Exponential scale function for a drift tied to diffusivity ($d{X}_t=f(X_t)d{B}_t-\frac{1}{2}f(X_t{)}^{2}dt,s(x)=e^x$)

For the displayed stochastic differential equation, the infinitesimal generator is $\mathcal{L}g=(f^2/2)(g^{\prime \prime }-g^{\prime })$, so $\mathcal{L}s=0$ 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 functionTaking $a\to -\infty$ gives ${\mathbb{P}}_x(T_b<\infty )=e^{x-b}$ for $b>x$. Passing to strict exceedances and then to their limits shows that ${\sup }_{t\ge 0}{X}_t-x$ 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.