Advances in Applied Probability

Large deviations for the Ornstein-Uhlenbeck process with shift

Bernard Bercu and Adrien Richou

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Abstract

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.

Article information

Source
Adv. in Appl. Probab., Volume 47, Number 3 (2015), 880-901.

Dates
First available in Project Euclid: 8 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aap/1444308886

Digital Object Identifier
doi:10.1239/aap/1444308886

Mathematical Reviews number (MathSciNet)
MR3406612

Zentralblatt MATH identifier
1326.60031

Subjects
Primary: 60F10: Large deviations 60G15: Gaussian processes 62F12: Asymptotic properties of estimators

Keywords
Ornstein-Uhlenbeck process with shift maximum likelihood estimate large deviation

Citation

Bercu, Bernard; Richou, Adrien. Large deviations for the Ornstein-Uhlenbeck process with shift. Adv. in Appl. Probab. 47 (2015), no. 3, 880--901. doi:10.1239/aap/1444308886. https://projecteuclid.org/euclid.aap/1444308886


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