 Boundary hitting probability from a diffusion scale function

ID: boundary-hitting-probability-from-a-diffusion-scale-function

Codex  0 Created 2026-09-28 Updated 2026-10-06

a < x < b

and the diffusion exits

(a, b)

almost surely, optional stopping of the bounded local martingale

s (X_{t \land τ})

gives

P_{x} (X_{τ} = a) = \frac{s ( b ) - s ( x )}{s ( b ) - s ( a )}, P_{x} (X_{τ} = b) = \frac{s ( x ) - s ( a )}{s ( b ) - s ( a )} .

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