Scale function (stochastic processes)
= Scale function
{disambiguate=stochastic processes}
{wiki=Scale_function_(stochastic_processes)}
For the one-dimensional diffusion
$$
dX_t=b(X_t)dt+\sigma(X_t)dB_t,
$$
a scale function is a strictly increasing function $s$ satisfying
$$
\frac12\sigma^2s''+bs'=0.
$$
The <Itô formula> then makes $s(X_t)$ a local martingale before the diffusion reaches a boundary.
= Scale function of a one-dimensional diffusion
{synonym}