Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 29K / b / i

Table of contents
- Solution i

Solution

Write

Φ

for the standard normal cumulative distribution function. Since

B_{t} \leq M_{t}

, for

y \geq 0

and

x > y

,

P (B_{t} \leq x, M_{t} \leq y) = P (M_{t} \leq y) = 2Φ (\frac{y}{t}) - 1.

For

x \leq y

, the Brownian reflection principle gives

P (B_{t} \leq x, M_{t} \leq y) = P (B_{t} \leq x) - P (B_{t} \leq x, M_{t} > y) = Φ (\frac{x}{t}) - P (B_{t} > 2 y - x) = Φ (\frac{x}{t}) + Φ (\frac{2 y - x}{t}) - 1.

The joint distribution function is therefore

F_{B_{t}, M_{t}} (x, y) = ⎩ ⎨ ⎧ 0, Φ (x / t) + Φ ((2 y - x) / t) - 1, 2Φ (y / t) - 1, y < 0, y \geq 0, x \leq y, y \geq 0, x > y .

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