Asymptotic Behaviour of the Trajectory Fitting Estimator for Reflected Ornstein–Uhlenbeck Processes
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The Ornstein–Uhlenbeck process with reflection, which has been the subject of an enormous body of literature, both theoretical and applied, is a process that returns continuously and immediately to the interior of the state space when it attains a certain boundary. In this work, we are mainly concerned with the study of the asymptotic behavior of the trajectory fitting estimator for nonergodic reflected Ornstein–Uhlenbeck processes, including strong consistency and asymptotic distribution. Moreover, we also prove that this kind of estimator for ergodic reflected Ornstein–Uhlenbeck processes does not possess the property of strong consistency.
KeywordsReflected Ornstein–Uhlenbeck processes Trajectory fitting estimator Nonergodic
Mathematics Subject Classification (2010)Primary 60F15 Secondary 62F12
The authors are grateful to the referees and associate editor for constructive comments which led to improvement of this work. We also thank Professor Hui Jiang at Nanjing University of Aeronautics and Astronautics for his comments on the revised version. This work was completed when the first author was visiting the University of Kansas in 2015; this author would like to thank Professor Yaozhong Hu in the Department of Mathematics at the University of Kansas for his warm hospitality. Zang acknowledges partial research support from National Natural Science Foundation of China (Grant Nos. 11326174 and 11401245), Natural Science Foundation of Jiangsu Province (Grant No. BK20130412), Natural Science Research Project of Ordinary Universities in Jiangsu Province (Grant No. 12KJB110003), China Postdoctoral Science Foundation (Grant No. 2014M551720), Jiangsu Government Scholarship for Overseas Studies, and Qing Lan project of Jiangsu Province (2016). Zhang acknowledges partial research support Zhang acknowledges partial research support from National Natural Science Foundation of China (Grant No. 11225104 and 11731012), Zhejiang Provincial Natural Science Foundation (Grant No. R6100119) and the Fundamental Research Funds for the Central Universities.
- 18.Goldstein, R.S., Keirstead, W.P.: On the term structure of interest rates in the presence of reflecting and absorbing boundaries (1997). https://doi.org/10.2139/ssrn.10.2139/ssrn.19840
- 20.Harrison, M.: Brownian Motion and Stochastic Flow Systems. Wiley, New York (1986)Google Scholar
- 23.Hu, Y., Lee, C., Lee, M.H., Song, J.: Parameter estimation for reflected Ornstein–Uhlenbeck processes with discrete observations Stat. Inference Stoch. Process. 18, 279–291 (2015)Google Scholar
- 30.Lee, C., Song, J.: On drift parameter estimation for reflected fractional Ornstein–Uhlenbeck processes (2013). arXiv:1303.6379
- 35.Mandjes, M., Spreij, P.: A note on the central limit theorem for a one-sided reflected Ornstein–Uhlenbeck process (2016). arXiv:1601.05653v1
- 43.Ward, W.: Stochastic-Process Limits. Springer Series in Operations Research. Springer, New York (2002)Google Scholar