Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 3 / 29K / d / Solution

By continuity of Brownian motion, the event that the last strict crossing of $b$ occurs by time $t$ isApply part (b) with $a$ and $b$ interchanged and its time parameter replaced by $1/t$. This turns the right-hand side intoPart (c), now with drift $b$, barrier $a$, and horizon $1/t$, givesThis is the distribution function of the last passage time above a level for Brownian motion with negative drift. Taking $t\downarrow 0$ givesthe probability that the negatively drifted Brownian path never exceeds $b$, consistent with the infinite-horizon crossing probability for Brownian motion with negative drift.

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