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The diffusion generator is $\mathcal{L}f=\frac{1}{2}{\sigma }^2{f}^{\prime \prime }+b{f}^{\prime }$. On a compact subinterval of $I$, the coefficient $s^{\prime }\sigma$ is bounded, and the Itô formula up to its exit time $\rho$ givesbecause $\mathcal{L}s=0$. Increasing compact subintervals proves the local-martingale property up to $T$. The notation $s(X_{t\wedge T})$ at an exit endpoint is understood with its continuous endpoint extension when needed; stopping on $[a,b]\subset I$ below is always well-defined. This is the martingale property of a scale function of a one-dimensional diffusion.

For the hitting ratio there are two necessary qualifications: $s(b)\ne s(a)$ and $\tau =T_a\wedge T_b<\infty$ almost surely. A usual scale function is strictly monotone, which ensures the first condition, but the printed hypotheses alone ensure neither. Under these conditions, $s(X_{t\wedge \tau })$ is a bounded local martingale and hence a uniformly integrable martingale. The optional stopping theorem givesContinuity prevents simultaneous hitting of distinct endpoints, and almost sure exit makes these two probabilities sum to one. Solving yields the boundary hitting probability from a diffusion scale function

The hypotheses for a diffusion scale hitting formula cannot be omitted here. Take $\sigma \equiv 0$, drift $b\equiv 0$, $s(y)=y$ and $X_t\equiv x\in (a,b)$. All printed coefficient and differential conditions hold, but neither endpoint is ever reached, so the left side is zero whereas the displayed ratio is strictly positive. A constant $s$ is another allowed solution of the printed differential equation and makes its denominator zero. In general the bounded stopped martingale has a limit $Y_{\infty }$, and optional stopping instead givesThe following parts have nondegenerate diffusion and strictly monotone powers, and their finite-exit properties are justified explicitly.