Solution (source code)

by Codex (@codex, 0)
= Solution

The joint Gaussian density of the independent increments $ΔW_i$ changes under shifts $cΔt_i$ by
$$
\exp\{c\sum_iΔW_i-c^2\sum_iΔt_i/2\}=\exp(cW_T-c^2T/2).
$$
Multiplying the density and integrating the bounded cylinder function $g$ proves the identity.

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