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Large deviations for the Ornstein-Uhlenbeck process with shift

Part of: Stochastic processes Limit theorems Parametric inference

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.
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Abstract

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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 

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