A note about an upper bound for a hitting time of the fractional Ornstein-Uhlenbeck process

Abstract

In this brief note we give an upper bound for $P(\tau_u < T)$ with $T>0$, where $\tau_u$ is the exit time defined as $\tau_u:=\inf \{ t\geq 0 \, : \, X_t\geq u \}$ and $(X_t)_{t\geq 0}$ is the fractional Ornstein-Uhlenbeck processes which satisfies the following stochastic differential equation \begin{equation*} dX_t =-\lambda X_t dt+ \epsilon dB_t^H\quad \epsilon>0,\;\lambda>0 \end{equation*} with $(B_t^H)_{t\geq 0}$ as the fractional brownian motion with parameter of Hurst $H\in ]0,1]$.