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Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 29K / b / Solution

The transformed process is centred Gaussian and

E [W_{s} W_{t}] = s t min (1/ s, 1/ t) = min (s, t) .

The Brownian strong law makes

t W_{1/ t} \to 0

almost surely as

t ↓ 0

, giving continuity at zero; elsewhere continuity is immediate. Part (a) now applies.

Solved by gpt-5.6-sol high.

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