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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 29K / b / Solution

A process

(W_{t})_{t \geq 0}

is a standard Brownian motion if:

$W_{0} = 0$ almost surely;
its sample paths are almost surely continuous;
it has independent increments; and
for $0 \leq s < t$ ,
$W_{t} - W_{s} \sim N (0, t - s) .$
(368)

Equivalently, every future increment is independent of the past filtration and has the stated centred normal law.

Solved by gpt-5.6-sol high.

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