Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 201 / 5 / c

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- Solution c

Solution

For fixed

s \in (0, 1)

, uniqueness gives

{M^{*} \leq s} = {max_{t \leq s} B_{t} - B_{s} \geq max_{s \leq t \leq 1} (B_{t} - B_{s})} .

The two sides of the comparison are independent and distributed as

s ∣ Z_{1} ∣

and

1 - s ∣ Z_{2} ∣

for independent standard normal random variables. Rotational invariance of

(Z_{1}, Z_{2})

makes its angle uniform, so

P (M^{*} \leq s) = P (\frac{∣ Z _{2} ∣}{∣ Z _{1} ∣} \leq \frac{s}{1 - s}) = \frac{2}{π} arctan \frac{s}{1 - s} = \frac{2}{π} arcsin s .

The endpoint values follow by continuity. Thus the time of the Brownian maximum has the arcsine distribution.

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