时变延迟随机微分方程的稳定性和有界性
Stability and Boundedness of Time-Changed Delay Stochastic Differential Equations
摘要: 本文研究了时变延迟随机微分方程的稳定性和有界性问题。本文分别讨论了当延迟函数为常数、有界函数和无界函数等三种情况的稳定性和有界性问题，利用李雅普诺夫函数和时变伊藤公式，得出了三种情况下相应解的稳定性和有界性的判别准则。最后，本文列举了一些例子来说明所得结果的有效性。
Abstract: In this paper, we study the stability and boundedness of time-changed delay stochastic differential equations. Three cases of the stability and boundedness of time-changed delay stochastic differen-tial equations are discussed separately, that is, the delay function is constant, bounded function and unbounded function. Using Lyapunov function method and time-changed Ito’s formula, the criteria of stability and boundedness of these three cases are obtained. Finally, three examples are listed to illustrate the effectiveness of our conclusion.
文章引用：吴承业. 时变延迟随机微分方程的稳定性和有界性[J]. 理论数学, 2023, 13(10): 2780-2793. https://doi.org/10.12677/PM.2023.1310286

参考文献

[1] Zhou, W., Yang, J., Yang, X., et al. (2015) pth Moment Exponential Stability of Stochastic Delayed Hybrid Systems with Lévy Noise. Applied Mathematical Modelling, 39, 5650-5658. [Google Scholar] [CrossRef] 
[2] Anguraj, A., Ravikumar, K. and Nieto, J.J. (2021) On Stability of Stochastic Differential Equations with Random Impulses Driven by Poisson Jumps. Stochastics, 93, 682-696. [Google Scholar] [CrossRef] 
[3] Ngoc, P.H.A. (2020) A Novel Approach to Mean Square Exponential Stability of Stochastic Delay Differential Equations. IEEE Transactions on Automatic Control, 66, 2351-2356. [Google Scholar] [CrossRef] 
[4] Ngoc, P.H.A. (2020) A New Approach to Mean Square Exponential Stability of Stochastic Functional Differential Equations. IEEE Control Systems Letters, 5, 1645-1650. [Google Scholar] [CrossRef] 
[5] Yuan, C. and Lygeros, J. (2006) Asymptotic Sta-bility and Boundedness of Delay Switching Diffusions. IEEE Transactions on Automatic Control, 51, 171-175. [Google Scholar] [CrossRef] 
[6] Magdziarz, M. and Schilling, R. (2015) Asymptotic Properties of Brownian Motion Delayed by Inverse Subordinators. Proceedings of the American Mathematical Society, 143, 4485-4501. [Google Scholar] [CrossRef] 
[7] Jarrow, R. and Protter, P. (2004) A Short History of Stochas-tic Integration and Mathematical Finance: The Early Years, 1880-1970. In: DasGupta, A., ed., A Festschrift for Herman Rubin, Vol. 45, Institute of Mathematical Statistics, Beachwood, OH, 75-91. [Google Scholar] [CrossRef] 
[8] Liao, X.X. and Mao, X. (1997) Almost Sure Exponential Stability of Neutral Stochastic Differential Difference Equations. Journal of Mathematical Analysis and Applications, 212, 554-570. [Google Scholar] [CrossRef] 
[9] 胡杨子, 黄乘明. 含多个函数时滞的随机延迟微分方程的矩稳定性[J]. 数学杂志, 2009, 29(6): 801-808.
[10] 王琦, 温洁嫦. 滞后型分段连续随机微分方程的稳定性[J]. 数学杂志, 2015, 35(2): 307-317.
[11] 袁志宏, 刘变红, 刘桂荣. 一类中立型随机泛函微分方程的分布稳定性[J]. 数学的实践与认识, 2018, 48(11): 252-259.
[12] 张慈. 时变布朗运动下带交易费的亚式期权定价[D]: [硕士学位论文]. 徐州: 中国矿业大学, 2015.
[13] Kobayashi, K. (2011) Stochastic Calculus for a Time-Changed Semimartingale and the Associated Stochastic Differential Equations. Journal of Theoretical Probability, 24, 789-820. [Google Scholar] [CrossRef] 
[14] Wu, Q. (2016) Stability of Stochastic Differential Equations with Respect to Time-Changed Brownian Motions.
https://arxiv.org/abs/1602.08160
[15] Zhu, M., Li, J. and Liu, D. (2021) Exponential Stability for Time-Changed Stochastic Differential Equations. Acta Mathematicae Applicatae Sinica, English Series, 37, 617-627. [Google Scholar] [CrossRef] 
[16] Hu, L., Mao, X. and Shen, Y. (2013) Stability and Boundedness of Nonlinear Hybrid Stochastic Differential Delay Equations. Systems & Control Letters, 62, 178-187. [Google Scholar] [CrossRef] 
[17] Bao, J., Hou, Z. and Yuan, C. (2009) Stability in Distribution of Neutral Stochastic Differential Delay Equations with Markovian Switching. Statistics & Probability Letters, 79, 1663-1673. [Google Scholar] [CrossRef] 
[18] Tan, L., Jin, W. and Suo, Y. (2015) Stability in Distri-bution of Neutral Stochastic Functional Differential Equations. Statistics & Probability Letters, 107, 27-36. [Google Scholar] [CrossRef] 
[19] Bao, J., Yin, G. and Yuan, C. (2017) Stationary Distributions for Retarded Stochastic Differential Equations without Dissipative. Stochastics, 89, 530-549. [Google Scholar] [CrossRef] 
[20] Hu, J., Mao, W. and Mao, X. (2023) Advances in Nonlinear Hybrid Stochastic Differential Delay Equations: Existence, Boundedness and Stability. Automatica, 147, Article ID: 110682. [Google Scholar] [CrossRef] 

为你推荐