IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v438y2023ics0096300322006476.html
   My bibliography  Save this article

Strong convergence and almost sure exponential stability of balanced numerical approximations to stochastic delay Hopfield neural networks

Author

Listed:
  • Rathinasamy, Anandaraman
  • Mayavel, Pichamuthu

Abstract

This paper deals with the balanced numerical schemes of the stochastic delay Hopfield neural networks. The balanced methods have a strong convergence rate of at least 12 and the balanced schemes have almost sure exponential stability under certain conditions. Under the Lipchitz and linear growth conditions, the balanced Euler methods are proved to have a strong convergence of order 12 in mean-square sense. Using the Lipchitz conditions on the various parameters of the model, based on the semimartingale convergence theorem and some reasonable assumptions, the balanced Euler methods of the stochastic delay Hopfield neural networks are proved to be almost sure exponentially stable. Numerical experiments are provided to illustrate the theoretical results which are derived in this paper. The computational efficiency of the balanced methods is demonstrated by numerical tests and compared to the Euler–Maruyama approximation scheme of the stochastic delay Hopfield neural networks. Furthermore, the obtained numerical results show that the balanced numerical methods of stochastic delay Hopfield neural networks are very efficient with the least error and have the best step size region for almost sure mean square stable.

Suggested Citation

  • Rathinasamy, Anandaraman & Mayavel, Pichamuthu, 2023. "Strong convergence and almost sure exponential stability of balanced numerical approximations to stochastic delay Hopfield neural networks," Applied Mathematics and Computation, Elsevier, vol. 438(C).
    • Handle: RePEc:eee:apmaco:v:438:y:2023:i:c:s0096300322006476
    DOI: 10.1016/j.amc.2022.127573
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300322006476
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2022.127573?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Qian Guo & Wenwen Xie & Taketomo Mitsui, 2013. "Convergence and Stability of the Split-Step -Milstein Method for Stochastic Delay Hopfield Neural Networks," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-12, April.
    2. Rathinasamy, A. & Narayanasamy, J., 2019. "Mean square stability and almost sure exponential stability of two step Maruyama methods of stochastic delay Hopfield neural networks," Applied Mathematics and Computation, Elsevier, vol. 348(C), pages 126-152.
    3. Tan, Jianguo & Tan, Yahua & Guo, Yongfeng & Feng, Jianfeng, 2020. "Almost sure exponential stability of numerical solutions for stochastic delay Hopfield neural networks with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    4. Liu, Linna & Zhu, Quanxin, 2015. "Almost sure exponential stability of numerical solutions to stochastic delay Hopfield neural networks," Applied Mathematics and Computation, Elsevier, vol. 266(C), pages 698-712.
    5. G. N. Milstein & Eckhard Platen & H. Schurz, 1998. "Balanced Implicit Methods for Stiff Stochastic Systems," Published Paper Series 1998-1, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Rathinasamy, Anandaraman & Mayavel, Pichamuthu, 2023. "The balanced split step theta approximations of stochastic neutral Hopfield neural networks with time delay and Poisson jumps," Applied Mathematics and Computation, Elsevier, vol. 455(C).
    2. Deng, Jie & Li, Hong-Li & Cao, Jinde & Hu, Cheng & Jiang, Haijun, 2023. "State estimation for discrete-time fractional-order neural networks with time-varying delays and uncertainties," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    3. Hualin Song & Cheng Hu & Juan Yu, 2022. "Stability and Synchronization of Fractional-Order Complex-Valued Inertial Neural Networks: A Direct Approach," Mathematics, MDPI, vol. 10(24), pages 1-23, December.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Rathinasamy, Anandaraman & Mayavel, Pichamuthu, 2023. "The balanced split step theta approximations of stochastic neutral Hopfield neural networks with time delay and Poisson jumps," Applied Mathematics and Computation, Elsevier, vol. 455(C).
