数学物理学报(英文版) ›› 2009, Vol. 29 ›› Issue (3): 599-608.doi: 10.1016/S0252-9602(09)60056-4

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ON THE SINGULARITY OF LEAST SQUARES ESTIMATOR FOR MEAN-REVERTING α-STABLE MOTIONS

胡耀忠,龙红卫   

  1. Department of Mathematics, University of Kansas, 605 Snow Hall, Lawrence, Kansas 66045-2142, USA;Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, Florida 33431-0991, USA
  • 收稿日期:2008-11-19 出版日期:2009-05-20 发布日期:2009-05-20
  • 基金资助:

    Hu is supported by the National Science Foundation under Grant No. DMS0504783; Long is supported by FAU Start-up funding at the C. E. Schmidt College of Science

ON THE SINGULARITY OF LEAST SQUARES ESTIMATOR FOR MEAN-REVERTING α-STABLE MOTIONS

Department of Mathematics, University of Kansas, 605 Snow Hall, Lawrence, Kansas 66045-2142, USA;Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, Florida 33431-0991, USA   

  • Received:2008-11-19 Online:2009-05-20 Published:2009-05-20
  • Supported by:

    Hu is supported by the National Science Foundation under Grant No. DMS0504783; Long is supported by FAU Start-up funding at the C. E. Schmidt College of Science

摘要:

We study the problem of parameter estimation for mean-reverting α-stable motion, dXt = (α0 − θ0Xt)dt + dZt, observed at discrete time instants. A least squares estimator is obtained and its asymptotics is discussed in the singular case (α0θ0) = (0, 0). If α0 = 0, then the mean-reverting α-stable motion becomes Ornstein-Uhlenbeck process and is studied in [7] in the ergodic case θ0 > 0. For the Ornstein-Uhlenbeck process, asymptotics of the least squares estimators for the singular case (θ0 = 0) and for ergodic case (θ0 > 0) are completely different.

关键词: asymptotic distribution of LSE, consistency of LSE, discrete observa-tion, least squares method, Ornstein-Uhlenbeck processes, mean-reverting processes, singularity,  α-stable processes, stable stochastic integrals

Abstract:

We study the problem of parameter estimation for mean-reverting α-stable motion, dXt = (α0 − θ0Xt)dt + dZt, observed at discrete time instants. A least squares estimator is obtained and its asymptotics is discussed in the singular case (α0θ0) = (0, 0). If α0 = 0, then the mean-reverting α-stable motion becomes Ornstein-Uhlenbeck process and is studied in [7] in the ergodic case θ0 > 0. For the Ornstein-Uhlenbeck process, asymptotics of the least squares estimators for the singular case (θ0 = 0) and for ergodic case (θ0 > 0) are completely different.

Key words: asymptotic distribution of LSE, consistency of LSE, discrete observa-tion, least squares method, Ornstein-Uhlenbeck processes, mean-reverting processes, singularity,  α-stable processes, stable stochastic integrals

中图分类号: 

  • 60G52