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Asymptotic Behaviour of the Trajectory Fitting Estimator for Reflected Ornstein–Uhlenbeck Processes

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Abstract

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.

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Acknowledgements

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.

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Authors and Affiliations

  1. School of Mathematical Science, Huaiyin Normal University, Huai’an, 223300, Jiangsu, People’s Republic of China

    Qingpei Zang

  2. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, Zhejiang, People’s Republic of China

    Lixin Zhang

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  1. Qingpei Zang
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Correspondence to Qingpei Zang.

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Zang, Q., Zhang, L. Asymptotic Behaviour of the Trajectory Fitting Estimator for Reflected Ornstein–Uhlenbeck Processes. J Theor Probab 32, 183–201 (2019). https://doi.org/10.1007/s10959-017-0796-7

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  • DOI: https://doi.org/10.1007/s10959-017-0796-7

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