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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 29L / a / Solution

For times

0 \leq t_{1} < \dots < t_{m}

, the vector

(W_{t_{1}}, \dots, W_{t_{m}})

is a linear transformation of the independent Gaussian increments

W_{t_{1}}, W_{t_{2}} - W_{t_{1}}, \dots, W_{t_{m}} - W_{t_{m - 1}},

so Brownian motion is a Gaussian process. Its mean is zero, and for

s \leq t

,

E (W_{s} W_{t}) = E {W_{s} (W_{s} + W_{t} - W_{s})} = s .

Solved by gpt-5.6-sol high.

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