分布不确定下随机微分方程参数最小二乘估计

数学物理学报 ›› 2019, Vol. 39 ›› Issue (6): 1499-1513.

分布不确定下随机微分方程参数最小二乘估计

费晨¹,费为银^2,*()

¹ 东华大学旭日工商管理学院上海 200051
² 安徽工程大学数理学院安徽芜湖 241000

收稿日期:2018-06-11 出版日期:2019-12-26 发布日期:2019-12-28
通讯作者: 费为银 E-mail:wyfei@ahpu.edu.cn
基金资助:
国家自然科学基金(71571001)

Consistency of Least Squares Estimation to the Parameter for Stochastic Differential Equations Under Distribution Uncertainty

Chen Fei¹,Weiyin Fei^2,*()

¹ Glorious Sun School of Business and Management, Donghua University, Shanghai 200051
² School of Mathematics and Physics, Anhui Polytechnic University, Anhui Wuhu 241000

Received:2018-06-11 Online:2019-12-26 Published:2019-12-28
Contact: Weiyin Fei E-mail:wyfei@ahpu.edu.cn
Supported by:
the NSFC(71571001)

摘要/Abstract

摘要：

在分布不确定性条件下，基于离散观察数据，研究了随机微分方程（SDE）参数最小二乘估计（LSE）的相合性，其中噪声特征为G-布朗运动.为了得到参数估计相合性的主要结果，利用次线性期望的随机微积分理论，给出了一些引理.结果表明，在一定的正则性条件下，基于分布不确定的最小二乘估计具有强相合性.最后，给出了一个算例说明理论的有效性.

关键词: G-随机微分方程(G-SDE), 次线性期望, 最小二乘估计量, 容度的指数鞅不等式, 强相合性

Abstract:

Under distribution uncertainty, on the basis of discrete observation data we investigate the consistency of the least squares estimator (LSE) of the parameter for the stochastic differential equation (SDE) where the noise are characterized by G-Brownian motion. In order to obtain our main result of consistency of parameter estimation, we provide some lemmas by the theory of stochastic calculus of sublinear expectation. The result shows that under some regularity conditions, the least squares estimator is strong consistent uniformly on the prior set. An illustrative example is discussed.

Key words: Stochastic differential equation disturbed by G-Brownian motiion (G-SDE), Sublinear expectation, Least squares estimator, Exponential martingale inequality for capacity, Strong consistency

中图分类号:

O211.6

费晨,费为银. 分布不确定下随机微分方程参数最小二乘估计[J]. 数学物理学报, 2019, 39(6): 1499-1513.

Chen Fei,Weiyin Fei. Consistency of Least Squares Estimation to the Parameter for Stochastic Differential Equations Under Distribution Uncertainty[J]. Acta mathematica scientia,Series A, 2019, 39(6): 1499-1513.

图/表 2

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$n $	$1 \times 10^4$	$ 2 \times 10^4$	$3 \times 10^4$	$4 \times 10^4$	$5 \times 10^4$
$\bar{\hat{\theta}}_{n, T}(10, 2^9) $	1.0313	1.0193	1.0144	1.0089	1.0057
$ \underline{\hat{\theta}}_{n, T}(10, 2^9) $	1.0104	1.0027	1.0014	0.9972	0.9978
$\bar{\hat{\theta}}_{n, T}(10, 2^9)- \underline{\hat{\theta}}_{n, T}(10, 2^9) $	0.0209	0.0166	0.0130	0.0117	0.0079

$J$	$2^{3}$	$2^4$	$2^5$	$2^6$	$2^7$
$\bar{\hat{\theta}}_{5 \times 10^3, T}(10, J)$	1.1108	1.0955	1.0903	1.0780	1.0672
$ \underline{\hat{\theta}}_{5 \times 10^3, T}(10, J) $	0.9718	0.9713	0.9879	0.9888	0.9995
$\bar{\hat{\theta}}_{5 \times 10^3, T}(10, J)-\underline{\hat{\theta}}_{5 \times 10^3, T}(10, J)$	0.1390	0.1241	0.1023	0.0892	0.0677

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