JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (11): 15-26.doi: 10.6040/j.issn.1671-9352.0.2022.317

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Parameter estimation for the sub-fractional Vasicek model based on discrete observation

Cuiyun ZHANG(),Jingjun GUO*(),Aiqin MA   

  1. School of Statistics, Lanzhou University of Finance and Economics, Lanzhou 730020, Gansu, China
  • Received:2022-05-30 Online:2023-11-20 Published:2023-11-07
  • Contact: Jingjun GUO E-mail:nwnu_zhangcuiyun@163.com;guojj@lzufe.edu.cn

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Abstract:

The problem of statistical analysis of the Vasicek model driven by sub-fractional Brownian motion is mainly investigated. Firstly, based on discrete observations, the estimation of drift parameters μ and θ in Vasicek model are given by the least square estimation method. Secondly, for the θ≠0 and θ=0 cases, the consistency and the asymptotic distribution of the estimators are obtained, respectively. Finally, simulations are performed with the Monte Carlo method to demonstrate the unbiasedness and validity of the estimates.

Key words: sub-fractional Brownian motion, least squares estimation, Vasicek model, consistency, asymptotic distribution

CLC Number: 

  • O212.1

Table 1

Mean value and standard deviation of the estimator $\hat{\mu}$"

H μ=0.6 μ=1 μ=1.5 μ=1.75
$\hat{\mu}$的均值 $\hat{\mu}$的标准差 $\hat{\mu}$的均值 $\hat{\mu}$的标准差 $\hat{\mu}$的均值 $\hat{\mu}$的标准差 $\hat{\mu}$的均值 $\hat{\mu}$的标准差
0.65 0.599 9 8.69×10-5 0.999 9 8.71×10-5 1.499 9 8.90×10-5 1.750 0 8.14×10-5
0.75 0.599 9 3.95×10-5 1.000 0 3.64×10-5 1.500 0 3.58×10-5 1.749 9 3.62×10-5
0.85 0.599 9 1.41×10-5 0.999 9 1.50×10-5 1.499 9 1.57×10-5 1.749 9 1.49×10-5

Table 2

Mean value and standard deviation of the estimator $\hat{\theta}$"

Hθ=-0.7θ=-0.8θ=-0.9θ=-0.95
$\hat{\theta}$的均值 $\hat{\theta}$的标准差 $\hat{\theta}$的均值 $\hat{\theta}$的标准差 $\hat{\theta}$的均值 $\hat{\theta}$的标准差 $\hat{\theta}$的均值 $\hat{\theta}$的标准差
0.55 -0.699 9 9.03×10-6 -0.799 9 9.16×10-6 -0.900 0 9.26×10-6 -0.949 9 8.86×10-6
0.65 -0.699 9 6.84×10-6 -0.800 0 6.33×10-6 -0.899 9 6.77×10-6 -0.949 9 7.06×10-6
0.75 -0.700 0 4.43×10-6 -0.800 0 4.89×10-6 -0.899 9 4.53×10-6 -0.949 9 4.68×10-6
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