Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-29T05:47:28.472Z Has data issue: false hasContentIssue false
  • English
  • Français

Large deviations for the Ornstein-Uhlenbeck process with shift

Part of: Limit theorems Parametric inference Stochastic processes

Published online by Cambridge University Press:  21 March 2016

Bernard Bercu  and
Adrien Richou
Bernard Bercu*
Affiliation:
Université de Bordeaux
Adrien Richou*
Affiliation:
Université de Bordeaux
*
∗ Postal address: Institut de Mathématiques de Bordeaux, Université de Bordeaux, UMR 5251, 351 Cours de la Libération, 33405 Talence cedex, France.
∗ Postal address: Institut de Mathématiques de Bordeaux, Université de Bordeaux, UMR 5251, 351 Cours de la Libération, 33405 Talence cedex, France.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We investigate the large deviation properties of the maximum likelihood estimators for the Ornstein-Uhlenbeck process with shift. We propose a new approach to establish large deviation principles which allows us, via a suitable transformation, to circumvent the classical nonsteepness problem. We estimate simultaneously the drift and shift parameters. On the one hand, we prove a large deviation principle for the maximum likelihood estimates of the drift and shift parameters. Surprisingly, we find that the drift estimator shares the same large deviation principle as the estimator previously established for the Ornstein-Uhlenbeck process without shift. Sharp large deviation principles are also provided. On the other hand, we show that the maximum likelihood estimator of the shift parameter satisfies a large deviation principle with a very unusual implicit rate function.

Keywords

Ornstein-Uhlenbeck process with shiftmaximum likelihood estimatelarge deviation

MSC classification

Primary: 60F10: Large deviations 60G15: Gaussian processes 62F12: Asymptotic properties of estimators
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 3 , September 2015 , pp. 880 - 901
Copyright
Copyright © Applied Probability Trust 2015 

References

Bercu, B. and Rouault, A. (2002). Sharp large deviations for the Ornstein–Uhlenbeck process. Theory Prob. Appl. 46, 1–19.CrossRefGoogle Scholar
Bercu, B. and Richou, A. (2014). Large deviations for the Ornstein–Uhlenbeck process with shift. Available at http://www.arxiv.org/abs/1311.7039.Google Scholar
Bercu, B., Coutin, L. and Savy, N. (2012). Sharp large deviations for the non-stationary Ornstein–Uhlenbeck process. Stoch. Process. Appl. 122, 3393–3424.Google Scholar
Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications (Appl. Math. (New York) 38), 2nd edn. Springer, New York.Google Scholar
Demni, N. and Zani, M. (2009). Large deviations for statistics of the Jacobi process. Stoch. Process. Appl. 119, 518–533.CrossRefGoogle Scholar
Donati-Martin, C. and Yor, M. (1993). On some examples of quadratic functionals of Brownian motion. Adv. Appl. Prob. 25, 570–584.CrossRefGoogle Scholar
Florens-Landais, D. and Pham, H. (1999). Large deviations in estimation of an Ornstein–Uhlenbeck model. J. Appl. Prob. 36, 60–77.Google Scholar
Gao, F. and Jiang, H. (2009). Deviation inequalities and moderate deviations for estimators of parameters in an Ornstein–Uhlenbeck process with linear drift. Electron. Commun. Prob. 14, 210–223.CrossRefGoogle Scholar
Janson, S. (1997). Gaussian Hilbert Spaces (Camb. Tracts Math. 129). Cambridge University Press.Google Scholar
Jeanblanc, M., Yor, M. and Chesney, M. (2009). Mathematical Methods for Financial Markets. Springer, London.CrossRefGoogle Scholar
Jiang, H. (2012). Berry–Esseen bounds and the law of the iterated logarithm for estimators of parameters in an Ornstein–Uhlenbeck process with linear drift. J. Appl. Prob. 49, 978–989.Google Scholar
Kutoyants, Y. A. (2004). Statistical Inference for Ergodic Diffusion Processes. Springer, London.CrossRefGoogle Scholar
Vasicek, O. (1977). An equilibrium characterization of the term structure. J. Financial Econom. 5, 177–188.Google Scholar
Zani, M. (2002). Large deviations for squared radial Ornstein–Uhlenbeck processes. Stoch. Process. Appl. 102, 25–42.CrossRefGoogle Scholar