    2. Tan, Jianguo & Tan, Yahua & Guo, Yongfeng & Feng, Jianfeng, 2020. "Almost sure exponential stability of numerical solutions for stochastic delay Hopfield neural networks with jumps," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 545(C).
    3. Liu, Yufen & Cao, Wanrong & Li, Yuelin, 2022. "Split-step balanced θ-method for SDEs with non-globally Lipschitz continuous coefficients," Applied Mathematics and Computation, Elsevier, vol. 413(C).
    4. Xiaoling Wang & Xiaofei Guan & Pei Yin, 2020. "A New Explicit Magnus Expansion for Nonlinear Stochastic Differential Equations," Mathematics, MDPI, vol. 8(2), pages 1-17, February.
    5. Zhenyu Wang & Qiang Ma & Xiaohua Ding, 2020. "Simulating Stochastic Differential Equations with Conserved Quantities by Improved Explicit Stochastic Runge–Kutta Methods," Mathematics, MDPI, vol. 8(12), pages 1-15, December.
    6. Kahl Christian & Schurz Henri, 2006. "Balanced Milstein Methods for Ordinary SDEs," Monte Carlo Methods and Applications, De Gruyter, vol. 12(2), pages 143-170, April.
    7. Nicola Bruti-Liberati & Eckhard Platen, 2005. "On the Strong Approximation of Jump-Diffusion Processes," Research Paper Series 157, Quantitative Finance Research Centre, University of Technology, Sydney.
    8. Li, Yan & Zhang, Qimin, 2020. "The balanced implicit method of preserving positivity for the stochastic SIQS epidemic model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 538(C).
    9. Xianming Sun & Siqing Gan, 2014. "An Efficient Semi-Analytical Simulation for the Heston Model," Computational Economics, Springer;Society for Computational Economics, vol. 43(4), pages 433-445, April.
    10. Li, Zhao-Yan & Shang, Shengnan & Lam, James, 2019. "On stability of neutral-type linear stochastic time-delay systems with three different delays," Applied Mathematics and Computation, Elsevier, vol. 360(C), pages 147-166.
    11. Yansheng Ma & Yong Li, 2012. "A uniform asymptotic expansion for stochastic volatility model in pricing multi‐asset European options," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 28(4), pages 324-341, July.
    12. Wang, Fen & Chen, Yuanlong, 2021. "Mean square exponential stability for stochastic memristor-based neural networks with leakage delay," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    13. Komori Yoshio, 1995. "Stahle ROW-Type Weak Scheme for Stochastic Differential Equations," Monte Carlo Methods and Applications, De Gruyter, vol. 1(4), pages 279-300, December.
    14. Halidias Nikolaos, 2016. "On the construction of boundary preserving numerical schemes," Monte Carlo Methods and Applications, De Gruyter, vol. 22(4), pages 277-289, December.
    15. Robert Elliott & Eckhard Platen, 1999. "Hidden Markov Chain Filtering for Generalised Bessel Processes," Research Paper Series 23, Quantitative Finance Research Centre, University of Technology, Sydney.
    16. Nikolaos Halidias & Ioannis Stamatiou, 2015. "Approximating explicitly the mean reverting CEV process," Papers 1502.03018, arXiv.org, revised May 2015.
    17. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1, July-Dece.
    18. Yang, Xiaochen & Yang, Zhanwen & Zhang, Chiping, 2023. "Numerical analysis of the Linearly implicit Euler method with truncated Wiener process for the stochastic SIR model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 1-14.
    19. H. A. Mardones & C. M. Mora, 2020. "First-Order Weak Balanced Schemes for Stochastic Differential Equations," Methodology and Computing in Applied Probability, Springer, vol. 22(2), pages 833-852, June.
    20. Yang, Xu & Zhao, Weidong, 2018. "Finite element methods and their error analysis for SPDEs driven by Gaussian and non-Gaussian noises," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 58-75.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:438:y:2023:i:c:s0096300322006476. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.