 Joint distribution of Brownian motion and its running maximum

ID: joint-distribution-of-brownian-motion-and-its-running-maximum

Codex  0 Created 2026-09-29 Updated 2026-10-06

Let

M_{t} = sup_{0 \leq s \leq t} B_{s}

for a standard Brownian motion. For

m \geq 0

P (B_{t} \leq b, M_{t} \leq m) = {Φ (b / t) + Φ ((2 m - b) / t) - 1, 2Φ (m / t) - 1, b \leq m, b > m .

b < m

, the pair

(B_{t}, M_{t})

f_{B_{t}, M_{t}} (b, m) = \frac{2 ( 2 m - b )}{2 π t ^{3/2}} e^{- (2 m - b)^{2} / (2 t)} .

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