$\boldsymbol{\alpha}$ -稳定过程驱动的非线性随机微分方程的参数估计: 非遍历情形

数学物理学报 ›› 2023, Vol. 43 ›› Issue (1): 249-260.

$\boldsymbol{\alpha}$ -稳定过程驱动的非线性随机微分方程的参数估计: 非遍历情形

张雪康^1,^*(),万山林¹(),舒慧生²()

¹安徽工程大学数理与金融学院安徽芜湖241000
²东华大学理学院上海201620

收稿日期:2021-04-22 修回日期:2022-07-05 出版日期:2023-02-26 发布日期:2023-03-07
通讯作者: ^*张雪康, E-mail: xkzhang@ahpu.edu.cn
作者简介:万山林, E-mail: 2221020107@stu.ahpu.edu.cn|舒慧生, E-mail: hsshu@dhu.edu.cn
基金资助:
国家自然科学基金(12101004);国家自然科学基金(62073071);国家自然科学基金(12271003);安徽工程大学引进人才科研启动基金(2020YQQ064);安徽省高端装备智能控制国际联合研究中心开放基金(IRICHE-05)

Parameter Estimation for Nonlinear Stochastic Differential Equations Driven by $\boldsymbol\alpha$-Stable Processes: Non-ergodic Case

Xuekang Zhang^1,^*(),Shanlin Wan¹(),Huisheng Shu²()

¹School of Mathematics-Physics and Finance, Anhui Polytechnic University, Anhui Wuhu 241000
²College of Science, Donghua University, Shanghai 201620

Received:2021-04-22 Revised:2022-07-05 Online:2023-02-26 Published:2023-03-07
Supported by:
The NSFC(12101004);The NSFC(62073071);The NSFC(12271003);Startup Foundation for Introducing Talent of Anhui Polytechnic University(2020YQQ064);Open Project of Anhui Province Center for International Reasearch of Intelligent Control of High-end Equipment(IRICHE-05)

摘要/Abstract

摘要：

该文研究了基于连续时间状态观测的 $\alpha$ -稳定过程驱动非线性随机微分方程的参数估计问题. 首先, 讨论了加权拟合估计量的相合性和收敛速率. 随后, 建立了估计量的渐近分布.

关键词: 非遍历情形, $\alpha$-稳定过程, 非线性随机微分方程, 相合性, 渐近分布

Abstract:

The present paper deals with the parameter estimation problem for nonlinear stochastic differential equations driven by $\alpha$-stable processes based on continuous-time observation. We first discuss the consistency and the rate of convergence of the weighted trajectory fitting estimator. Then, we have established the asymptotic distribution of the estimator.

Key words: Non-ergodic case, $\alpha$-stable processes, Nonlinear stochastic differential equations, Consistency, Asymptotic distribution

中图分类号:

O211

张雪康, 万山林, 舒慧生. $\boldsymbol{\alpha}$ -稳定过程驱动的非线性随机微分方程的参数估计: 非遍历情形[J]. 数学物理学报, 2023, 43(1): 249-260.

Xuekang Zhang, Shanlin Wan, Huisheng Shu. Parameter Estimation for Nonlinear Stochastic Differential Equations Driven by $\boldsymbol\alpha$-Stable Processes: Non-ergodic Case[J]. Acta mathematica scientia,Series A, 2023, 43(1): 249-260.

