Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 32 / 3 / c

The Brownian reflection principle, applied to the negative increments of the Brownian motion started at $a$, giveswhere $\Phi$ is the standard normal distribution function. Hence $T_0<\infty$ almost surely. The stopped Brownian motion $M_t=B_{t\wedge T_0}$ is a continuous nonnegative martingale, starts at $a$, and eventually vanishes. The maximal identity for a continuous nonnegative local martingale tending to zero therefore gives, for $H={\sup }_{0\le t\le T_0}{B}_t$,Equivalently,There is no atom at $a$. This is the Pareto distribution of shape one, or the translated version of the maximum before a lower Brownian barrier.