Search Advanced

AIMS Mathematics

2022, Volume 7, Issue 7: 12952-12961. doi: 10.3934/math.2022717
Previous Article Next Article
Research article

Parameter estimation for partially observed stochastic differential equations driven by fractional Brownian motion

  • School of Mathematics and Statistics, Anyang Normal University, Anyang, 455000, China
  • Received: 07 January 2022 Revised: 06 April 2022 Accepted: 24 April 2022 Published: 07 May 2022

  • MSC : 60H10, 62F12

  • This paper is concerned with parameter estimation for partially observed stochastic differential equations driven by fractional Brownian motion. Firstly, the state estimation equation is given and the parameter estimator is derived. Then, the strong consistency and asymptotic normality of the maximum likelihood estimator are derived by applying the strong law of large numbers for continuous martingales and the central limit theorem for stochastic integrals with respect to Gaussian martingales. Finally, an example is provided to verify the results.

    Citation: Chao Wei. Parameter estimation for partially observed stochastic differential equations driven by fractional Brownian motion[J]. AIMS Mathematics, 2022, 7(7): 12952-12961. doi: 10.3934/math.2022717

    Related Papers:

  • This paper is concerned with parameter estimation for partially observed stochastic differential equations driven by fractional Brownian motion. Firstly, the state estimation equation is given and the parameter estimator is derived. Then, the strong consistency and asymptotic normality of the maximum likelihood estimator are derived by applying the strong law of large numbers for continuous martingales and the central limit theorem for stochastic integrals with respect to Gaussian martingales. Finally, an example is provided to verify the results.


