The distribution of the first hit for stable and asymptotically stable walks on an interval

Let $\{\xi(t);t\ge0\}$ be a strongly stable process, $\tau=\inf\{t\colon\xi(t)\notin[0,1]\}$.
Formulas for $\mathbf P\{1\le\xi(\tau)<1+y\mid\xi(0)=x\}$, and $\mathbf P\{0\ge\xi(\tau)>-y\mid\xi(0)=x\}$, $0\le x\le1$, $y\ge0$, are derived and applied to random walks.