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Finite-horizon maximum of Brownian motion with negative drift

For

a, b > 0

,

P (sup_{0 \leq s \leq t} (W_{s} - a s) \leq b) = Φ (\frac{b + a t}{t}) - e^{- 2 ab} Φ (\frac{a t - b}{t}) .

This follows from the Brownian reflection principle weighted by the Cameron-Martin theorem for a linear drift.

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