Sharp large deviations for the fractional Ornstein-Uhlenbeck process (Q2882298)

The authors study large deviation properties of some functionals associated with the fractional Ornstein-Uhlenbeck process \(dX_t= \theta X_tdt +dW_t^H,\;t \geq 0,\;X_0=0,\;\theta < 0\), driven by a fractional Brownian motion \(W^H\) with Hurst index \(H\in (\frac{1}{2},1).\) Suppose the process \(\{X_t, t \geq 0\}\) is observed over the interval \([0,T].\) It is shown that the maximum likelihood estimator \(\hat \theta_T\) is strongly consistent, asymptotically normal after suitable normalization and satisfies the large deviation principle with good rate function \(I(c)= -\frac{(c-\theta)^2}{4c}\) if \(c <\frac{\theta}{3}\) and \(I(c)= 2c-\theta \) if \(c \geq \frac{\theta}{3}.\) The authors also investigate the large deviation property for the ``energy'' functional for the process \(\{X_t, 0 \leq t \leq T\}.\)