Scale function (stochastic processes)

For the one-dimensional diffusiona scale function is a strictly increasing function $s$ satisfyingThe Itô formula then makes $s(X_t)$ a local martingale before the diffusion reaches a boundary.

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