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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.

Let $S$ denote this maximum and put $\tau =\tau (-b)$. Brownian motion reaches $-b$ almost surely: the Brownian reflection principle gives crossing probability $2\mathbb{P}(W_t<-b)\to 1$. Continuity gives $W_{\tau }=-b$, with the strict-crossing infimum interpreted as in part (b). The processis a continuous nonnegative local martingale starting at one and tending to zero. Its maximum is $1+S/b$. Apply part (b) at $a=1+x/b$, for $x>0$:Differentiating gives the maximum before a lower Brownian barrier densityThe tail tends to one as $x\downarrow 0$, so there is no atom at zero. The density integrates to one.