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.

For independent Brownian motions $W,B$, the Itô formula cancels the drift after the arctangent transform. Normalizing the two noise coefficients gives a continuous local martingale $Z$ with quadratic variation $t$, hence another Brownian motion by the Lévy characterization of Brownian motion.

A diffusion remaining in $(-\pi /2,\pi /2)$ with the displayed equation is a bounded martingale and has a terminal limit. Its expected total quadratic variation is finite by its bounded second moments. An interior limit would leave its squared diffusion coefficient bounded away from zero and force infinite quadratic variation. Thus it converges to an endpoint, and the preserved mean determines the two endpoint probabilities.

For $dX=b(X)dt+dW$ with continuous drift, integrate the positive derivative displayed above. The resulting scale function of a one-dimensional diffusion solves $g^{\prime \prime }=-2b{g}^{\prime }$. The Itô formula removes the drift of $g(X)$, giving $dY=h(Y)dW$, where $h=g^{\prime }\circ g^{-1}$ on the interval $g(\mathbb{R})$. If the drift is bounded, $h^{\prime }=-2b\circ g^{-1}$ is bounded, so $h$ is Lipschitz on that interval.

The zero extension of a scale diffusion coefficient at finite endpoints makes the transformed equation globally Lipschitz. Inverting its solution gives the original additive-noise equation. The pathwise bound $|X_t|\le |x|+{\sup }_{s\le T}|W_s|+\Vert b{\Vert }_{\infty }T$ prevents the inverse scale from reaching infinity at any finite time. This proves global strong existence and pathwise uniqueness even when the original drift is not Lipschitz.

The diffusion coefficient obtained from a scale transform for an additive-noise diffusion need only be defined on an open interval. If it is Lipschitz, its limit at a finite endpoint is zero: a positive limit would bound the derivative $1/h$ of the inverse scale and prevent the inverse from diverging. Extending it by zero beyond finite endpoints therefore preserves global Lipschitz continuity.

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

The equation $\frac{1}{2}{\sigma }^2{s}^{\prime \prime }+b{s}^{\prime }=0$ 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.