Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 31 / 6 / b / Solution

The deterministic shift has derivative $g_s=-1_{[0,1]}(s)$ and finite total energy one. The infinite-horizon Cameron-Martin formula therefore givesThe integral of this density is one by the moment-generating function of $B_1\sim N(0,1)$. It is strictly positive, so the shifted law and Wiener measure are equivalent on the whole half-line. The density depends only on the path through time one; after that time the shift is constant.