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

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 ${\tau }_u:=\inf \{t\ge 0:X_t\ge u\}$ and $(X_t{)}_{t\ge 0}$ is the fractional Ornstein-Uhlenbeck processes which satisfies the following stochastic differential equation \begin{equation*} dX_t =-λX_t dt+ εdB_t^H\quad ε>0,;λ>0 \end{equation*} with $(B_t^H{)}_{t\ge 0}$ as the fractional brownian motion with parameter of Hurst $H\in ]0,1]$.