THE LEAST SQUARES ESTIMATOR FOR AN ORNSTEIN-UHLENBECK PROCESS DRIVEN BY A HERMITE PROCESS WITH A PERIODIC MEAN

doi:10.1007/s10473-021-0215-0

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (2): 517-534.doi: 10.1007/s10473-021-0215-0

THE LEAST SQUARES ESTIMATOR FOR AN ORNSTEIN-UHLENBECK PROCESS DRIVEN BY A HERMITE PROCESS WITH A PERIODIC MEAN

申广君¹, 余迁^1,2, 唐正^1,3

1. Department of Mathematics, Anhui Normal University, Wuhu 241000, China;
2. School of Statistics, East China Normal University, Shanghai 200062, China;
3. School of Mathematics and Finance, Chuzhou Universty, Chuzhou 239012, China

收稿日期:2020-02-14 修回日期:2020-05-11 出版日期:2021-04-25 发布日期:2021-04-29
通讯作者: Qian YU E-mail:qyumath@163.com
作者简介:Guangjun SHEN,E-mail:gjshen@163.com;Zheng TANG,E-mail:tzheng8889@126.com
基金资助:
This work was supported by National Natural Science Foundation of China (12071003).

THE LEAST SQUARES ESTIMATOR FOR AN ORNSTEIN-UHLENBECK PROCESS DRIVEN BY A HERMITE PROCESS WITH A PERIODIC MEAN

Guangjun SHEN¹, Qian YU^1,2, Zheng TANG^1,3

1. Department of Mathematics, Anhui Normal University, Wuhu 241000, China;
2. School of Statistics, East China Normal University, Shanghai 200062, China;
3. School of Mathematics and Finance, Chuzhou Universty, Chuzhou 239012, China

Received:2020-02-14 Revised:2020-05-11 Online:2021-04-25 Published:2021-04-29
Contact: Qian YU E-mail:qyumath@163.com
About author:Guangjun SHEN,E-mail:gjshen@163.com;Zheng TANG,E-mail:tzheng8889@126.com
Supported by:
This work was supported by National Natural Science Foundation of China (12071003).

摘要/Abstract

摘要： We consider the least square estimator for the parameters of Ornstein-Uhlenbeck processes $${\rm d}Y_s=\Big (\sum\limits_{j=1}^{k}\mu_j \phi_j (s)- \beta Y_s\Big){\rm d}s + {\rm d}Z_s^{q,H},$$ driven by the Hermite process $Z_s^{q,H}$ with order $q \geq 1$ and a Hurst index $H \in (\frac12,1)$, where the periodic functions $\phi_j(s), j=1,\ldots,k$ are bounded, and the real numbers $\mu_j, j=1,\ldots, k$ together with $\beta>0$ are unknown parameters. We establish the consistency of a least squares estimation and obtain the asymptotic behavior for the estimator. We also introduce alternative estimators, which can be looked upon as an application of the least squares estimator. In terms of the fractional Ornstein-Uhlenbeck processes with periodic mean, our work can be regarded as its non-Gaussian extension.

关键词: Least squares estimator, consistency, asymptotic distribution, Ornstein-Uhlenbeck processes, Hermite processes

Abstract: We consider the least square estimator for the parameters of Ornstein-Uhlenbeck processes $${\rm d}Y_s=\Big (\sum\limits_{j=1}^{k}\mu_j \phi_j (s)- \beta Y_s\Big){\rm d}s + {\rm d}Z_s^{q,H},$$ driven by the Hermite process $Z_s^{q,H}$ with order $q \geq 1$ and a Hurst index $H \in (\frac12,1)$, where the periodic functions $\phi_j(s), j=1,\ldots,k$ are bounded, and the real numbers $\mu_j, j=1,\ldots, k$ together with $\beta>0$ are unknown parameters. We establish the consistency of a least squares estimation and obtain the asymptotic behavior for the estimator. We also introduce alternative estimators, which can be looked upon as an application of the least squares estimator. In terms of the fractional Ornstein-Uhlenbeck processes with periodic mean, our work can be regarded as its non-Gaussian extension.

Key words: Least squares estimator, consistency, asymptotic distribution, Ornstein-Uhlenbeck processes, Hermite processes

中图分类号:

60G18

申广君, 余迁, 唐正. THE LEAST SQUARES ESTIMATOR FOR AN ORNSTEIN-UHLENBECK PROCESS DRIVEN BY A HERMITE PROCESS WITH A PERIODIC MEAN[J]. 数学物理学报(英文版), 2021, 41(2): 517-534.

Guangjun SHEN, Qian YU, Zheng TANG. THE LEAST SQUARES ESTIMATOR FOR AN ORNSTEIN-UHLENBECK PROCESS DRIVEN BY A HERMITE PROCESS WITH A PERIODIC MEAN[J]. Acta mathematica scientia,Series B, 2021, 41(2): 517-534.

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THE LEAST SQUARES ESTIMATOR FOR AN ORNSTEIN-UHLENBECK PROCESS DRIVEN BY A HERMITE PROCESS WITH A PERIODIC MEAN

THE LEAST SQUARES ESTIMATOR FOR AN ORNSTEIN-UHLENBECK PROCESS DRIVEN BY A HERMITE PROCESS WITH A PERIODIC MEAN

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