= Hypotheses for a diffusion scale hitting formula
{title2=$s(b)\ne s(a),\quad T_a\wedge T_b<\infty\text{ a.s.}$}
The equation $\frac12\sigma^2s''+bs'=0$ makes the scale transform a <local martingale> but does not alone prove the usual two-boundary hitting ratio. Stopped on $[a,b]$, it is bounded, so optional stopping gives that ratio when exit is almost surely finite and the endpoint scale values differ. If exit can be infinite, the terminal <martingale> has a further contribution on the non-exit event. For example, $\sigma=b=0$, $X_t=x\in(a,b)$ and $s(y)=y$ satisfy the differential equation but never hit either endpoint; the claimed ratio would nevertheless be positive. A constant solution for $s$ also gives an undefined denominator and is excluded by the usual strict-monotonicity convention for scale.
Back to article page
