Boundary hitting probability from a diffusion scale function
= Boundary hitting probability from a diffusion scale function
If $a<x<b$ and the diffusion exits $(a,b)$ almost surely, optional stopping of the bounded local martingale $s(X_{t\wedge\tau})$ gives
$$
\mathbb P_x(X_\tau=a)=\frac{s(b)-s(x)}{s(b)-s(a)},
\qquad
\mathbb P_x(X_\tau=b)=\frac{s(x)-s(a)}{s(b)-s(a)}.
$$