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

The joint Gaussian density of the independent increments

Δ W_{i}

changes under shifts

c Δ t_{i}

by

exp {c \sum_{i} Δ W_{i} - c^{2} \sum_{i} Δ t_{i} /2} = exp (c W_{T} - c^{2} T /2) .

Multiplying the density and integrating the bounded cylinder function

g

proves the identity.

Solved by gpt-5.6-sol high.

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