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Least squares estimators for discretely observed stochastic processes driven by small Lévy noises

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  • Long, Hongwei
  • Shimizu, Yasutaka
  • Sun, Wei

Abstract

We study the problem of parameter estimation for discretely observed stochastic processes driven by additive small Lévy noises. We do not impose any moment condition on the driving Lévy process. Under certain regularity conditions on the drift function, we obtain consistency and rate of convergence of the least squares estimator (LSE) of the drift parameter when a small dispersion coefficient ε→0 and n→∞ simultaneously. The asymptotic distribution of the LSE in our general setting is shown to be the convolution of a normal distribution and a distribution related to the jump part of the Lévy process. Moreover, we briefly remark that our methodology can be easily extended to the more general case of semi-martingale noises.

Suggested Citation

  • Long, Hongwei & Shimizu, Yasutaka & Sun, Wei, 2013. "Least squares estimators for discretely observed stochastic processes driven by small Lévy noises," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 422-439.
    • Handle: RePEc:eee:jmvana:v:116:y:2013:i:c:p:422-439
    DOI: 10.1016/j.jmva.2013.01.012
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    2. Ma, Chunhua & Yang, Xu, 2014. "Small noise fluctuations of the CIR model driven by α-stable noises," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 1-11.
    3. Yang, Xu, 2017. "Maximum likelihood type estimation for discretely observed CIR model with small α-stable noises," Statistics & Probability Letters, Elsevier, vol. 120(C), pages 18-27.
    4. Zhao, Huiyan & Zhang, Chongqi, 2019. "Minimum distance parameter estimation for SDEs with small α-stable noises," Statistics & Probability Letters, Elsevier, vol. 145(C), pages 301-311.
    5. Yusuke Kaino & Masayuki Uchida, 2018. "Hybrid estimators for small diffusion processes based on reduced data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 81(7), pages 745-773, October.
    6. Guangjun Shen & Qian Yu, 2019. "Least squares estimator for Ornstein–Uhlenbeck processes driven by fractional Lévy processes from discrete observations," Statistical Papers, Springer, vol. 60(6), pages 2253-2271, December.
    7. Shu, Huisheng & Jiang, Ziwei & Zhang, Xuekang, 2023. "Parameter estimation for integrated Ornstein–Uhlenbeck processes with small Lévy noises," Statistics & Probability Letters, Elsevier, vol. 199(C).
    8. Qian Yu, 2021. "Least squares estimator of fractional Ornstein–Uhlenbeck processes with periodic mean for general Hurst parameter," Statistical Papers, Springer, vol. 62(2), pages 795-815, April.
    9. Long, Hongwei & Ma, Chunhua & Shimizu, Yasutaka, 2017. "Least squares estimators for stochastic differential equations driven by small Lévy noises," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1475-1495.
    10. Ren, Panpan & Wu, Jiang-Lun, 2021. "Least squares estimation for path-distribution dependent stochastic differential equations," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    11. Yiying Cheng & Yaozhong Hu & Hongwei Long, 2020. "Generalized moment estimators for $$\alpha $$α-stable Ornstein–Uhlenbeck motions from discrete observations," Statistical Inference for Stochastic Processes, Springer, vol. 23(1), pages 53-81, April.
    12. Xuekang Zhang & Huisheng Shu & Haoran Yi, 2023. "Parameter Estimation for Ornstein–Uhlenbeck Driven by Ornstein–Uhlenbeck Processes with Small Lévy Noises," Journal of Theoretical Probability, Springer, vol. 36(1), pages 78-98, March.
    13. Yasutaka Shimizu, 2017. "Threshold Estimation for Stochastic Processes with Small Noise," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(4), pages 951-988, December.
    14. Zhang, Xuekang & Yi, Haoran & Shu, Huisheng, 2019. "Nonparametric estimation of the trend for stochastic differential equations driven by small α-stable noises," Statistics & Probability Letters, Elsevier, vol. 151(C), pages 8-16.

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