Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 27 / 5 / c / 2 / Solution

Use the positive exponential Brownian martingaleThe strong law for Brownian motion, $W_t/t\to 0$, makes its exponent tend to $-\infty$ and hence $M_t\to 0$. If $S={\sup }_{t\ge 0}(W_t-\lambda t)$, then $\sup M=e^{2\lambda S}$. Part (b) gives, for $x>0$,Thus the maximum is exponentially distributed with rate $2\lambda$:There is no atom at zero, by letting $x\downarrow 0$ in the tail. This is the infinite-horizon crossing probability for Brownian motion with negative drift.