分布不确定下随机微分方程参数最小二乘估计

Abstract

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

CLC Number:

O211.6

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.

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References 30

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

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