  We investigate the large deviation properties of the maximum likelihood
estimators for the Ornstein-Uhlenbeck process with shift. We estimate
simultaneously the drift and shift parameters. On the one hand, we establish 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 one 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.
