Distribution of a one-sided Brownian hitting time

Fix the measurable full-measure event $A$ specified in the statement. Replacing the original path by zero on $A^c$ preserves every finite-dimensional law and makes every path continuous with $B_0=0$. Each coordinate remains measurable because $A$ is measurable. Two such choices agree on the intersection of their events, so the resulting measurable hitting times agree there and have the same distribution. By [F1] applied to the closed singleton $\{a\}$, ${\tau }_a$ is a stopping time for the chosen process's own raw natural filtration, hence an extended nonnegative measurable random variable. No stopping-time claim for the original raw filtration is used.

Define $g(s)=a(2\pi s^3{)}^{-1\mathrm{/}2}{e}^{-a^2\mathrm{/}(2s)}$ for $s>0$. For $0<\epsilon <t$, the substitution $u=a\mathrm{/}\sqrt{s}$ on $[\epsilon ,t]$, whose derivative $-a\mathrm{/}(2s^{3\mathrm{/}2})$ is continuous and $2\phi$ is continuous, gives ${\int }_{\epsilon }^{t}g(s)ds={\int }_{a\mathrm{/}\sqrt{t}}^{a\mathrm{/}\sqrt{\epsilon }}2\phi (u)du=2(\mathrm{\Phi }(a\mathrm{/}\sqrt{\epsilon })-\mathrm{\Phi }(a\mathrm{/}\sqrt{t}))$ by oriented substitution, the compact Riemann/Lebesgue bridge, and the density-integral identity for increments of $\mathrm{\Phi }$.

For $t>0$, if ${\tau }_a\le t$, the first hit is attained by continuity (as in the closed-set hitting lemma), so $M_t\ge a$. Conversely $M_t\ge a$ gives a time $s\in [0,t]$ with $B_s=M_t$ by [F5]; since $B_0=0<a\le B_s$, the intermediate value theorem gives a hit by time $s$. Hence $\{{\tau }_a\le t\}=\{M_t\ge a\}$ as exact measurable events for this representative. Continuity at zero also gives ${\tau }_a>0$ on every path, since $B_0=0<a$.

The normal CDF obeys $\mathrm{\mid }\mathrm{\Phi }(v)-\mathrm{\Phi }(u)\mathrm{\mid }\le \mathrm{\mid }v-u\mathrm{\mid }\mathrm{/}\sqrt{2\pi }$, by its density bound, hence is continuous. Symmetry and total mass one give $\mathrm{\Phi }(0)=1\mathrm{/}2$; monotone convergence of density integrals gives $\mathrm{\Phi }(x)\to 1$ as $x\to \mathrm{\infty }$. Put $x_n=a(1-1\mathrm{/}n)$ for $n\ge 1$. The measurable events $\{M_t\le x_n\}$ increase to $\{M_t<a\}$, so [F3] and [F1] give $P(M_t<a)={\lim }_n(2\mathrm{\Phi }(x_n\mathrm{/}\sqrt{t})-1)=2\mathrm{\Phi }(a\mathrm{/}\sqrt{t})-1=P(M_t\le a)$. Thus $P(M_t=a)=0$, and step 2.1 yields $P({\tau }_a\le t)=2(1-\mathrm{\Phi }(a\mathrm{/}\sqrt{t}))$. Taking integer $t\to \mathrm{\infty }$ and monotone convergence of the events $\{{\tau }_a\le t\}$ gives $P({\tau }_a<\mathrm{\infty })=1$.

Let $\epsilon =t\mathrm{/}n$ in step 1.2 and let integers $n\ge 2$ tend to infinity. The nonnegative integrals increase to ${\int }_0^{t}g(s)ds$, and $\mathrm{\Phi }(a\mathrm{/}\sqrt{\epsilon })\to 1$, so ${\int }_0^{t}g(s)ds=2(1-\mathrm{\Phi }(a\mathrm{/}\sqrt{t}))=P({\tau }_a\le t)$. Letting integer $t\to \mathrm{\infty }$ now gives ${\int }_0^{\mathrm{\infty }}g=1$.

Extend $g$ by zero on $(-\mathrm{\infty },0]$. It is nonnegative Borel measurable, and [F5] and step 4.1 make its density measure a Borel probability measure on $\mathbb{R}$. To use [F4] with a real random variable, replace ${\tau }_a=\mathrm{\infty }$ by the value $1$ on its measurable null event, obtaining ${\tilde{\tau }}_a$. This leaves every finite-time distribution probability unchanged, and ${\tilde{\tau }}_a>0$ by step 2.1. The density measure and ${\tilde{\tau }}_a$ have CDF zero for nonpositive arguments, and the same CDF at every positive argument by step 4.1. Thus [F4] identifies the laws. In particular the original extended hitting time has density $g$ on $(0,\mathrm{\infty })$, no atom there or at zero, and no mass at infinity.

The parameter $a>0$ and compact substitution bounds $0<\epsilon <t$ ensure every denominator is positive. At $t=0$, step 2.1 gives $P({\tau }_a=0)=0$. The infinity limit and total density mass were proved in steps 3.1 and 4.1. AC supplies the Countable Choice hypotheses of both the compact integration bridge and [F4], and the Brownian and hitting-time suppliers. The event equality uses the declared continuous representative throughout.