= Exponential scale function for a drift tied to diffusivity
{title2=$dX_t=f(X_t)dB_t-\tfrac12f(X_t)^2dt,\quad s(x)=e^x$}
For the displayed <stochastic differential equation>, the <infinitesimal generator> is $\mathcal Lg=(f^2/2)(g''-g')$, so $\mathcal Ls=0$ and the <Itô formula> makes $e^{X_t}$ a positive <local martingale>. If $f$ is bounded, strictly positive, and <globally Lipschitz>, the equation has a unique global <strong solution of a stochastic differential equation>. For $a<x<b$ and $T_y=\inf\{t:X_t=y\}$, the <Itô isometry> for the bounded stopped exponential gives finite mean exit time: the bracket density $e^{2z}f(z)^2$ has a positive minimum on $[a,b]$. The <optional stopping theorem> then yields the <boundary hitting probability from a diffusion scale function>
$$
\mathbb P_x(T_b<T_a)=\frac{e^x-e^a}{e^b-e^a}.
$$
Taking $a\to-\infty$ gives $\mathbb P_x(T_b<\infty)=e^{x-b}$ for $b>x$. Passing to strict exceedances and then to their limits shows that $\sup_{t\geq0}X_t-x$ has the <exponential distribution> of rate one. The <probability distribution> of the maximum is independent of the particular $f$; changing the speed of motion preserves this <scale function>.
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