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Sharp large deviations for the non-stationary Ornstein-Uhlenbeck process. (English) Zbl 1316.60035

Summary: For the Ornstein-Uhlenbeck process, the asymptotic behavior of the maximum likelihood estimator of the drift parameter is totally different in the stable, unstable, and explosive cases. Notwithstanding this trichotomy, we investigate sharp large deviation principles for this estimator in the three situations. In the explosive case, we exhibit a very unusual rate function with a shaped flat valley and an abrupt discontinuity point at its minimum.

MSC:

60F10 Large deviations
60J60 Diffusion processes
62F12 Asymptotic properties of parametric estimators
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References:

[1] Bercu, B., On large deviations in the Gaussian autoregressive process: stable unstable and explosive cases, Bernoulli, 7, 2, 299-316 (2001) · Zbl 0981.62072
[2] Bercu, B.; Coutin, L.; Savy, N., Sharp large deviations for the fractional Ornstein-Uhlenbeck process, Theory Probab. Appl., 55, 4, 575-610 (2011) · Zbl 1252.60026
[3] Bercu, B.; Gamboa, F.; Lavielle, M., Sharp large deviations for Gaussian quadratic forms with applications, ESAIM Probab. Stat., 4, 1-24 (2000) · Zbl 0939.60013
[4] Bercu, B.; Rouault, A., Sharp large deviations for the Ornstein-Uhlenbeck process, Theory Probab. Appl., 46, 1, 1-19 (2002) · Zbl 1101.60320
[5] Brown, B. M.; Hewitt, J. I., Asymptotic likelihood theory for diffusion processes, J. Appl. Probab., 12, 228-238 (1975) · Zbl 0314.62036
[6] Bryc, W.; Dembo, A., Large deviations for quadratic functionals of Gaussian processes, J. Theoret. Probab., 10, 2, 307-332 (1997), Dedicated to Murray Rosenblatt · Zbl 0894.60026
[7] Dembo, A.; Zeitouni, O., (Large Deviations Techniques and Applications. Large Deviations Techniques and Applications, Applications of Mathematics (New York), vol. 38 (1998), Springer-Verlag: Springer-Verlag New York) · Zbl 0896.60013
[8] Dietz, H. M.; Kutoyants, Y. A., Parameter estimation for some non-recurrent solutions of SDE, Statist. Decisions, 21, 1, 29-45 (2003) · Zbl 1046.62081
[9] Feigin, P. D., Maximum likelihood estimation for continuous-time stochastic processes, Adv. in Appl. Probab., 8, 4, 712-736 (1976) · Zbl 0355.62086
[10] Feigin, P. D., Some comments concerning a curious singularity, J. Appl. Probab., 16, 2, 440-444 (1979) · Zbl 0409.62082
[11] Florens-Landais, D.; Pham, H., Large deviations in estimation of an Ornstein-Uhlenbeck model, J. Appl. Probab., 36, 1, 60-77 (1999) · Zbl 0978.62070
[12] Hu, Y.; Long, H., Least squares estimator for Ornstein-Uhlenbeck processes driven by \(\alpha \)-stable motions, Stochastic Process. Appl., 119, 8, 2465-2480 (2009) · Zbl 1171.62045
[13] Kutoyants, Y. A., (Statistical Inference for Ergodic Diffusion Processes. Statistical Inference for Ergodic Diffusion Processes, Springer Series in Statistics (2004), Springer-Verlag London Ltd.: Springer-Verlag London Ltd. London) · Zbl 1038.62073
[14] Lebedev, N. N., Special Functions and their Applications (1965), Prentice-Hall Inc.: Prentice-Hall Inc. Englewood Cliffs, NJ, Translated and edited by Richard A. Silverman · Zbl 0131.07002
[15] Liptser, R. S.; Shiryaev, A. N., (Statistics of Random Processes. II. Statistics of Random Processes. II, Applications of Mathematics (New York), vol. 6 (2001), Springer-Verlag: Springer-Verlag Berlin), Applications, Translated from the 1974 Russian original by A. B. Aries, Stochastic Modelling and Applied Probability
[16] Shimizu, Y., Notes on drift estimation for certain non-recurrent diffusion processes from sampled data, Statist. Probab. Lett., 79, 20, 2200-2207 (2009) · Zbl 1171.62341
[17] Zani, M., Large deviations for squared radial Ornstein-Uhlenbeck processes, Stoch. Proc. Appl., 102, 25-42 (2002) · Zbl 1075.62535
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