    加载中


    [1] H. Chen, Z. D. Wang, J. L. Liang, M. Z. Li, State estimation for stochastic time-varying Boolean networks, IEEE T. Automat. Contr., 65 (2020), 5480–5487. https://doi.org/10.1109/TAC.2020.2973817 doi: 10.1109/TAC.2020.2973817
    [2] Y. Chen, H. Zhou, Parameter estimation for an Ornstein-Uhlenbeck process driven by a general Gaussian noise, Acta Math. Sci., 41 (2021), 573–595. https://doi.org/10.1007/s10473-021-0218-x doi: 10.1007/s10473-021-0218-x
    [3] M. Dai, J. Duan, J. Liao, X. J. Wang, Maximum likelihood estimation of stochastic differential equations with random effects driven by fractional Brownian motion, Appl. Math. Comput., 397 (2021), 1–13. https://doi.org/10.1016/j.amc.2020.125927 doi: 10.1016/j.amc.2020.125927
    [4] F. Ding, D. Meng, J. Dai, Q. S. Li, A. Alsaedi, T. Hayat, Least squares based iterative parameter estimation algorithm for stochastic dynamical systems with ARMA noise using the model equivalence, Int. J. Control Autom., 16 (2018), 630–639. https://doi.org/10.1007/s12555-017-0001-x doi: 10.1007/s12555-017-0001-x
    [5] H. L. Dong, X. Bu, N. Hou, Y. R. Liu, F. E. Alsaadi, T. Hayate, Event-triggered distributed state estimation for a class of time-varying systems over sensor networks with redundant channels, Inform. Fusion, 36 (2017), 243–250. https://doi.org/10.1016/j.inffus.2016.12.005 doi: 10.1016/j.inffus.2016.12.005
    [6] Y. Z. Hu, D. Nualart, H. Zhou, Drift parameter estimation for nonlinear stochastic differential equations driven by fractional Brownian motion, Stochastics, 91 (2019), 1067–1091. https://doi.org/10.1080/17442508.2018.1563606 doi: 10.1080/17442508.2018.1563606
    [7] M. Imani, U. M. Braga-Neto, Maximum-likelihood adaptive filter for partially observed Boolean dynamical systems, IEEE T. Signal Pr., 65 (2016), 359–371. https://doi.org/10.1109/TSP.2016.2614798 doi: 10.1109/TSP.2016.2614798
    [8] Y. Ji, X. Jiang, L. Wan, Hierarchical least squares parameter estimation algorithm for two-input Hammerstein finite impulse response systems, J. Franklin Inst., 357 (2020), 5019–5032. https://doi.org/10.1016/j.jfranklin.2020.03.027 doi: 10.1016/j.jfranklin.2020.03.027
    [9] Z. H. Li, C. H. Ma, Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model, Stoch. Proc. Appl., 125 (2015), 3196–3233.
    [10] S. Li, Y. Dong, Parametric estimation in the Vasicek-type model driven by sub-fractional Brownian motion, Algorithms, 11 (2018), 5–18. https://doi.org/10.3390/a11120197 doi: 10.3390/a11120197
    [11] J. Li, Z. D. Wang, H. L. Dong, W. Y. Fei, Delay-distribution-dependent state estimation for neural networks under stochastic communication protocol with uncertain transition probabilities, Neural. Netw., 130 (2020), 143–151. https://doi.org/10.1016/j.neunet.2020.06.023 doi: 10.1016/j.neunet.2020.06.023
    [12] H. Liu, Z. D. Wang, B. Shen, X. Liu, Event-triggered $H_{\infty}$ state estimation for delayed stochastic memristive neural networks with missing measurements: The discrete time case, IEEE T. Neur. Net. Lear., 29 (2018), 3726–3737. https://doi.org/10.1109/TNNLS.2017.2728639 doi: 10.1109/TNNLS.2017.2728639
    [13] X. R. Mao, Stochaastic differential equations and applications, Horwood Publishing Limited, Second Edition, UK, 2008.
    [14] B. Onsy, K. Es-Sebaiy, F. Viens, Parameter estimation for a partially observed Ornstein-Uhlenbeck process with long-memory noise, Stochastics, 89 (2017), 431–468. https://doi.org/10.1080/17442508.2016.1248967 doi: 10.1080/17442508.2016.1248967
    [15] V. Paxson, Fast, approximate synthesis of fractional Gaussian noise for generating self-similar network traffic, Comput. Commun. Rev., 27 (1997), 5–18. https://doi.org/10.1145/269790.269792 doi: 10.1145/269790.269792
    [16] B. L. S. Prakasa Rao, Instrumental variable estimation for a linear stochastic differential equation driven by a mixed fractional Brownian motion, Stoch. Anal. Appl., 35 (2017), 943–953. https://doi.org/10.1080/07362994.2017.1338577 doi: 10.1080/07362994.2017.1338577
    [17] B. L. S. Prakasa Rao, Parametric estimation for linear stochastic differential equations driven by mixed fractional Brownian motion, Stoch. Anal. Appl., 36 (2018), 767–781. https://doi.org/10.1080/07362994.2018.1462714 doi: 10.1080/07362994.2018.1462714
    [18] B. L. S. Prakasa Rao, Nonparametric estimation of trend for stochastic differential equations driven by fractional Lévy process, J. Stat. Theory Pract., 15 (2021), 1–12. https://doi.org/10.1007/s42519-020-00138-z doi: 10.1007/s42519-020-00138-z
    [19] B. L. S. Prakasa Rao, Parametric inference for stochastic differential equations driven by a mixed fractional Brownian motion with random effects based on discrete observations, Stoch. Anal. Appl., 45 (2021), 1–12. https://doi.org/10.1080/07362994.2021.1902352 doi: 10.1080/07362994.2021.1902352
    [20] M. Rathinam, M. Yu, State and parameter estimation from exact partial state observation in stochastic reaction networks, J. Chem. Phys., 154 (2021), 034103. https://doi.org/10.1063/5.0032539 doi: 10.1063/5.0032539
    [21] G. J. Shen, Q. B. Wang, X. W. Yin, Parameter estimation for the discretely observed Vasicek model with small fractional Lévy noise, Acta Math. Sin., 36 (2020), 443–461. https://doi.org/10.1007/s10114-020-9121-y doi: 10.1007/s10114-020-9121-y
    [22] Y. Wang, F. Ding, M. Wu, Recursive parameter estimation algorithm for multivariate output-error systems, J. Franklin Inst., 355 (2018), 5163–5181. https://doi.org/10.1016/j.jfranklin.2018.04.013 doi: 10.1016/j.jfranklin.2018.04.013
    [23] C. Wei, H. S. Shu, Maximum likelihood estimation for the drift parameter in diffusion processes, Stochastics, 88 (2016), 699–710. https://doi.org/10.1080/17442508.2015.1124879 doi: 10.1080/17442508.2015.1124879
    [24] C. Wei, Estimation for incomplete information stochastic systems from discrete observations, Adv. Differ. Equ., 227 (2019), 1–16. https://doi.org/10.1186/s13662-019-2169-2 doi: 10.1186/s13662-019-2169-2
    [25] C. Wei, Y. Wei, Y. Y. Zhou, Least squares estimation for discretely observed stochastic Lotka-Volterra model driven by small $\alpha$-stable noises, Discrete Dyn. Nat. Soc., 2020 (2020), 1–11. https://doi.org/10.1155/2020/8837689 doi: 10.1155/2020/8837689
    [26] X. Yan, D. Tong, Q. Chen, W. N. Zhou, Y. H. Xu, Adaptive state estimation of stochastic delayed neural networks with fractional Brownian motion, Neural Process. Lett., 50 (2019), 2007–2020. https://doi.org/10.1007/s11063-018-9960-z doi: 10.1007/s11063-018-9960-z
  • Reader Comments
  • © 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.4

Metrics

Article views(1200) PDF downloads(94) Cited by(2)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(1)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog