Sequential Maximum Likelihood Estimation for the Squared Radial Ornstein-Uhlenbeck Process
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Sequential Maximum Likelihood Estimation for the Squared Radial Ornstein-Uhlenbeck Process Huantian Xie1
· Nenghui Kuang2
Received: 19 November 2019 / Revised: 31 May 2020 / Accepted: 2 September 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we study the properties of a sequential maximum likelihood estimator of the unknown parameter for the squared radial Ornstein-Uhlenbeck process. The estimator is proved to be closed, unbiased, normally distributed and strongly consistent. Lastly a simulation study is presented to illustrate the efficiency of the estimators. Keywords Sequential maximum likelihood estimator · Squared radial Ornstein-Uhlenbeck process · Unbiasedness · Mean squared error · Efficiency. Mathematics Subject Classification (2010) 60H10 · 62F99
1 Introduction In this paper, we study a sequential maximum likelihood estimator (SMLE) of the unknown parameter for the following squared radial Ornstein-Uhlenbeck process: (1.1) dXt = (δ + 2βXt )dt + 2σ Xt dWt ; X0 = x0 , where x0 ≥ 0, {Wt , t ≥ 0} is a standard Wiener process on a complete filtered probability space (, F, (Ft ; t ≥ 0), P ), where the filtration (Ft ; t ≥ 0) satisfies the usual conditions. σ > 0 and δ ≥ 0 are known parameters, β is an unknown parameter. The squared radial Research in part supported by the NSF of Shandong Province (Grants nos. ZR2018LA008 and 2019KJI003), the National Natural Science Foundation of China (Grants nos. 11771195 and 11701252), and the Education Department Foundation of Hunan Province(Grant no. 14C0456). Huantian Xie
[email protected] Nenghui Kuang [email protected] 1
School of Mathematics and Statistics, Linyi University, Linyi, Shandong 276005, People’s Republic of China
2
School of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, People’s Republic of China
Methodology and Computing in Applied Probability
Ornstein-Uhlenbeck process has received a lot of attention these days(see, e.g., Clifford and Wei (1993), Elworthy et al. (1999), Zani (2002), Aquilina and Rogers (2004), Ye (2008), and Gao and Jiang (2009)). Sequential maximum likelihood estimation in discrete time processes has been studied in Lai and Siegmund (1983) and the idea dates back to Anscombe (1952). In continuous time processes, a SMLE was first studied by Novikov (1972). The results of his work appeared in Liptser and Shiryaev (2001). Since then there has been extensions to semimartingales and stochastic partial differential equations (SPDEs). Lee et al. (2012) obtained a SMLE for reflected Ornstein-Uhlenbeck processes. Bo and Yang (2012) investigated a SMLE of the unknown drift parameter for a class of reflected generalized Ornstein-Uhlenbeck processes driven by spectrally positive Levy ´ processes. Kuang and Xie (2015) studied a SMLE for the hyperbolic diffusion process. Kuang et al. (2015) obtained a SMLE for the parameter of the linear drift term of the Rayleigh diffusion process. The aim of this paper is to study the statistical
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