, 郭爽. 半线性分段连续型随机微分方程数值解的收敛性和稳定性[J]. 黑龙江大学自然科学学报, 2015, 32(2): 201-207.
[14] 巩全壹, 张玲, 王文丽, 等. 线性分段连续型随机微分方程数值解的收敛性和稳定性[J]. 哈尔滨师范大学自然科学学报, 2015, 31(2): 40-44.
[15] 胡琳. 几类带泊松跳随机微分方程数值方法的收敛性与稳定性[D]: [博士学位论文]. 长沙: 中南大学, 2012.
[16] 戴红玉, 刘明珠. 分段连续型随机微分方程的均方稳定性分析[J]. 黑龙江大学自然科学学报, 2008(5): 625-629.

为你推荐