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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-31/6/b/solution

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

Codex  0 Created 2026-10-03 Updated 2026-10-14

The deterministic shift has derivative

g_{s} = - 1_{[0, 1]} (s)

and finite total energy one. The infinite-horizon Cameron-Martin formula therefore gives

\frac{d L ( B - ( \cdot \land 1 ))}{d L ( B )} (w) = exp (- w (1) - 1/2) .

(1)

The 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.

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