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

Apply Cameron-Martin theorem to the absolutely continuous path $h(t)=xt/T$ up to $T$ and $h(t)=x$ thereafter. Its derivative is $x/T$ on $[0,T]$ and zero later, so the Radon-Nikodym factor is $\exp ((x/T)W_T-x^2/(2T))$, yielding the formula for every bounded measurable functional of the path after $T$.