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

doi:10.1016/S0252-9602(09)60056-4

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

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

胡耀忠,龙红卫

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/Abstract

摘要：

We study the problem of parameter estimation for mean-reverting α-stable motion, dX_t = (α₀ − θ₀X_t)dt + dZ_t, observed at discrete time instants. A least squares estimator is obtained and its asymptotics is discussed in the singular case (α₀, θ₀) = (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:

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

胡耀忠, 龙红卫. ON THE SINGULARITY OF LEAST SQUARES ESTIMATOR FOR MEAN-REVERTING α-STABLE MOTIONS[J]. 数学物理学报(英文版), 2009, 29(3): 599-608.

HU Yao-Zhong, LONG Gong-Wei. ON THE SINGULARITY OF LEAST SQUARES ESTIMATOR FOR MEAN-REVERTING α-STABLE MOTIONS[J]. Acta mathematica scientia,Series B, 2009, 29(3): 599-608.

参考文献

[1] Ahn S K, Fotopoulos S B, He L. Unit root tests with infinite variance errors. Econometric Reviews, 2001,
20: 461–483

[2] Chan N H, Tran L T. On the first-order autoregressive process with infinite variance. Econometric Theory,
1989, 5: 354–362

[3] Dietz H M. Asymptotic behaviour of trajectory fitting estimators for certain non-ergodic SDE. Stat Infer-
ence Stoch Process, 2001, 4: 249–258

[4] Feigin P D. Some comments concerning a curious singularity. J Appl Probab, 1979, 16: 440–444

[5] Fouque J -P, Papanicolaou G, Sircar K R. Derivatives in Financial Markets with Stochastic Volatility.
Cambridge: Cambridge University Press, 2000

[6] Fristedt B. Sample functions of stochastic processes with stationary, independent increments//Ney P, Port
S, eds. Advances in Probability and Related Topics, Vol 3. New York: Marcel Dekker, 1974: 241–396

[7] Hu Y, Long H. Least squares estimator for Ornstein-Uhlenbeck processes driven by α-stable motions. To
appear in Stochastic Processes and Their Applications

[8] Jain N C. A Donsker-Varadhan type of invariance principle. Z Wahrscheinlichkeitstheorie Verw Gebiete,
1982, 59: 117–138

[9] Janicki A, Weron A. Simulation and Chaotic Behavior of -stable Stochastic Processes. New York: Marcel
Dekker, 1994

[10] Kallenberg O. On the existence and path properties of stochastic integrals. Ann Probab, 1975, 3: 262–280

[11] Kallenberg O. Some time change representations of stable integrals, via predictable transformations of
local martingales. Stochastic Process Appl, 1992, 40: 199–223

[12] Phillips P C B. Time series regression with a unit root and infinite-variance errors. Econometric Theory,
1990, 6: 44–62

[13] Rosinski J, Woyczynski W A. On Ito stochastic integration with respect to p-stable motion: inner clock,
integrability of sample paths, double and multiple integrals. Ann Probab, 1986, 14: 271–286

[14] Samorodnitsky G, Taqqu M S. Stable non-Gaussian Random Processes: Stochastic Models with Infinite
Variance. New York, London: Chapman & Hall, 1994

[15] Sato K I. L´evy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge University Press,
1999

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

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

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