参考文献 29

[1]	Applebaum D. Lévy Processes and Stochastic Calculus. United Kingdom: Cambridge University Press, 2009
[2]	Basawa I, Scott D. Asymptotic Optimal inference for Non-ergodic Models. New York: Springer, 1983
[3]	Bercu B, ProÏa F, Savy N. On Ornstein-Uhlenbeck driven by Ornstein-Uhlenbeck processes. Statist Probab Lett, 2014, 85: 36-44 doi: 10.1016/j.spl.2013.11.002
[4]	Bo L, Wang Y, Yang X, Zhang G. Maximum likelihood estimation for reflected Ornstein-Uhlenbeck processes. J Statist Plann Inference, 2011, 141: 588-596 doi: 10.1016/j.jspi.2010.07.001
[5]	Dietz H. Asymptotic behavior of trajectory fitting estimators for certain non-ergodic SDE. Stat Inference Stoch Process, 2001, 4: 249-258 doi: 10.1023/A:1012254332474
[6]	Dietz H, Kutoyants Y. A class of minimum-distance estimators for diffusion processes with ergodic properties. Stat Decis, 1997, 15: 211-227
[7]	Dietz H, Kutoyants Y. Parameter estimation for some non-recurrent solutions of SDE. Stat Decis, 2003, 21: 29-45
[8]	Fama E. Mandelbrot and the stable Paretian hypothesis. J Business, 1963, 36: 420-429 doi: 10.1086/294633
[9]	Hu Y, Long H. Parameter estimation for Ornstein-Uhlenbeck processes driven by $\alpha$-stable Lévy motions. Commun Stoch Anal, 2007, 1: 175-192
[10]	Hu Y, Long H. Least squares estimator for Ornstein-Uhlenbeck processes driven by $\alpha$-stable motions. Stochastic Process Appl, 2009, 119: 2465-2480 doi: 10.1016/j.spa.2008.12.006
[11]	Hu Y, Long H. On the singularity of least squares estimator for mean-reverting $\alpha$-stable motions. Acta Math Sci, 2009, 29B(3): 599-608
[12]	Janicki A, Weron A. Simulation and Chaotic Behavior of $\alpha$-stable Stochastic Processes. New York: Marcel Dekker, 1994
[13]	Jeong H, Tomber B, Albert R, et al. The large-scale organization of metabolic networks. Nature, 2000, 407: 378-382
[14]	Jiang H, Dong X. Parameter estimation for the non-stationary Ornstein-Uhlenbeck process with linear drift. Stat Papers, 2015, 56: 257-268 doi: 10.1007/s00362-014-0580-z
[15]	Kessler M. Estimation of an ergodic diffusion from discrete observations. Scand J Statist, 1997, 24: 211-229 doi: 10.1111/1467-9469.00059
[16]	Kutoyants Y. Statistical Inference for Ergodic Diffusion Processes. Heidelberg: Springer-Verlag, 2004
[17]	Long H. Parameter estimation for a class of stochastic differential equations driven by small stable noises from discrete observations. Acta Math Sci, 2010, 30B(3): 645-663
[18]	Mandelbrot B. The Pareto-Lévy law and the distribution of income. Int Econ Rev, 1960, 1: 79-106 doi: 10.2307/2525289
[19]	Pan Y, Yan L. The least squares estimation for the $\alpha$-stable Ornstein-Uhlenbeck process with constant drift. Methodol Comput Appl Probab, 2019, 21: 1165-1182 doi: 10.1007/s11009-018-9654-z
[20]	Sato K. Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press, 1999
[21]	Shimizu Y. Local asymptotic mixed normality for discretely observed non-recurrent Ornstein-Uhlenbeck processes. Ann Inst Stat Math, 2012, 64: 193-211 doi: 10.1007/s10463-010-0307-4
[22]	Xu W, Wu C, Dong Y, Xiao W. Modeling Chinese stock returns with stable distribution. Math Comput Model, 2011, 54: 610-617 doi: 10.1016/j.mcm.2011.03.004
[23]	Zang Q, Zhang L. Asymptotic behaviour of the trajectory fitting estimator for reflected Ornstein-Uhlenbeck Processes. J Theor Probab, 2019, 32: 183-201 doi: 10.1007/s10959-017-0796-7
[24]	Zang Q, Zhu C. Asymptotic behaviour of parametric estimation for nonstationary reflected Ornstein-Uhlenbeck processes. J Math Anal Appl, 2016, 444: 839-851 doi: 10.1016/j.jmaa.2016.06.067
[25]	Zhang S, Zhang X. A least squares estimator for discretely observed Ornstein-Uhlenbeck processes driven by symmetric $\alpha$-stable motions. Ann Inst Statist Math, 2013, 65: 89-103 doi: 10.1007/s10463-012-0362-0
[26]	Zhang X, Yi H, Shu H. Nonparametric estimation of the trend for stochastic differential equations driven by small $\alpha$-stable noises. Statist Probab Lett, 2019, 151: 8-16 doi: 10.1016/j.spl.2019.03.012
[27]	Zhang X, Yi H, Shu H. Parameter estimation for non-stationary reflected Ornstein-Uhlenbeck processes driven by $\alpha$-stable noises. Statist Probab Lett, 2020, 156: 108617 doi: 10.1016/j.spl.2019.108617
[28]	Zhang X, Yi H, Shu H. Parameter estimation for certain nonstationary processes driven by $\alpha$-stable motions. Comm Statist Theory Methods, 2021, 50: 95-104 doi: 10.1080/03610926.2019.1630436
[29]	Zolotarev V. One-Dimensional Stable Distribution. Providence: American Mathematical Society, 1986